Accepted Manuscript
Analytical solutions for lined circular tunnels in viscoelastic rock considering various interface conditions Fei Song , Huaning Wang , Mingjing Jiang PII: DOI: Reference:
S0307904X(17)306480 10.1016/j.apm.2017.10.031 APM 12032
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
20 April 2017 29 August 2017 23 October 2017
Please cite this article as: Fei Song , Huaning Wang , Mingjing Jiang , Analytical solutions for lined circular tunnels in viscoelastic rock considering various interface conditions, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.10.031
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ACCEPTED MANUSCRIPT
Highlights An analytical solution is proposed for lined circular tunnels in viscoelastic rock.
All the possible interface conditions of rockliner and linerliner are considered.
Any viscoelastic models and installation times of two liners can be considered.
The influences of installation time of second liner are investigated.
An analyticallybased method for design of the second liner is provided.
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Analytical solutions for lined circular tunnels in viscoelastic rock considering various interface conditions
1
School of Aerospace Engineering and Applied Mechanics,
Tongji University, Shanghai, China, 200092;
State Key Laboratory of Disaster Reduction in Civil Engineering,
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2
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Fei Song1, Huaning Wang*1, 2, Mingjing Jiang2, 3
Tongji University, Shanghai, China, 200092; 3
Department of Geotechnical Engineering, College of Civil Engineering,
*
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Tongji University, Shanghai, China, 200092.
Corresponding author: Huaning Wang, Professor & Ph.D.
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Email address:
[email protected] (Wang HN).
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Abstract The time dependency of ground deformation and stresses in tunneling is mainly due to the rock rheology and tunnel construction process. This study provides an efficient analytical approach to predict the timedependent displacement and stress fields around deeply buried circular tunnels. The viscoelasticity of host rock, sequential installation of primary and secondary liners, various interface
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conditions and nonhydrostatic initial stress field are all taken into account in the analysis.
The complex variable method, Laplace transformation technique, and extension of correspondence principle are employed in the analytical derivation. Timedependent potentials
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expressed as Laurent series are addressed by satisfying all the governing equations in Cases (1)(4), where all possible conditions at rock–liner and liner–liner interfaces are considered. The timedependent analytical displacement and stress fields of rock and liners are proposed for the entire tunneling process. As a validation step, the analytical solutions in Case (1) agree well with the
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results of the FEM numerical simulations.
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Parametric analyses are then carried out to investigate the influences of installation time of the second liner in Case (1) and (2), and an analyticallybased method for design of the second liner is
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also presented. The proposed solutions in this paper provide a quick and simple alternative method for preliminary tunnel design, any viscoelastic models and any installation time of the first and the
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second liners can be considered.
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Keywords: analytical solution; circular tunnel; liner; viscoelastic rock
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1. Introduction The tunnels supported with a system comprising several liners are commonly encountered in hydraulic, traffic and mining engineering. The primary support (the first liner), typically in the form of shotcrete liner or steel set, is installed immediately after tunnel excavation to seal the rock and
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withstand the loads that may arise during the excavation. Later on, the secondary support (the second liner) is put in place to ensure the longterm stability of the tunnel [1]. Owing to the rock rheology, the tunnel convergence, as well as the support pressure due to the presence of liners, increases with time [2]. The final displacements and stresses of the rock and liners, which are crucial in tunnel
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design, are strongly affected by the rock viscosity, the installation time of liners and their stiffness [3, 4]. Therefore, a proper simulation of the entire process of excavation and support installation is quite necessary to obtain a reliable tool for the determination of the optimal values of the tunneling parameters, and to achieve optimal designs [35].
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This problem can be tackled either numerically or analytically. Numerical methods, such as finite
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element, finite difference, discrete element and boundary element methods, can provide some results useful to many complex underground projects [611]. However, long computational time is
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required, especially when complete parametric analyses need to be performed. As an alternative, analytical solutions provide an efficient and quick approach to gain insights into the nature of the
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problem [12]; in addition, they enable analyses of a wide range of parameter values so that the
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physics of the problem can be better understood. Regarding the rock–liner interactions, some analytical solutions have been developed for tunnels
constructed in elastic rock [1315]. By using complex variable method, and assuming that the rock is elastic, Wang and Li [13] presented the analytical solutions to the stresses and displacements around a lined circular tunnel, considering the rock–liner misfit. While By introducing a relaxation factor was introduced to account for the initial stress relief prior to installation of the liners [16, 17] or the effect of longitudinal tunnel advancement [17], an elastic solution was proposed for circular 4 / 56
ACCEPTED MANUSCRIPT singleliner tunnels. Naggar et al. [15] presented a closedform solution for tunnels with composite liners in an infinite elastic medium. The partial gap prior to the liner installation and two interface conditions between the first and second liners were considered. For a tunnel supported by two liners subjected to a uniform shear stress at infinity, its elastic solutions were provided by Mason and Abelman [18]. In these references, the timedependent behaviors of the rock and the sequential
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installation of liners were neglected ignored.
