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An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure
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An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure Ali Reza Kargar , Hadi Haghgouei PII: DOI: Reference:
S0307-904X(19)30555-4 https://doi.org/10.1016/j.apm.2019.09.025 APM 13029
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
12 May 2019 8 August 2019 12 September 2019
Please cite this article as: Ali Reza Kargar , Hadi Haghgouei , An analytical solution for timedependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.09.025
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Highlights
An analytical solution is proposed for stress field around lined circular tunnels surrounded by viscoelastic rock mass. The solution is able to predict stress field accurately both in short-term and long-term periods of time. The solution is based on complex potential functions and correspondence principle, in a step wise procedure. The method which employs time discretization approach could be applied to any rheological model.
An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure Ali Reza Kargar*, Hadi Haghgouei School of Mining Engineering, College of Engineering, University of Tehran, Iran *
Corresponding author. Tel.: +98-9171388825;
E-mail address: [email protected] (A.R. Kargar)
Abstract Time-dependent behavior of surrounding mass plays a significant role in designing underground constructions. Considering simple configuration of lined circular tunnels, a lot of solution have been proposed to this problem. However, many assume hydrostatic initial stress field, and other solutions are only applicable to simple rheological models and could not account for viscosity effect in long-term time periods. In this study, an analytical plane strain solution is proposed for lined circular tunnels under non-hydrostatic initial stress field, assuming rock mass as a viscoelastic material obeying Burgers model, while concrete lining is supposed to have linear elastic behavior. The solution which employs complex variable method combined with correspondence principle benefits from time discretization approach enabling the solution to take into account the viscosity effect in the both short-term and long-term periods of time, while predicting stress components accurately. The results obtained by the proposed solution were compared with those predicted by finite element COMSOL software which exhibited a close agreement. It was found that by increasing time both the proposed analytical solution and finite element numerical method tend to an oblique asymptote due to viscosity effect of Maxwell body in the Burgers model. Finally, a parametric analysis was performed with respect to Burgers model coefficients which showed different behavior for short and long periods of time. Keywords: Time-dependent behavior; Lined circular tunnels; Complex variable method; Correspondence principle; Time discretization approach
1. Introduction Once a tunnel is excavated, the ground pressure on the lining support increases with time due to the rheological properties of surrounding rock, effected by tunnel advancement and rock-support interaction [1-5]. This phenomenon has a substantial impact on support design especially in sedimentary rocks, in which excess pressure on support system may be produced over time. This pressure could develop due to tunnel face progress which controls the rate of loading on support, and rock-support interaction inducing time-dependent stress and deformation in lining and rock material. In this study, the excavation rate is assumed infinitely large that the effect of loading rate on support behavior is ignored, and time-dependent analysis is performed only due to rocksupport interaction. A lot of attention has been paid to time-dependent interaction of rock-support system [1-11]. Generally, this problem is encountered through two main approaches: numerical and analytical methods. Although the numerical approach benefits from a great strength of being applicable to many sophisticated problems with various complexities from boundary conditions to material constitutive models, the time of solving process could be prolonged, in addition to the fact that their prediction also is not very accurate compared to analytical solutions and is affected by different parameters such as mesh size, time steps length, boundaries of the model and etc. However, analytical methods predict the solution exceedingly more accurate compared to numerical approaches, and present a measure to assess the validity of numerical methods [12]. A majority of researches have been conducted in this field considering opening as a circular hole, supported by lining with constant thickness, in a homogenous isotropic viscoelastic rock mass under a hydrostatic initial stress field [1, 2, 4-6, 9, 13, 14]. Gnrik and Johnson determined deformation field analytically around a circular shaft with and without lining support, considering viscoelastic behavior for surrounding rock mass[6]. Sakurai, by introducing an equivalent initial stress, developed an analytical solution and obtained pressure acting on the tunnel supports vs. time and tunnel advancement [1]. Panet also explored the time-dependent deformation in underground works considering different rheological models in this regard, as well as their limitations [2]. Ladanyi and Grill investigated the effect of long-term rock deformation on lining pressure through analytical methods[9]. Pan and Dong proposed a solution for time-dependent tunnel convergence by employing tunnel advancement effect and support installation in a viscoelastic medium[4, 5]. Fahimifar et al. suggested a closed form solution for excavation of circular tunnels in viscoelastic burgers’ rock masses, in case of both lined and unlined tunnels[13]. Moreover, Nikomos et al. derived a solution for tunnels in viscoelastic materials considering support installation time[14]. However, In case that the in situ stress constitutes a non-hydrostatic stress field, rarely any researches were done. Lo and Yuen presented a solution for lined circular tunnels under non-slip and full-slip liningrock interface condition considering general Kelvin model dominates rock mass behavior [15]. Song et al. also investigated the problem in case that the tunnel was supported with two circular linings installed at different times by using correspondence principle along with convolution integral to model different boundary loads applied at different times [16, 17]. Zhang et al also presented a time-dependent solution for ground-support interaction of a pile adjacent to an
excavation through applying Mindlin’s solution along with corresponding principle, as well as employing Pasternak’s foundation model assuming surrounding soil follows Boltzmann viscoelastic constitutive model [18]. However, with respect to long-term time periods, especially when viscos element controls rock mass behavior (such as Burgers model), correspondence principle could not give correct solution without time discretization. This problem could lead to a wrong prediction of stress and deformation field within lining and surrounding mass. In this study, a new solution is proposed for time-dependent stress and deformation field of lining and surrounding rock mass for circular tunnels, considering Lining support and surrounding mass as homogenous isotropic linear elastic and viscoelastic materials, respectively, in which Burgers model dominates the viscoelastic behavior. The solution is achieved through applying complex variable method, along with time discretization, in a stepwise procedure. For simplicity, it is assumed tunnel advancement rate is infinitely large that the viscos deformation during excavation could be neglected, and consequently induced stress field varies over time only owing to lining support interaction. By applying time discretization, the unloading phenomenon is modelled in each time step with respect to complex potential functions.
2. Problem statement and assumptions Solving the problem of time-dependent behavior of lined circular tunnels under non-hydrostatic initial stress field, several assumptions are considered as follows: 1. The surrounding rock mass is an isotropic homogenous medium with viscoelastic behavior, obeying Burger’s constitutive model. 2. The in situ stress field is non-hydrostatic. 3. The tunnel is excavated in such a great depth that the gradient of stress field could be neglected (in other words, the tunnel is a deep-buried tunnel). 4. The lining is made of linear-elastic material. 5. The rate of excavation is infinitely large that no viscous behavior exists during this period. The solution is derived for two cases. First the case when the lining is installed instantly behind tunnel face, and second, once the lining is installed with a distance from the face considering an elapsed time before support installation. The rate of excavation is assumed so large that no viscos behavior is evident in this period, and initial equilibrium between lining structure and surrounding rock mass is achieved immediately. Subsequently, time-dependent interaction between support system and surrounding ground begins. This interaction is investigated through a stepwise procedure which allows incorporating elastic unloading phenomenon occurred with time in surrounding rock. Since the initial stress field is anisotropic, and the problem deals with a multi-region model of materials with viscoelastic behavior, complex variable method introduced by Muskhelishvili, along with correspondence principle is implemented in a stepwise procedure.