Unlike the case of linear elastic materials with constitutive equations in the form of algebraic equations, linear viscoelastic materials have their constitutive relations expressed by a set of operator equations. Since it is very difficult to address analytical solutions under complex conditions,
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most of the solutions considering the rock viscoelasticity/plasticity presented in the current references were derived for circular singleliner tunnels subjected to isotropic initial stresses (axisymmetric problems). Sulem et al. [] presented analytical solutions to address axisymmetric problems in order to
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determine the radial displacements and pressures applied to tunnel liners by characterizing the surrounding rock as a KelvinVoigt model. Fahimifar et al. [19] provided an analytical solution using
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a correspondence principle to predict the timedependent tunnel convergence, which considered the rock viscoelasticity and single liner, as well as the effects of the tunnel’s advancement, along with a
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Burger model of the rock. By assuming the Burger’s viscoelastic model of the rock around an
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axisymmetric tunnel with one elastic liner, Nomikos et al. [20] derived the solutions to deal with stresses and displacements, and performed a parametric study to investigate the effects of the liner
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parameters. However, these solutions only apply to singleliner tunnels. Recently, Wang et al. [3, 4] provided analytical solutions for circular tunnels with two/nliner system under isotropic initial stresses, considering rock rheology, excavation process, and installation time of liners. For lined tunnels, the interface conditions at rockliner and linerliner interfaces, have a significant effect on mechanical fields of the rock and liners [15, 21]. For a circular singleliner tunnel, The expressions for liner pressure, stresses, thrusts, and moments were given by Lo and 5 / 56
ACCEPTED MANUSCRIPT Yuen [21] for a circular singleliner tunnel under two interface conditions, taken into account the liner installation time as well as the viscoelastic properties of the rock and liner. In summary, the analytical solution of lined tunnels was mostly concerned with elastic problems, or viscoelastic problem with single liner under axisymmetric conditions. Instead in this study, we attempts to present a new analytical model for circular tunnels with two liners, with consideration of:
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(1) various interface conditions, i.e., the fullslip and noslip conditions at rockliner and linerliner interfaces; (2) the installation times of the first and second liners; (3) viscoelasticity of surrounding rock, and (4) nonhydrostatic initial stress state. The timedependent analytical solutions are proposed below for the most generalized model of deeply buried lined circular tunnels, which will
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be used to enable quick and simple preliminary design of tunnels.
2. Assumptions and Problem Definition
This paper considers the construction of a deeply buried circular tunnel in rheological rock,
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sequentially installed with two liners. The following assumptions apply throughout the analyses: (a) The tunnel is of circular section, excavated instantaneously at t=0. The initial stress field around
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the tunnel is idealized as p0 for the vertical pressure, and λp0 for the horizontal pressure, where
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λ is a horizontal to vertical stress ratio;
(b) The tunnel is deeply buried such that no linear variation of the stresses with depth needs to be
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considered;
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(c) The rock mass is homogeneous, isotropic, which has small deformation properties. Its rheology properties can be described as linear viscoelasticity, modelled by the Hookean elastic springs and Newtonian viscous dashpots in series or parallel connection. as shown in Figure 1 to model a variety of rheological characteristics of the rock mass； (d) The tunnel is supported by two elastic liners installed instantaneously at t=t1 and t2(t2≥t1), respectively. The outer and inner radii of the primary (first) liner are R1 and R2 , respectively,
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while those of the secondary (second) liner are R2 and R3 , respectively; (e) The uniform pressure qwat (t ) , acting on the boundary of the second liner at time t=t3 (t3≥t2), is assumed to represent the water pressure of hydraulic tunnels during operational period. In the analysis, the effect of the tunnel advancement along the longitudinal direction is not considered. This means that the crosssection considered in this analysis is at a sufficient distance
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from the tunnel face so that stresses and strains are unaffected by threedimensional effects. According to the aforementioned assumptions, the equivalent plane–strain problem in the tunnel tra crosssection plane can be illustrated in Figure 2, where σ tra ρ and σ ρθ , applied to the boundary of the
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tunnel after the excavation, are tractions induced by the initial stresses.