In order to solve the problem, first time-dependent behavior and complex variable method are briefly introduced in the following sections. 2.1. Viscoelastic behavior Viscoelastic behavior, in brief, represents viscous, associated with fluids, and elastic, related to solids, behaviors, in which the former is modeled by dashpot, and the latter is modeled by spring in rheological models. The simplest viscoelastic models are known as Maxwell and Kelvin bodies, which are shown in Fig. 1. However, these two models are too elementary to model complicated behavior of rocks. Consequently, the Burger model was introduced which is a series combination of Maxwell and Kelvin bodies. This rheological model, which is shown in Fig. 1, was better able to model time-dependent rocks behavior. Generally, time-dependent deformation is determined from deviatoric stress components, while hydrostatic stress component only contributes to elastic dilation. Therefore, stress- strain relations in rheological models may be written as follows [19]: m k k e q S ij ij k t k t k k 0 k 0 ii 3K ii n
pk
(1)
t represents the derivative operator with respect to time, S ij and e ij also denote deviatoric parts of
Where K is the bulk modulus, ij and ij are components of stress and strain tensors,
and , respectively, according to the following equation: ij
ij
ij ij ij ij
kk 3
kk 3
e ij (2)
S ij
Considering that
S ij 2Ge ij
(3)
And comparing this equation with Eq. (1), 2G may be defined as the following operator in viscoelastic problems:
k p k t k 2G k m0 k q k t k k 0 n
(4)
In the case of creep problem, where stress field remains constant with respect to time, the coefficient G could be obtained as follows:
1 G t G k
G kt 1 e k
1 t G m m
1
(5)
Where G k and k are Kelvin shear modulus and viscosity, G m and m are shear modulus and viscosity of Maxwell model. The viscoelastic models are generally employed within stress and strain analysis through correspondence principle approach. In fact, this method emerges from the analogues between the original basic elasticity equations and their Laplace transformation with respect to time. According to this method the coefficient G in elastic solutions is replaced with
Q in transformed space, in which Q and P are obtained from Laplace transformation of Eq. P (1). The final viscoelastic solution in time space is obtained by Laplace inverse transformation of this solution. In the next section, the well-known elasticity solution, namely complex variable method, proposed by Muskhelsihvili, is introduced. 2.2. Complex variable method According to complex variable method, there exist two holomorphic complex potential functions, named z and z , in plane theory of elasticity from which stress components could be obtained as follows [20]:
xx yy 2 ' z ' z
yy xx 2i xy 2 z z z ''
'
(6)
Where z x iy is the coordinate of the material point in z-plane, for which i 2 1 . Moreover, the displacement components may also be determined based on aforementioned functions z and z as follows:
2G ux iu y k z z ' z z
(7)
where G is the shear modulus, and k is a parameter related to Poisson’s ratio which is given by: 3 4 k 3 1
plane strain plane stress
(8)
Generally, the holomorphic functions z and z are obtained according to the boundary conditions. In the case of first fundamental problems where traction vectors constitute the boundary conditions, these functions are calculated from the following equation:
i
z
z0
p iq ds z z ' z z C
(9)
where p and q are components of the surface traction vector in x and y directions, respectively, and C is the constant of integration. In the case of second fundamental problems, in which displacements along boundaries contribute to the boundary conditions, the functions z and z are determined from Eq. (7). 3. Derivation of the solution In order to find a solution, at the primary stage the conventional solution for lined circular tunnel in elastic medium is introduced [21-23]. Subsequently, the solution for lined circular tunnels surrounded in a viscoelastic medium is proposed, considering two cases of instant support installation adjacent to tunnel face, or installing support system with delay behind the face. 3.1. Solution for lined circular tunnels in elastic medium Since the problem focuses on the interaction between lining support and surrounding rock mass in circular tunnels, this issue could be employed to the solving process through considering the model as a multi-region medium. Therefore, for each region, namely rock mass and concrete lining support, complex potential functions r z , r z , and c z , c z are defined, respectively, which are introduced as follows:
r z Γz 0 z r z Γz 0 z c z R1 z R 2 z
(10)
c z Q1 z Q 2 z Where 0 z and 0 z are holomorphic functions defined in rock mass medium, while
R1 z , R 2 z , Q1 z and Q 2 z are those defined in concrete material. The coefficients Γ and Γ are real constants determined with regard to the uniform stress state at infinity from the following equation [20]:
1 1 H 1 2 4 4 1 1 Γ 1 2 H 2 2
(11)
Where H and are the depth of the hole, and unit weight of the rock mass, respectively. also represents the ratio of horizontal to vertical in situ stresses. By applying conformal mapping, the circular hole and lining are mapped into two concentric circles with unit and R 0 radius in plane (Fig. 2), according to the following equation: z R
(12)
Consequently, Eq. 10 is rewritten as follows in -plane:
r Γ R 0 r Γ R 0
(13)
c R1 R 2 c Q1 Q 2 Where
0 ( ) a j j , j 0
Q1 q j1 j , j 0
Q 2 q j 2 j , j 1
0 ( ) b j j j 0
R1 r j1 j
(14)
j 0
R 2 r j 2 j j 1
These unknown functions are determined through applying boundary condition followed by 1 d Cauchy integration ( ) along the unit circle. The boundary conditions are zero 2 i traction vector along inner boundary of the lining support, along with identical displacement and traction vector along rock-concrete interface, as follows:
k kr 1 1 0 0' 0 c c c' c Gr Gr Gc Gc
r r' r c c' c
(15)
c R 0 R 0c' R 0 c R 0 0 Where k r , G r and k c , G c are the coefficients k , G related to rock mass and concrete regions, R1 . After taking Cauchy integration of Eq. (15) for inside R and outside of the unit circle, a system of 6 equations with 6 unknown functions, i.e. 0 ( ) ,
respectively, e i , and R o
0 ( ) , Q1 , Q 2 , R1 and R 2 , are obtained, which may be solved by series solution method. Consequently, following determination of unknown functions, complex potential
functions r , r , and c , c are obtained from Eq. (13), and corresponding stress and deformation field could be calculated based on Eqs. (6) and (7) considering z R , as follows [20]:
2 ' ' R 2e 2i 2i '' ' R
(16)
2G u x iu y k '
Where and are complex potential functions, and , , as well as are radial, circumferential, and shear stress components associated with the studied region. 3.2. Solution for lined circular tunnels in viscoelastic medium considering lining installation adjacent to the tunnel face Since the rate of excavation is infinitely large, initial equilibrium between surrounding rock and support system is achieved immediately, whose stress and deformation fields could be determined according to section 3.1. Note that due to extremely short period of time, rock mass behaves as an elastic material. Subsequently, rock mass viscoelastic behavior begins in which, for each time increment, there exists the corresponding rock mass deformation inducing stress in lining material. This induced stress field could be determined by defining the complex potential functions within lining region in incremental form as follows: c , t R1 , t R 2 , t
(17)
c , t Q1 , t Q 2 , t
Where
Q1 , t q j1 t j ,
R1 , t r j1 t j
j 0
Q 2 , t q j
2
t
j
j 1
j 0
,
R 2 , t r j
2
t
(18) j
j 1
The time-dependent rock mass deformation could affect the stress and deformation field of the lining material through considering its deformation along rock-concrete interface, as the boundary condition of induced displacement for lining region. This fact could be employed to the solution for each time step through the following equation: kc 1 c , t t 2Gc 2Gc
c , t t c , t t u xr i u yr
(19)
Where u xr and u yr are the rock displacement increments in x and y directions along rockconcrete interface, which is calculated as follows: 1 k t t k r t u xr i u yr r Γ R 0 , t 2 G r t t G r t (20) ΓR 1 1 1 0 , t 0 , t ΓR 2 G r t t G r t
G r t is calculated from Eq. 5, and k r t is obtained from Eq. 8 considering
3K 2G t
6K 2G t
,
in which K is the Bulk modulus. 0 ,t and 0 ,t are holomorphic potential functions associated with deformation and stress fields in rock mass region, according to Eq. (14). It is obvious that 0 , 0 and 0 , 0 denote those functions determined based on initial equilibrium described in section 3.1. On the other hand, the traction vector must vanish along inner lining surface, which in turn constitutes the other boundary condition for complex functions c , t and c , t , as follows:
c R 0 , t t R 0
c R 0 , t t c R 0 , t t 0
Where e i . After multiplying both sides of Eqs. (19) and (21) by
(21)
d and integrating 2 i 1
along unit-radii circle, a system of 4 equations with 4 unknown functions, i.e. R1 , R 2 ,
Q1 , and Q 2 , are concluded which after some manipulation, and applying series 2 solution, the following equation set with respect to unknown coefficients r j1 and r j is
obtained: s1
s1
n 1
n 1
r1
r1
A n1,m rm1 t t A n 2,m rm 2 t t B n1 (22)
A r t t A r t t B n 1
3 n ,m
Where
1 m
n 1
4 n ,m
2 m
2
n
A n1,m R 0n 2 R 0n
n 2
n 2, m
1 A n 2,m m k c R 0m n ,m R0 1 k A n 3,m R 0n cn n ,m R 0 1,n n ,m R0 R0 1 1 A n 4,m n 2 n n 2 n ,3 n 2,m R0 R0 k t t k r t n 1 1 B n1 G c r R 0 an t G c ' R 0 R n ,1 G t t G t G t t G t r r r r k t t k r t R G 1 1 R B n 2 cn b n t n 2 n ,3an 2 t n ,1 G c r G t t G t R n ,1 R 0 G r t t G r t R0 r r 0 (23) Moreover, the functions i , j and i , j are defined in preceding equation as follows:
1 i j 0 i j
i , j
1 i j 0 i j
i , j
(24)
By solving Eq. (23), the functions R1 , t t and R 2 , t t are obtained, and consequently, the unknown coefficients of Q1 , t t and Q 2 , t t are calculated according to the following equations:
k t t k r t q j1 t t G c r a t k r 2 t t G t t G t j c j r r 1 1 ' Gc R j ,1 j 2 r j12 t t G t t G t r r 1 1 q j 2 t t G c j 2 a j 2 t b j t R j ,1 k c r j1 t t G t t G t r r k t t k r t Gc r R j ,1 j ,1r j1 t t j 2 j ,3r j22 t t G t t G t r r (25) Therefore, the unknown complex potential functions could be computed at the time t t as follows:
Q1 , t t Q1 , t Q1 , t t Q 2 , t t Q 2 , t Q 2 , t t R1 , t t R1 , t R1 , t t
(26)
R 2 , t t R 2 , t R 2 , t t Note that Q1 , 0 , Q 2 , 0 , R1 , 0 , and R 2 , 0 are Q1 , Q 2 , R1 , and R 2 determined from initial equilibrium in section 3.1, respectively. Subsequently, 0 ,t t and 0 ,t t are obtained from equilibrium condition of traction vectors from rock and concrete sides, as follows:
0 , t t
0 , t t 0 , t t c , t t c , t t c , t t
(27)
After applying Cauchy integral along unit-radii circle, the unknown coefficients of 0 ,t t and 0 ,t t are determined as well:
a j t t r j 2 t t j 2 r j12 t t q j1 t t b j t t j 2 j ,3a j 2 t t 1 j ,1 r j1 t t
(28)
j 2 j ,3r j22 q j 2 t t Similarly, the unknown complex potential functions 0 and 0 could be calculated at t t as follows:
0 , t t 0 , t 0 , t t 0 , t t 0 , t 0 , t t
(29)
Applying this method in a stepwise procedure, the ultimate complex potential functions could be determined at a specific time t from following equation:
0 , t 0 0( n ) , t ' N
n 1
0 , t 0 0 n , t ' N
n 1
N
Q1 , t Q1 Q1 n , t ' n 1 N
Q 2 , t Q 2 Q 2
n
n 1
N
, t
N
R 2 , t R 2 R 2 n , t ' n 1
Where t ' n t , t
(30)
'
R1 , t R1 R1 n , t ' n 1
t , and N denotes the number of time steps. N
After determination of complex potential functions for the corresponding regions, stress and deformation field may be obtained through applying Eq. 16. 3.3. installation of lining support system with delay behind the tunnel face In case the lining is installed with a specific distance from the tunnel face, and an elapsed time exists before support installation, the problem may be modeled through applying convergenceconfinement concept. The solution process includes two stages, indeed. First before support installation, once the stress and deformation field occurs only due to in situ stresses and the support effect of tunnel face, along with rheological properties of rock mass, and next, after installing lining support when the interaction between lining and surrounding ground begins. Therefore, complex potential functions concerning rock mass and concrete lining behavior may be defined as follows based on superposition principle:
r , t r1 r 2 , t r , t r1 r 2 , t c , t R1 , t R 2 , t
(31)
c , t Q1 , t Q 2 , t Where subscript r and c are related to rock mass and concrete, while superscript (1) and (2) are 1 1 2 associated with stages 1 and 2, respectively. The functions r , r , r , t and
r 2 , t are defined as the following equations:
r1 Γ R 01 r1 Γ R 01 r 2 , t 1 Γ R 0 2 , t
(32)
r 2 , t 1 Γ R 0 2 , t Where is convergence parameter obtained based on current and ultimate tunnel wall displacement with respect to the distance between studied section and tunnel face [8]. Moreover, 01 , 01 , and 0 2 ,t , 0 2 ,t are complex potential functions corresponding to induced stress at stages 1 and 2, respectively, while the terms Γ R , Γ R , and
1 Γ R , 1 Γ R are related to initial stresses at stages 1 and 2, considering tunnel face effect, respectively.
01 and 01 are obtained through applying the boundary condition of zero traction vector along tunnel periphery as follows:
r1
1 r r1 0
(33)
And implementing Cauchy integration along unit-radii circle. Consequently, a system of 2 equations with 2 unknown functions, i.e. 01 and 01 are concluded, from which:
01 R ' 1
(34)
01 R 2 1 ' 3
Considering the elapsed time as t 0 , initial displacement due to excavation before support installation is determined as follows:
u x 0 iu y 0
k r t 0
2G r t 0
01
1 1 0 0 2G r t 0 1
(35)
After installing lining, due to exceedingly large rate of excavation, initial equilibrium is achieved immediately, according to section 3.1, as follows:
k r 0
G r 0
0 2 , t 0
2 k 1 0 , t 0 0 2 , t 0 c c , t 0 c , t 0 c , t 0 G r 0 G c Gc 1
2 r , t 0 r 2 , t 0 c , t 0 c , t 0 c , t 0 c R 0 , t 0 c R 0 , t 0 c R 0 , t 0 0
r 2 , t 0
(36) From that point on, time-dependent interaction between lining and surrounding rock mass begins which could be computed according to section 3.1. In fact, R1 , t t , R 2 , t t ,
Q1 , t t , Q 2 , t t , 0(2) ,t t , and 0(2) ,t t are determined based on Eqs. (22), (25), and (28) which finally leads to finding unknown complex potential functions R1 , R 2 , Q1 , Q 2 , 0(2) , and 0(2) at the time t from Eq. 30. 4. Discussion In this section, first the number of time steps is determined for the following analysis through an example, in such a manner that the convergence of the solution is achieved with high accuracy. Subsequently, a comparison is provided between the proposed analytical solution and COMSOL finite element code prediction in rock mass and lining regions. Finally, by applying the proposed solution, a parametric analysis is carried out with respect to the Burgers model coefficients. The example involves a circular tunnel driven in 300m depth in rock salt with the unit weigh of
0.021MN
m3
. The input data for rock mass and lining properties, as well as tunnel geometry,
are presented in Table 1. Note that due to circular tunnel configuration, it is concluded after solving linear equation systems in section 3 that only the coefficients a1 , b1 , b3 , r1(1) , r3 , r1 , 1
2
q1(1) , q1 , q 3 exist and other coefficients vanish. 2
2
4.1. Number of time steps for analysis In order to apply the solution, first the number of time steps should be selected such that the required convergence is obtained. To this purpose, absolute relative approximation error of circumferential stress vs. the number of time steps is provided for different rheological times in Tables 2, 3 and 4. It is demonstrated that for the number of steps larger than 50, the error reduced to less than 0.01% which is of great accuracy. Therefore, the analyses in the following sections are implemented considering 50 time steps, according to section 3. 4.2. Comparison of the analytical solution with COMSOL finite element code In order to verify the solution, a FEM numerical model is made by COMSOL software. Note that the comparison is drawn for the rheological time of 1000 days, although the results are the same for other alternatives. For convenience, the compression and tension are assumed as positive and negative quantities in this section.
The tunnel cross-section grid mesh along with boundary geometries is presented in Fig. 3. Fig. 4 shows the magnitude of circumferential stress along internal lining periphery predicted by the analytical solution and COMSOL finite element software. Figs. 5 and 6 also depict circumferential stress distribution along rock-concrete interface from concrete and rock sides, respectively. Moreover, Fig. 7 shows radial stress obtained by the analytical solution and COMSOL software along rock-concrete interface. As observed, a close agreement is evident between the analytical and numerical solution. Fig. 8 shows radial and shear stress components along internal lining periphery. It is seen that COMSOL predicts finite values while this solution predicts almost zero radial and tangential stress components, demonstrating its higher accuracy. Fig. 9 depicts circumferential stress along rock-concrete interface from rock side, at the wall and roof of the tunnel, with respect to rheological time. As seen, the circumferential stress component tends to initial stress magnitudes in the long-term, as expected due to presence of Maxwell body in rheological model, for both analytical and COMSOL numerical simulation. Furthermore, Figs 10 and 11 also illustrate circumferential stress along rock-concrete interface and internal lining periphery, respectively, at the wall and roof of the tunnel, with respect to rheological time. It is demonstrated that circumferential stress component tends to an oblique asymptote in both figures, for the analytical and numerical simulation due to steady deformation of rock mass with respect to time resulted from the viscose effect of Maxwell body. 4.3. Parametric analysis of Burgers model coefficients In this section, a parametric analysis is carried out with respect to the Burgers model coefficients. This analysis is performed for both short-term and long-term periods of time, in which 10 and 1000 days are assumed as those time periods, respectively. Since the wall and roof of the tunnel are critical points in stress analysis around the hole, these points are selected for parametric analysis. Furthermore, the preceding example is used as input data. 4.3.1 The influence of shear moduli G m and G k The parameters G m and G k vary from 10 to 100 Gpa in this analysis. Figs. 12 and 13, (a) and (b), represent the impact of the shear moduli on induced circumferential stress at the wall and roof of the tunnel in concrete and rock materials, along internal lining periphery and rock-concrete interface, for the short-term time period of 10 days. It is shown that circumferential stress reduces more sharply in lining region for low values of shear modulus, approximately less than 40 Gpa, and it flattens for large values. However, this trend is not the same for the long-term time period of 1000 days, as shown in Figs 14 and 15, (a) and (b). It is seen that circumferential stress component remains approximately the same by increasing shear modulus. On the other hand, Figs. 12, 13, 14, and 15 demonstrate that circumferential stress is about the same in the rock region by increasing shear modulus, for the both long and short periods of time. 4.3.2 The influence of viscosity coefficients m and k
The parameter m varies from 50 to 1000 Gpa.day in this section. As seen in Figs. 16 (a) and (b), circumferential stress reduces more sharply in lining region for lower values of the viscosity coefficient m , approximately less than 400 Gpa.day, and it flattens for large values, for the short-term time period of 10 days. However, with respect to the long-term time period of 1000 days circumferential stress in concrete region experiences an almost decrease based on the Figs. 17 (a) and (b), except for low values along internal lining periphery at the roof, and rockconcrete interface at the wall of the opening, where the curves exhibit a peak around the magnitude of 100 Gpa.day. On the other hand, in the rock region, Figs 16 (a) and (b) denote almost constant values for circumferential stress at the roof and wall of the tunnel for the short-term time period. However, with respect to long period of time, Figs. 17 (a) and (b) show a monotonous gradual increase in circumferential stress values. The parametric analysis of the coefficient k is also done over the range of 10 to 100 Gpa.day for circumferential stress in rock and concrete regions, along internal periphery and rock-concrete interface. It is observed according to Figs. 18 and 19, (a) and (b), that the circumferential stress quantities, in rock and concrete regions, are approximately constant over this domain for both short and long periods of time, at the wall and roof of the tunnel.
5. Conclusion In this study, an analytical solution was proposed for stress field around lined circular tunnels and in their lining support. The rock mass was supposed as a viscoelastic material obeying Burgers model, while lining was assumed as a linear elastic material. The solution was based on complex potential functions, combined with correspondence principle in a stepwise procedure. The proposed method was able to predict stress and deformation components in the long-term specifically when the viscosity element of Maxwell body dominated surrounding rock mass behavior inducing non-stop deformation. The solution was compared with finite element COMSOL software prediction which exhibited a close agreement. However, the solution benefits from some prominences as follows: 1. The solution determines stress field more accurately 2. The solving-time is immensely less than that for COMSOL numerical simulation software. This fact also makes the solution executable for parametric analysis of the ground and support structure properties in the long-term.
Finally, a parametric analysis was carried out with respect to Burgers model coefficients for short-term and long-term time periods of 10 and 1000 days, respectively. By using the preceding example as input data, the analysis demonstrated different behavior for each parameter, with respect to short and long periods of time. It was observed that circumferential stress reduces with increasing the value of G m and G k parameters in lining region, while remains approximately the same for rock region along rock-concrete interface at roof and wall of the tunnel, for short period
of time. With respect to long-term time period, circumferential stress is about the same by increasing shear moduli, G m and G k , in rock and lining regions. It was also observed that increasing the parameter m generally reduces circumferential stress in concrete region at wall and roof of the opening, while it doesn’t have significant influence on circumferential stress in rock region, and only makes a small enhance with respect to the long period of time. However, considering the variation of the parameter k , circumferential stress remains approximately the same in rock and lining regions, along internal periphery and rock-concrete interface, at wall and roof of the opening. To sum up, applying complex variable method in combined with correspondence principle in step-wise procedure coms up to be an effective approach in solving viscoelastic problems not only for Burgers model but also with any rheological constitutive behavior.
References [1] S. Sakurai, Approximate time‐dependent analysis of tunnel support structure considering progress of tunnel face, International Journal for Numerical and Analytical Methods in Geomechanics, 2 (1978) 159175. [2] M. Panet, Time-dependent deformations in underground works, 4th ISRM Congress, International Society for Rock Mechanics, 1979. [3] J. Sulem, M. Panet, A. Guenot, An analytical solution for time-dependent displacements in a circular tunnel, International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1987, pp. 155-164. [4] Y.-W. Pan, J.-J. Dong, Time-dependent tunnel convergence—II. Advance rate and tunnel-support interaction, International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1991, pp. 477-488. [5] Y.-W. Pan, J.-J. Dong, Time-dependent tunnel convergence—I. Formulation of the model, International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1991, pp. 469-475. [6] P.F. Gnirk, R.E. Johnson, The deformational behavior of a circular mine shaft situated in a viscoelastic medium under hydrostatic stress, The 6th US Symposium on Rock Mechanics (USRMS), American Rock Mechanics Association, 1964. [7] G. Gioda, A finite element solution of non-linear creep problems in rocks, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Elsevier, 1981, pp. 35-46. [8] M. Penet, A. Guenot, Analysis of Convergence Behind the Face of a Tunnel, Int, Symposium Tunneling, 1982, pp. 197-204. [9] B. Ladanyi, D. Gill, Design of tunnel linings in a creeping rock, International Journal of Mining and Geological Engineering, 6 (1988) 113-126. [10] A. Cividini, G. Gioda, A. Carini, A finite element analysis of the time dependent behaviour of underground openings, Beer, Booker, Carter (Eds.), Computer Methods and Advances in Geomechanics, (1991). [11] C. Paraskevopoulou, M. Diederichs, Analysis of time-dependent deformation in tunnels using the Convergence-Confinement Method, Tunnelling and Underground Space Technology, 71 (2018) 62-80. [12] J.C. Jaeger, N.G. Cook, R. Zimmerman, Fundamentals of rock mechanics, John Wiley & Sons2009.
[13] A. Fahimifar, F.M. Tehrani, A. Hedayat, A. Vakilzadeh, Analytical solution for the excavation of circular tunnels in a visco-elastic Burger’s material under hydrostatic stress field, Tunnelling and Underground Space Technology, 25 (2010) 297-304. [14] P. Nomikos, R. Rahmannejad, A. Sofianos, Supported axisymmetric tunnels within linear viscoelastic Burgers rocks, Rock Mechanics and Rock Engineering, 44 (2011) 553-564. [15] K. Lo, C.M. Yuen, Design of tunnel lining in rock for long term time effects, Canadian Geotechnical Journal, 18 (1981) 24-39. [16] F. Song, H. Wang, M. Jiang, Analytically-based simplified formulas for circular tunnels with two liners in viscoelastic rock under anisotropic initial stresses, Construction and Building Materials, 175 (2018) 746-767. [17] F. Song, H. Wang, M. Jiang, Analytical solutions for lined circular tunnels in viscoelastic rock considering various interface conditions, Applied Mathematical Modelling, 55 (2018) 109-130. [18] Z. Zhang, M. Huang, C. Zhang, K. Jiang, M. Lu, Time-domain analyses for pile deformation induced by adjacent excavation considering influences of viscoelastic mechanism, Tunnelling and Underground Space Technology, 85 (2019) 392-405. [19] G.T. Mase, G.E. Mase, Continuum mechanics for engineers, CRC press1999. [20] N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Springer Science & Business Media2013. [21] S.-c. Li, M.-b. Wang, Elastic analysis of stress–displacement field for a lined circular tunnel at great depth due to ground loads and internal pressure, Tunnelling and Underground Space Technology, 23 (2008) 609-617. [22] S.-c. Li, M.-b. Wang, An elastic stress–displacement solution for a lined tunnel at great depth, International Journal of Rock Mechanics and Mining Sciences, 45 (2008) 486-494. [23] M.B. Wang, S.C. Li, A complex variable solution for stress and displacement field around a lined circular tunnel at great depth, International journal for numerical and analytical methods in geomechanics, 33 (2009) 939-951. [24] H. Zhang, Z. Wang, Y. Zheng, P. Duan, S. Ding, Study on tri-axial creep experiment and constitutive relation of different rock salt, Safety science, 50 (2012) 801-805.
Tables Table 1. Input data including rock mass and concrete properties, as well as tunnel crosssection geometry
Input data for Burgers model [24]
G k (MPa)
k (MPa day)
G m (MPa)
6.73E 4
1.99E 4
1.99E 4
m (MPa*day) 1.39E 5
Properties of concrete lining and tunnel geometry
E c (MPa)
c
R (m)
R in (m)
1.65E 4
0.2
4.5
4.3
K (MPa) 4.3116e+04 Ratio of Horizontal to vertical in situ stress components
0.5
Table 2. Average of absolute relative approximation error of circumferential stress along internal lining periphery with respect to the number of time steps for different rheological times Number of time steps
2
10
20
50
100
500
1000
0.162
0.038
0.006
0.001 0.000 0.000
478.092 0.085
0.031
0.006
0.001 0.000 0.000
167.746 140.639 216.063 0.002
0.000 0.000 0.000
Time (days)
10.184
100 1000 10000
Table 3. Average of absolute relative approximation error of circumferential stress along rock-lining interface from lining side with respect to number of time steps for different rheological times Number of time steps Time (days)
2
10
20
50
100
500
1000
10.524
100 1000 10000
0.158
0.037
0.006
0.001 0.000 0.000
362.695 0.071
0.026
0.005
0.001 0.000 0.000
140.639 216.063 0.002
0.001 0.000 0.000
97.102
Table 4. Average of absolute relative approximation error of circumferential stress along rock-lining interface from rock side with respect to number of time steps for different rheological times Number of time steps
2
10
20
50
100
500
1000
0.079
0.018
0.003
0.001 0.000 0.000
181.843 0.072
0.026
0.005
0.001 0.000 0.000
106.436 140.641 216.063 0.000
0.000 0.000 0.000
Time (days)
100 1000 10000
Figures
7.727
(b)
(a)
(c) Fig. 1. Rheological models: a) Kelvin, b) Maxwell, c) Burgers
Fig. 2. Conformal mapping of tunnel cross-section in z-plane into two concentric circles in plane
Fig 3. COMSOL grid mesh for tunnel cross-section, and boundaries geometry
Fig. 4. Circumferential stress along internal lining periphery predicted by the proposed analytical solution and numerical COMSOL software
Fig. 5. Circumferential stress along rock-concrete interface from concrete side
Fig. 6. Circumferential stress along rock-concrete interface from rock side
Fig. 7. Radial stress along rock-concrete interface predicted by the proposed analytical solution and numerical COMSOL software
(a)
(b)
Fig. 8. Analytical solution and numerical COMSOL software prediction of a) normal stress and b) shear stress along inner lining periphery
Fig. 9. Circumferential stress along rock-concrete interface from rock side, at the wall and roof of the tunnel, with respect to rheological time
Fig. 10. Circumferential stress along rock-concrete interface from concrete side, at the wall and roof of the tunnel, with respect to rheological time
Fig. 11. Circumferential stress along internal lining periphery, at the wall and roof of the tunnel, with respect to rheological time
(a)
(b)
Fig. 12. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G m for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 13. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G k for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 14. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G m for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 15. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G k for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 16. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient m for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 17. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient m for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 18. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient k for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 19. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient k for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure Ali Reza Kargar , Hadi Haghgouei PII: DOI: Reference:
S0307-904X(19)30555-4 https://doi.org/10.1016/j.apm.2019.09.025 APM 13029
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
12 May 2019 8 August 2019 12 September 2019
Please cite this article as: Ali Reza Kargar , Hadi Haghgouei , An analytical solution for timedependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.09.025
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Highlights
An analytical solution is proposed for stress field around lined circular tunnels surrounded by viscoelastic rock mass. The solution is able to predict stress field accurately both in short-term and long-term periods of time. The solution is based on complex potential functions and correspondence principle, in a step wise procedure. The method which employs time discretization approach could be applied to any rheological model.
An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure Ali Reza Kargar*, Hadi Haghgouei School of Mining Engineering, College of Engineering, University of Tehran, Iran *
Corresponding author. Tel.: +98-9171388825;
E-mail address: [email protected] (A.R. Kargar)
Abstract Time-dependent behavior of surrounding mass plays a significant role in designing underground constructions. Considering simple configuration of lined circular tunnels, a lot of solution have been proposed to this problem. However, many assume hydrostatic initial stress field, and other solutions are only applicable to simple rheological models and could not account for viscosity effect in long-term time periods. In this study, an analytical plane strain solution is proposed for lined circular tunnels under non-hydrostatic initial stress field, assuming rock mass as a viscoelastic material obeying Burgers model, while concrete lining is supposed to have linear elastic behavior. The solution which employs complex variable method combined with correspondence principle benefits from time discretization approach enabling the solution to take into account the viscosity effect in the both short-term and long-term periods of time, while predicting stress components accurately. The results obtained by the proposed solution were compared with those predicted by finite element COMSOL software which exhibited a close agreement. It was found that by increasing time both the proposed analytical solution and finite element numerical method tend to an oblique asymptote due to viscosity effect of Maxwell body in the Burgers model. Finally, a parametric analysis was performed with respect to Burgers model coefficients which showed different behavior for short and long periods of time. Keywords: Time-dependent behavior; Lined circular tunnels; Complex variable method; Correspondence principle; Time discretization approach
1. Introduction Once a tunnel is excavated, the ground pressure on the lining support increases with time due to the rheological properties of surrounding rock, effected by tunnel advancement and rock-support interaction [1-5]. This phenomenon has a substantial impact on support design especially in sedimentary rocks, in which excess pressure on support system may be produced over time. This pressure could develop due to tunnel face progress which controls the rate of loading on support, and rock-support interaction inducing time-dependent stress and deformation in lining and rock material. In this study, the excavation rate is assumed infinitely large that the effect of loading rate on support behavior is ignored, and time-dependent analysis is performed only due to rocksupport interaction. A lot of attention has been paid to time-dependent interaction of rock-support system [1-11]. Generally, this problem is encountered through two main approaches: numerical and analytical methods. Although the numerical approach benefits from a great strength of being applicable to many sophisticated problems with various complexities from boundary conditions to material constitutive models, the time of solving process could be prolonged, in addition to the fact that their prediction also is not very accurate compared to analytical solutions and is affected by different parameters such as mesh size, time steps length, boundaries of the model and etc. However, analytical methods predict the solution exceedingly more accurate compared to numerical approaches, and present a measure to assess the validity of numerical methods [12]. A majority of researches have been conducted in this field considering opening as a circular hole, supported by lining with constant thickness, in a homogenous isotropic viscoelastic rock mass under a hydrostatic initial stress field [1, 2, 4-6, 9, 13, 14]. Gnrik and Johnson determined deformation field analytically around a circular shaft with and without lining support, considering viscoelastic behavior for surrounding rock mass[6]. Sakurai, by introducing an equivalent initial stress, developed an analytical solution and obtained pressure acting on the tunnel supports vs. time and tunnel advancement [1]. Panet also explored the time-dependent deformation in underground works considering different rheological models in this regard, as well as their limitations [2]. Ladanyi and Grill investigated the effect of long-term rock deformation on lining pressure through analytical methods[9]. Pan and Dong proposed a solution for time-dependent tunnel convergence by employing tunnel advancement effect and support installation in a viscoelastic medium[4, 5]. Fahimifar et al. suggested a closed form solution for excavation of circular tunnels in viscoelastic burgers’ rock masses, in case of both lined and unlined tunnels[13]. Moreover, Nikomos et al. derived a solution for tunnels in viscoelastic materials considering support installation time[14]. However, In case that the in situ stress constitutes a non-hydrostatic stress field, rarely any researches were done. Lo and Yuen presented a solution for lined circular tunnels under non-slip and full-slip liningrock interface condition considering general Kelvin model dominates rock mass behavior [15]. Song et al. also investigated the problem in case that the tunnel was supported with two circular linings installed at different times by using correspondence principle along with convolution integral to model different boundary loads applied at different times [16, 17]. Zhang et al also presented a time-dependent solution for ground-support interaction of a pile adjacent to an
excavation through applying Mindlin’s solution along with corresponding principle, as well as employing Pasternak’s foundation model assuming surrounding soil follows Boltzmann viscoelastic constitutive model [18]. However, with respect to long-term time periods, especially when viscos element controls rock mass behavior (such as Burgers model), correspondence principle could not give correct solution without time discretization. This problem could lead to a wrong prediction of stress and deformation field within lining and surrounding mass. In this study, a new solution is proposed for time-dependent stress and deformation field of lining and surrounding rock mass for circular tunnels, considering Lining support and surrounding mass as homogenous isotropic linear elastic and viscoelastic materials, respectively, in which Burgers model dominates the viscoelastic behavior. The solution is achieved through applying complex variable method, along with time discretization, in a stepwise procedure. For simplicity, it is assumed tunnel advancement rate is infinitely large that the viscos deformation during excavation could be neglected, and consequently induced stress field varies over time only owing to lining support interaction. By applying time discretization, the unloading phenomenon is modelled in each time step with respect to complex potential functions.
2. Problem statement and assumptions Solving the problem of time-dependent behavior of lined circular tunnels under non-hydrostatic initial stress field, several assumptions are considered as follows: 1. The surrounding rock mass is an isotropic homogenous medium with viscoelastic behavior, obeying Burger’s constitutive model. 2. The in situ stress field is non-hydrostatic. 3. The tunnel is excavated in such a great depth that the gradient of stress field could be neglected (in other words, the tunnel is a deep-buried tunnel). 4. The lining is made of linear-elastic material. 5. The rate of excavation is infinitely large that no viscous behavior exists during this period. The solution is derived for two cases. First the case when the lining is installed instantly behind tunnel face, and second, once the lining is installed with a distance from the face considering an elapsed time before support installation. The rate of excavation is assumed so large that no viscos behavior is evident in this period, and initial equilibrium between lining structure and surrounding rock mass is achieved immediately. Subsequently, time-dependent interaction between support system and surrounding ground begins. This interaction is investigated through a stepwise procedure which allows incorporating elastic unloading phenomenon occurred with time in surrounding rock. Since the initial stress field is anisotropic, and the problem deals with a multi-region model of materials with viscoelastic behavior, complex variable method introduced by Muskhelishvili, along with correspondence principle is implemented in a stepwise procedure.
In order to solve the problem, first time-dependent behavior and complex variable method are briefly introduced in the following sections. 2.1. Viscoelastic behavior Viscoelastic behavior, in brief, represents viscous, associated with fluids, and elastic, related to solids, behaviors, in which the former is modeled by dashpot, and the latter is modeled by spring in rheological models. The simplest viscoelastic models are known as Maxwell and Kelvin bodies, which are shown in Fig. 1. However, these two models are too elementary to model complicated behavior of rocks. Consequently, the Burger model was introduced which is a series combination of Maxwell and Kelvin bodies. This rheological model, which is shown in Fig. 1, was better able to model time-dependent rocks behavior. Generally, time-dependent deformation is determined from deviatoric stress components, while hydrostatic stress component only contributes to elastic dilation. Therefore, stress- strain relations in rheological models may be written as follows [19]: m k k e q S ij ij k t k t k k 0 k 0 ii 3K ii n
pk
(1)
t represents the derivative operator with respect to time, S ij and e ij also denote deviatoric parts of
Where K is the bulk modulus, ij and ij are components of stress and strain tensors,
and , respectively, according to the following equation: ij
ij
ij ij ij ij
kk 3
kk 3
e ij (2)
S ij
Considering that
S ij 2Ge ij
(3)
And comparing this equation with Eq. (1), 2G may be defined as the following operator in viscoelastic problems:
k p k t k 2G k m0 k q k t k k 0 n
(4)
In the case of creep problem, where stress field remains constant with respect to time, the coefficient G could be obtained as follows:
1 G t G k
G kt 1 e k
1 t G m m
1
(5)
Where G k and k are Kelvin shear modulus and viscosity, G m and m are shear modulus and viscosity of Maxwell model. The viscoelastic models are generally employed within stress and strain analysis through correspondence principle approach. In fact, this method emerges from the analogues between the original basic elasticity equations and their Laplace transformation with respect to time. According to this method the coefficient G in elastic solutions is replaced with
Q in transformed space, in which Q and P are obtained from Laplace transformation of Eq. P (1). The final viscoelastic solution in time space is obtained by Laplace inverse transformation of this solution. In the next section, the well-known elasticity solution, namely complex variable method, proposed by Muskhelsihvili, is introduced. 2.2. Complex variable method According to complex variable method, there exist two holomorphic complex potential functions, named z and z , in plane theory of elasticity from which stress components could be obtained as follows [20]:
xx yy 2 ' z ' z
yy xx 2i xy 2 z z z ''
'
(6)
Where z x iy is the coordinate of the material point in z-plane, for which i 2 1 . Moreover, the displacement components may also be determined based on aforementioned functions z and z as follows:
2G ux iu y k z z ' z z
(7)
where G is the shear modulus, and k is a parameter related to Poisson’s ratio which is given by: 3 4 k 3 1
plane strain plane stress
(8)
Generally, the holomorphic functions z and z are obtained according to the boundary conditions. In the case of first fundamental problems where traction vectors constitute the boundary conditions, these functions are calculated from the following equation:
i
z
z0
p iq ds z z ' z z C
(9)
where p and q are components of the surface traction vector in x and y directions, respectively, and C is the constant of integration. In the case of second fundamental problems, in which displacements along boundaries contribute to the boundary conditions, the functions z and z are determined from Eq. (7). 3. Derivation of the solution In order to find a solution, at the primary stage the conventional solution for lined circular tunnel in elastic medium is introduced [21-23]. Subsequently, the solution for lined circular tunnels surrounded in a viscoelastic medium is proposed, considering two cases of instant support installation adjacent to tunnel face, or installing support system with delay behind the face. 3.1. Solution for lined circular tunnels in elastic medium Since the problem focuses on the interaction between lining support and surrounding rock mass in circular tunnels, this issue could be employed to the solving process through considering the model as a multi-region medium. Therefore, for each region, namely rock mass and concrete lining support, complex potential functions r z , r z , and c z , c z are defined, respectively, which are introduced as follows:
r z Γz 0 z r z Γz 0 z c z R1 z R 2 z
(10)
c z Q1 z Q 2 z Where 0 z and 0 z are holomorphic functions defined in rock mass medium, while
R1 z , R 2 z , Q1 z and Q 2 z are those defined in concrete material. The coefficients Γ and Γ are real constants determined with regard to the uniform stress state at infinity from the following equation [20]:
1 1 H 1 2 4 4 1 1 Γ 1 2 H 2 2
(11)
Where H and are the depth of the hole, and unit weight of the rock mass, respectively. also represents the ratio of horizontal to vertical in situ stresses. By applying conformal mapping, the circular hole and lining are mapped into two concentric circles with unit and R 0 radius in plane (Fig. 2), according to the following equation: z R
(12)
Consequently, Eq. 10 is rewritten as follows in -plane:
r Γ R 0 r Γ R 0
(13)
c R1 R 2 c Q1 Q 2 Where
0 ( ) a j j , j 0
Q1 q j1 j , j 0
Q 2 q j 2 j , j 1
0 ( ) b j j j 0
R1 r j1 j
(14)
j 0
R 2 r j 2 j j 1
These unknown functions are determined through applying boundary condition followed by 1 d Cauchy integration ( ) along the unit circle. The boundary conditions are zero 2 i traction vector along inner boundary of the lining support, along with identical displacement and traction vector along rock-concrete interface, as follows:
k kr 1 1 0 0' 0 c c c' c Gr Gr Gc Gc
r r' r c c' c
(15)
c R 0 R 0c' R 0 c R 0 0 Where k r , G r and k c , G c are the coefficients k , G related to rock mass and concrete regions, R1 . After taking Cauchy integration of Eq. (15) for inside R and outside of the unit circle, a system of 6 equations with 6 unknown functions, i.e. 0 ( ) ,
respectively, e i , and R o
0 ( ) , Q1 , Q 2 , R1 and R 2 , are obtained, which may be solved by series solution method. Consequently, following determination of unknown functions, complex potential
functions r , r , and c , c are obtained from Eq. (13), and corresponding stress and deformation field could be calculated based on Eqs. (6) and (7) considering z R , as follows [20]:
2 ' ' R 2e 2i 2i '' ' R
(16)
2G u x iu y k '
Where and are complex potential functions, and , , as well as are radial, circumferential, and shear stress components associated with the studied region. 3.2. Solution for lined circular tunnels in viscoelastic medium considering lining installation adjacent to the tunnel face Since the rate of excavation is infinitely large, initial equilibrium between surrounding rock and support system is achieved immediately, whose stress and deformation fields could be determined according to section 3.1. Note that due to extremely short period of time, rock mass behaves as an elastic material. Subsequently, rock mass viscoelastic behavior begins in which, for each time increment, there exists the corresponding rock mass deformation inducing stress in lining material. This induced stress field could be determined by defining the complex potential functions within lining region in incremental form as follows: c , t R1 , t R 2 , t
(17)
c , t Q1 , t Q 2 , t
Where
Q1 , t q j1 t j ,
R1 , t r j1 t j
j 0
Q 2 , t q j
2
t
j
j 1
j 0
,
R 2 , t r j
2
t
(18) j
j 1
The time-dependent rock mass deformation could affect the stress and deformation field of the lining material through considering its deformation along rock-concrete interface, as the boundary condition of induced displacement for lining region. This fact could be employed to the solution for each time step through the following equation: kc 1 c , t t 2Gc 2Gc
c , t t c , t t u xr i u yr
(19)
Where u xr and u yr are the rock displacement increments in x and y directions along rockconcrete interface, which is calculated as follows: 1 k t t k r t u xr i u yr r Γ R 0 , t 2 G r t t G r t (20) ΓR 1 1 1 0 , t 0 , t ΓR 2 G r t t G r t
G r t is calculated from Eq. 5, and k r t is obtained from Eq. 8 considering
3K 2G t
6K 2G t
,
in which K is the Bulk modulus. 0 ,t and 0 ,t are holomorphic potential functions associated with deformation and stress fields in rock mass region, according to Eq. (14). It is obvious that 0 , 0 and 0 , 0 denote those functions determined based on initial equilibrium described in section 3.1. On the other hand, the traction vector must vanish along inner lining surface, which in turn constitutes the other boundary condition for complex functions c , t and c , t , as follows:
c R 0 , t t R 0
c R 0 , t t c R 0 , t t 0
Where e i . After multiplying both sides of Eqs. (19) and (21) by
(21)
d and integrating 2 i 1
along unit-radii circle, a system of 4 equations with 4 unknown functions, i.e. R1 , R 2 ,
Q1 , and Q 2 , are concluded which after some manipulation, and applying series 2 solution, the following equation set with respect to unknown coefficients r j1 and r j is
obtained: s1
s1
n 1
n 1
r1
r1
A n1,m rm1 t t A n 2,m rm 2 t t B n1 (22)
A r t t A r t t B n 1
3 n ,m
Where
1 m
n 1
4 n ,m
2 m
2
n
A n1,m R 0n 2 R 0n
n 2
n 2, m
1 A n 2,m m k c R 0m n ,m R0 1 k A n 3,m R 0n cn n ,m R 0 1,n n ,m R0 R0 1 1 A n 4,m n 2 n n 2 n ,3 n 2,m R0 R0 k t t k r t n 1 1 B n1 G c r R 0 an t G c ' R 0 R n ,1 G t t G t G t t G t r r r r k t t k r t R G 1 1 R B n 2 cn b n t n 2 n ,3an 2 t n ,1 G c r G t t G t R n ,1 R 0 G r t t G r t R0 r r 0 (23) Moreover, the functions i , j and i , j are defined in preceding equation as follows:
1 i j 0 i j
i , j
1 i j 0 i j
i , j
(24)
By solving Eq. (23), the functions R1 , t t and R 2 , t t are obtained, and consequently, the unknown coefficients of Q1 , t t and Q 2 , t t are calculated according to the following equations:
k t t k r t q j1 t t G c r a t k r 2 t t G t t G t j c j r r 1 1 ' Gc R j ,1 j 2 r j12 t t G t t G t r r 1 1 q j 2 t t G c j 2 a j 2 t b j t R j ,1 k c r j1 t t G t t G t r r k t t k r t Gc r R j ,1 j ,1r j1 t t j 2 j ,3r j22 t t G t t G t r r (25) Therefore, the unknown complex potential functions could be computed at the time t t as follows:
Q1 , t t Q1 , t Q1 , t t Q 2 , t t Q 2 , t Q 2 , t t R1 , t t R1 , t R1 , t t
(26)
R 2 , t t R 2 , t R 2 , t t Note that Q1 , 0 , Q 2 , 0 , R1 , 0 , and R 2 , 0 are Q1 , Q 2 , R1 , and R 2 determined from initial equilibrium in section 3.1, respectively. Subsequently, 0 ,t t and 0 ,t t are obtained from equilibrium condition of traction vectors from rock and concrete sides, as follows:
0 , t t
0 , t t 0 , t t c , t t c , t t c , t t
(27)
After applying Cauchy integral along unit-radii circle, the unknown coefficients of 0 ,t t and 0 ,t t are determined as well:
a j t t r j 2 t t j 2 r j12 t t q j1 t t b j t t j 2 j ,3a j 2 t t 1 j ,1 r j1 t t
(28)
j 2 j ,3r j22 q j 2 t t Similarly, the unknown complex potential functions 0 and 0 could be calculated at t t as follows:
0 , t t 0 , t 0 , t t 0 , t t 0 , t 0 , t t
(29)
Applying this method in a stepwise procedure, the ultimate complex potential functions could be determined at a specific time t from following equation:
0 , t 0 0( n ) , t ' N
n 1
0 , t 0 0 n , t ' N
n 1
N
Q1 , t Q1 Q1 n , t ' n 1 N
Q 2 , t Q 2 Q 2
n
n 1
N
, t
N
R 2 , t R 2 R 2 n , t ' n 1
Where t ' n t , t
(30)
'
R1 , t R1 R1 n , t ' n 1
t , and N denotes the number of time steps. N
After determination of complex potential functions for the corresponding regions, stress and deformation field may be obtained through applying Eq. 16. 3.3. installation of lining support system with delay behind the tunnel face In case the lining is installed with a specific distance from the tunnel face, and an elapsed time exists before support installation, the problem may be modeled through applying convergenceconfinement concept. The solution process includes two stages, indeed. First before support installation, once the stress and deformation field occurs only due to in situ stresses and the support effect of tunnel face, along with rheological properties of rock mass, and next, after installing lining support when the interaction between lining and surrounding ground begins. Therefore, complex potential functions concerning rock mass and concrete lining behavior may be defined as follows based on superposition principle:
r , t r1 r 2 , t r , t r1 r 2 , t c , t R1 , t R 2 , t
(31)
c , t Q1 , t Q 2 , t Where subscript r and c are related to rock mass and concrete, while superscript (1) and (2) are 1 1 2 associated with stages 1 and 2, respectively. The functions r , r , r , t and
r 2 , t are defined as the following equations:
r1 Γ R 01 r1 Γ R 01 r 2 , t 1 Γ R 0 2 , t
(32)
r 2 , t 1 Γ R 0 2 , t Where is convergence parameter obtained based on current and ultimate tunnel wall displacement with respect to the distance between studied section and tunnel face [8]. Moreover, 01 , 01 , and 0 2 ,t , 0 2 ,t are complex potential functions corresponding to induced stress at stages 1 and 2, respectively, while the terms Γ R , Γ R , and
1 Γ R , 1 Γ R are related to initial stresses at stages 1 and 2, considering tunnel face effect, respectively.
01 and 01 are obtained through applying the boundary condition of zero traction vector along tunnel periphery as follows:
r1
1 r r1 0
(33)
And implementing Cauchy integration along unit-radii circle. Consequently, a system of 2 equations with 2 unknown functions, i.e. 01 and 01 are concluded, from which:
01 R ' 1
(34)
01 R 2 1 ' 3
Considering the elapsed time as t 0 , initial displacement due to excavation before support installation is determined as follows:
u x 0 iu y 0
k r t 0
2G r t 0
01
1 1 0 0 2G r t 0 1
(35)
After installing lining, due to exceedingly large rate of excavation, initial equilibrium is achieved immediately, according to section 3.1, as follows:
k r 0
G r 0
0 2 , t 0
2 k 1 0 , t 0 0 2 , t 0 c c , t 0 c , t 0 c , t 0 G r 0 G c Gc 1
2 r , t 0 r 2 , t 0 c , t 0 c , t 0 c , t 0 c R 0 , t 0 c R 0 , t 0 c R 0 , t 0 0
r 2 , t 0
(36) From that point on, time-dependent interaction between lining and surrounding rock mass begins which could be computed according to section 3.1. In fact, R1 , t t , R 2 , t t ,
Q1 , t t , Q 2 , t t , 0(2) ,t t , and 0(2) ,t t are determined based on Eqs. (22), (25), and (28) which finally leads to finding unknown complex potential functions R1 , R 2 , Q1 , Q 2 , 0(2) , and 0(2) at the time t from Eq. 30. 4. Discussion In this section, first the number of time steps is determined for the following analysis through an example, in such a manner that the convergence of the solution is achieved with high accuracy. Subsequently, a comparison is provided between the proposed analytical solution and COMSOL finite element code prediction in rock mass and lining regions. Finally, by applying the proposed solution, a parametric analysis is carried out with respect to the Burgers model coefficients. The example involves a circular tunnel driven in 300m depth in rock salt with the unit weigh of
0.021MN
m3
. The input data for rock mass and lining properties, as well as tunnel geometry,
are presented in Table 1. Note that due to circular tunnel configuration, it is concluded after solving linear equation systems in section 3 that only the coefficients a1 , b1 , b3 , r1(1) , r3 , r1 , 1
2
q1(1) , q1 , q 3 exist and other coefficients vanish. 2
2
4.1. Number of time steps for analysis In order to apply the solution, first the number of time steps should be selected such that the required convergence is obtained. To this purpose, absolute relative approximation error of circumferential stress vs. the number of time steps is provided for different rheological times in Tables 2, 3 and 4. It is demonstrated that for the number of steps larger than 50, the error reduced to less than 0.01% which is of great accuracy. Therefore, the analyses in the following sections are implemented considering 50 time steps, according to section 3. 4.2. Comparison of the analytical solution with COMSOL finite element code In order to verify the solution, a FEM numerical model is made by COMSOL software. Note that the comparison is drawn for the rheological time of 1000 days, although the results are the same for other alternatives. For convenience, the compression and tension are assumed as positive and negative quantities in this section.
The tunnel cross-section grid mesh along with boundary geometries is presented in Fig. 3. Fig. 4 shows the magnitude of circumferential stress along internal lining periphery predicted by the analytical solution and COMSOL finite element software. Figs. 5 and 6 also depict circumferential stress distribution along rock-concrete interface from concrete and rock sides, respectively. Moreover, Fig. 7 shows radial stress obtained by the analytical solution and COMSOL software along rock-concrete interface. As observed, a close agreement is evident between the analytical and numerical solution. Fig. 8 shows radial and shear stress components along internal lining periphery. It is seen that COMSOL predicts finite values while this solution predicts almost zero radial and tangential stress components, demonstrating its higher accuracy. Fig. 9 depicts circumferential stress along rock-concrete interface from rock side, at the wall and roof of the tunnel, with respect to rheological time. As seen, the circumferential stress component tends to initial stress magnitudes in the long-term, as expected due to presence of Maxwell body in rheological model, for both analytical and COMSOL numerical simulation. Furthermore, Figs 10 and 11 also illustrate circumferential stress along rock-concrete interface and internal lining periphery, respectively, at the wall and roof of the tunnel, with respect to rheological time. It is demonstrated that circumferential stress component tends to an oblique asymptote in both figures, for the analytical and numerical simulation due to steady deformation of rock mass with respect to time resulted from the viscose effect of Maxwell body. 4.3. Parametric analysis of Burgers model coefficients In this section, a parametric analysis is carried out with respect to the Burgers model coefficients. This analysis is performed for both short-term and long-term periods of time, in which 10 and 1000 days are assumed as those time periods, respectively. Since the wall and roof of the tunnel are critical points in stress analysis around the hole, these points are selected for parametric analysis. Furthermore, the preceding example is used as input data. 4.3.1 The influence of shear moduli G m and G k The parameters G m and G k vary from 10 to 100 Gpa in this analysis. Figs. 12 and 13, (a) and (b), represent the impact of the shear moduli on induced circumferential stress at the wall and roof of the tunnel in concrete and rock materials, along internal lining periphery and rock-concrete interface, for the short-term time period of 10 days. It is shown that circumferential stress reduces more sharply in lining region for low values of shear modulus, approximately less than 40 Gpa, and it flattens for large values. However, this trend is not the same for the long-term time period of 1000 days, as shown in Figs 14 and 15, (a) and (b). It is seen that circumferential stress component remains approximately the same by increasing shear modulus. On the other hand, Figs. 12, 13, 14, and 15 demonstrate that circumferential stress is about the same in the rock region by increasing shear modulus, for the both long and short periods of time. 4.3.2 The influence of viscosity coefficients m and k
The parameter m varies from 50 to 1000 Gpa.day in this section. As seen in Figs. 16 (a) and (b), circumferential stress reduces more sharply in lining region for lower values of the viscosity coefficient m , approximately less than 400 Gpa.day, and it flattens for large values, for the short-term time period of 10 days. However, with respect to the long-term time period of 1000 days circumferential stress in concrete region experiences an almost decrease based on the Figs. 17 (a) and (b), except for low values along internal lining periphery at the roof, and rockconcrete interface at the wall of the opening, where the curves exhibit a peak around the magnitude of 100 Gpa.day. On the other hand, in the rock region, Figs 16 (a) and (b) denote almost constant values for circumferential stress at the roof and wall of the tunnel for the short-term time period. However, with respect to long period of time, Figs. 17 (a) and (b) show a monotonous gradual increase in circumferential stress values. The parametric analysis of the coefficient k is also done over the range of 10 to 100 Gpa.day for circumferential stress in rock and concrete regions, along internal periphery and rock-concrete interface. It is observed according to Figs. 18 and 19, (a) and (b), that the circumferential stress quantities, in rock and concrete regions, are approximately constant over this domain for both short and long periods of time, at the wall and roof of the tunnel.
5. Conclusion In this study, an analytical solution was proposed for stress field around lined circular tunnels and in their lining support. The rock mass was supposed as a viscoelastic material obeying Burgers model, while lining was assumed as a linear elastic material. The solution was based on complex potential functions, combined with correspondence principle in a stepwise procedure. The proposed method was able to predict stress and deformation components in the long-term specifically when the viscosity element of Maxwell body dominated surrounding rock mass behavior inducing non-stop deformation. The solution was compared with finite element COMSOL software prediction which exhibited a close agreement. However, the solution benefits from some prominences as follows: 1. The solution determines stress field more accurately 2. The solving-time is immensely less than that for COMSOL numerical simulation software. This fact also makes the solution executable for parametric analysis of the ground and support structure properties in the long-term.
Finally, a parametric analysis was carried out with respect to Burgers model coefficients for short-term and long-term time periods of 10 and 1000 days, respectively. By using the preceding example as input data, the analysis demonstrated different behavior for each parameter, with respect to short and long periods of time. It was observed that circumferential stress reduces with increasing the value of G m and G k parameters in lining region, while remains approximately the same for rock region along rock-concrete interface at roof and wall of the tunnel, for short period
of time. With respect to long-term time period, circumferential stress is about the same by increasing shear moduli, G m and G k , in rock and lining regions. It was also observed that increasing the parameter m generally reduces circumferential stress in concrete region at wall and roof of the opening, while it doesn’t have significant influence on circumferential stress in rock region, and only makes a small enhance with respect to the long period of time. However, considering the variation of the parameter k , circumferential stress remains approximately the same in rock and lining regions, along internal periphery and rock-concrete interface, at wall and roof of the opening. To sum up, applying complex variable method in combined with correspondence principle in step-wise procedure coms up to be an effective approach in solving viscoelastic problems not only for Burgers model but also with any rheological constitutive behavior.
References [1] S. Sakurai, Approximate time‐dependent analysis of tunnel support structure considering progress of tunnel face, International Journal for Numerical and Analytical Methods in Geomechanics, 2 (1978) 159175. [2] M. Panet, Time-dependent deformations in underground works, 4th ISRM Congress, International Society for Rock Mechanics, 1979. [3] J. Sulem, M. Panet, A. Guenot, An analytical solution for time-dependent displacements in a circular tunnel, International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1987, pp. 155-164. [4] Y.-W. Pan, J.-J. Dong, Time-dependent tunnel convergence—II. Advance rate and tunnel-support interaction, International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1991, pp. 477-488. [5] Y.-W. Pan, J.-J. Dong, Time-dependent tunnel convergence—I. Formulation of the model, International journal of rock mechanics and mining sciences & geomechanics abstracts, Elsevier, 1991, pp. 469-475. [6] P.F. Gnirk, R.E. Johnson, The deformational behavior of a circular mine shaft situated in a viscoelastic medium under hydrostatic stress, The 6th US Symposium on Rock Mechanics (USRMS), American Rock Mechanics Association, 1964. [7] G. Gioda, A finite element solution of non-linear creep problems in rocks, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Elsevier, 1981, pp. 35-46. [8] M. Penet, A. Guenot, Analysis of Convergence Behind the Face of a Tunnel, Int, Symposium Tunneling, 1982, pp. 197-204. [9] B. Ladanyi, D. Gill, Design of tunnel linings in a creeping rock, International Journal of Mining and Geological Engineering, 6 (1988) 113-126. [10] A. Cividini, G. Gioda, A. Carini, A finite element analysis of the time dependent behaviour of underground openings, Beer, Booker, Carter (Eds.), Computer Methods and Advances in Geomechanics, (1991). [11] C. Paraskevopoulou, M. Diederichs, Analysis of time-dependent deformation in tunnels using the Convergence-Confinement Method, Tunnelling and Underground Space Technology, 71 (2018) 62-80. [12] J.C. Jaeger, N.G. Cook, R. Zimmerman, Fundamentals of rock mechanics, John Wiley & Sons2009.
[13] A. Fahimifar, F.M. Tehrani, A. Hedayat, A. Vakilzadeh, Analytical solution for the excavation of circular tunnels in a visco-elastic Burger’s material under hydrostatic stress field, Tunnelling and Underground Space Technology, 25 (2010) 297-304. [14] P. Nomikos, R. Rahmannejad, A. Sofianos, Supported axisymmetric tunnels within linear viscoelastic Burgers rocks, Rock Mechanics and Rock Engineering, 44 (2011) 553-564. [15] K. Lo, C.M. Yuen, Design of tunnel lining in rock for long term time effects, Canadian Geotechnical Journal, 18 (1981) 24-39. [16] F. Song, H. Wang, M. Jiang, Analytically-based simplified formulas for circular tunnels with two liners in viscoelastic rock under anisotropic initial stresses, Construction and Building Materials, 175 (2018) 746-767. [17] F. Song, H. Wang, M. Jiang, Analytical solutions for lined circular tunnels in viscoelastic rock considering various interface conditions, Applied Mathematical Modelling, 55 (2018) 109-130. [18] Z. Zhang, M. Huang, C. Zhang, K. Jiang, M. Lu, Time-domain analyses for pile deformation induced by adjacent excavation considering influences of viscoelastic mechanism, Tunnelling and Underground Space Technology, 85 (2019) 392-405. [19] G.T. Mase, G.E. Mase, Continuum mechanics for engineers, CRC press1999. [20] N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Springer Science & Business Media2013. [21] S.-c. Li, M.-b. Wang, Elastic analysis of stress–displacement field for a lined circular tunnel at great depth due to ground loads and internal pressure, Tunnelling and Underground Space Technology, 23 (2008) 609-617. [22] S.-c. Li, M.-b. Wang, An elastic stress–displacement solution for a lined tunnel at great depth, International Journal of Rock Mechanics and Mining Sciences, 45 (2008) 486-494. [23] M.B. Wang, S.C. Li, A complex variable solution for stress and displacement field around a lined circular tunnel at great depth, International journal for numerical and analytical methods in geomechanics, 33 (2009) 939-951. [24] H. Zhang, Z. Wang, Y. Zheng, P. Duan, S. Ding, Study on tri-axial creep experiment and constitutive relation of different rock salt, Safety science, 50 (2012) 801-805.
Tables Table 1. Input data including rock mass and concrete properties, as well as tunnel crosssection geometry
Input data for Burgers model [24]
G k (MPa)
k (MPa day)
G m (MPa)
6.73E 4
1.99E 4
1.99E 4
m (MPa*day) 1.39E 5
Properties of concrete lining and tunnel geometry
E c (MPa)
c
R (m)
R in (m)
1.65E 4
0.2
4.5
4.3
K (MPa) 4.3116e+04 Ratio of Horizontal to vertical in situ stress components
0.5
Table 2. Average of absolute relative approximation error of circumferential stress along internal lining periphery with respect to the number of time steps for different rheological times Number of time steps
2
10
20
50
100
500
1000
0.162
0.038
0.006
0.001 0.000 0.000
478.092 0.085
0.031
0.006
0.001 0.000 0.000
167.746 140.639 216.063 0.002
0.000 0.000 0.000
Time (days)
10.184
100 1000 10000
Table 3. Average of absolute relative approximation error of circumferential stress along rock-lining interface from lining side with respect to number of time steps for different rheological times Number of time steps Time (days)
2
10
20
50
100
500
1000
10.524
100 1000 10000
0.158
0.037
0.006
0.001 0.000 0.000
362.695 0.071
0.026
0.005
0.001 0.000 0.000
140.639 216.063 0.002
0.001 0.000 0.000
97.102
Table 4. Average of absolute relative approximation error of circumferential stress along rock-lining interface from rock side with respect to number of time steps for different rheological times Number of time steps
2
10
20
50
100
500
1000
0.079
0.018
0.003
0.001 0.000 0.000
181.843 0.072
0.026
0.005
0.001 0.000 0.000
106.436 140.641 216.063 0.000
0.000 0.000 0.000
Time (days)
100 1000 10000
Figures
7.727
(b)
(a)
(c) Fig. 1. Rheological models: a) Kelvin, b) Maxwell, c) Burgers
Fig. 2. Conformal mapping of tunnel cross-section in z-plane into two concentric circles in plane
Fig 3. COMSOL grid mesh for tunnel cross-section, and boundaries geometry
Fig. 4. Circumferential stress along internal lining periphery predicted by the proposed analytical solution and numerical COMSOL software
Fig. 5. Circumferential stress along rock-concrete interface from concrete side
Fig. 6. Circumferential stress along rock-concrete interface from rock side
Fig. 7. Radial stress along rock-concrete interface predicted by the proposed analytical solution and numerical COMSOL software
(a)
(b)
Fig. 8. Analytical solution and numerical COMSOL software prediction of a) normal stress and b) shear stress along inner lining periphery
Fig. 9. Circumferential stress along rock-concrete interface from rock side, at the wall and roof of the tunnel, with respect to rheological time
Fig. 10. Circumferential stress along rock-concrete interface from concrete side, at the wall and roof of the tunnel, with respect to rheological time
Fig. 11. Circumferential stress along internal lining periphery, at the wall and roof of the tunnel, with respect to rheological time
(a)
(b)
Fig. 12. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G m for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 13. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G k for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 14. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G m for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 15. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the shear modulus G k for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 16. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient m for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 17. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient m for the long-term time period of 1000 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 18. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient k for the short-term time period of 10 days at a) wall and b) roof of the tunnel
(a)
(b)
Fig. 19. Circumferential stress, along internal lining periphery and rock-concrete interface, vs. variation of the viscos coefficient k for the long-term time period of 1000 days at a) wall and b) roof of the tunnel