The tunneling construction process can be divided into three stages: (1) excavation stage, spanning from time t=0 to t=t1; in this stage, the rock is free to deformation since no liners are supported; (2) the first liner stage, spanning from time t=t1 to t=t2, when the tunnel is only
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supported by the first liner; and (3) the second liner stage, spanning from t=t2 onwards, when the
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tunnel is supported by both the first and second liners. In the following analysis, sign conventions are defined as positive for tension. Both polar and Cartesian coordinates are employed in the
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derivations.
Fullslip condition is usually assumed for the rock–liner/liner–liner interfaces involving
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waterproof layer, while the noslip condition is set for other interface situations. In this study,
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analytical solutions are derived for Cases (1)(4), where all the possible interface conditions at the rock–liner and liner–liner interfaces are considered: Case (1): Noslip conditions at both rock–liner and liner–liner interfaces; Case (2): Noslip condition at the rock–liner interface and fullslip condition at the liner–liner interface; Case (3): Fullslip condition at the rock–liner interface and noslip condition at the liner–liner interface; and 7 / 56
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3. Derivation of Analytical Solutions 3.1 Extension of the correspondence principle for general viscoelastic problems [22] Correspondence principle gives the relationship between elastic and viscoelastic solutions when the distribution of boundary stresses is independent of time [23]. However, the supporting forces are
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timedependent in this study, and boundary forces, e.g., tractions, supporting forces, and water pressure, are applied at different time spans.
By using the complex potential theory [24], as well as the Laplace transformation and the inversion
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with respect to time, Wang et al. [22] presented the extension of correspondence principle to solid media with timevarying boundaries subjected to loads applied at different time spans. If the kth load (k=1,2,…., l ) is applied to the structure at time tbk and removed at tmk , the total displacements induced in the ground at the generic time t can be expressed in terms of elastic potentials as the
υ( k ) ( z, τ ) 1 l Tk 1 l T (k ) ψ ( k ) ( z, τ ) dτ tbk I (t τ )υ ( z, τ )dτ tbkk H (t τ ) z 2 k 1 2 k 1 z
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uxv ( z, t ) iu vy ( z, t )
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following:
(1)
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where uxv ( z, t ) and u vy ( z, t ) are horizontal and vertical displacements at time t, respectively;
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υ( k ) ( z, t ) and ψ ( k ) ( z, t ) are the two complex potentials in the case only subjected to the kth load;
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e( z, t ) is the conjugate of the complex function e e( z, t ) ; Tk min{tmk , t} and H (t ) L
1
1 ˆ and I (t ) L sG ( s)
1
μ( s ) ˆ sG ( s)
(2)
In Eq. (2), fˆ ( s) (or L [ f (t )] ) is defined in the Laplace transformation of function f (t ) ; L
1
[ g (s)] indicates the inverse Laplace transformation of g(s); and μ( s)
3Kˆ ( s) 7Gˆ ( s) is for plane 3Kˆ ( s) Gˆ ( s)
strain problems, where G(t ) and K (t ) represent the tangential and bulk relaxation moduli of a
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ACCEPTED MANUSCRIPT viscoelastic model, respectively. The expressions of G(t ) , H(t) and I(t) for the four viscoelastic models are provided in Tables 1 and 2, respectively. The stresses can be obtained by utilizing the principle of superposition [22]: υ k ( z , t ) 2 υ k ( z, t ) ψ k ( z, t ) l Re z , 2 v 2 k 1 σ y ( z, t ) z z z σ xv ( z, t )
(3)
2 υ k ( z, t ) ψ k ( z, t ) σ xyv ( z , t ) Im z . k 1 z 2 z
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l
where Re[] and Im[] denote the real and imaginary components of a generic complex variable [] , respectively.
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3.2 Displacements and stresses of the rock and liners expressed by potentials
The potentials of the rock and liners can be expanded as Laurent series as follows [24]:
For the rock: υijR ( z, t ) akij (t ) z k , ψijR ( z, t ) bkij (t ) z k with i 1, 2; j 1, 2,3, 4 k 1
k 1
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S υmij ( z, t ) ckmij (t ) z k d kmij (t ) z k k 1
For the liners:
ψ ( z, t ) e (t ) z k 1
mij k
k
f
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S mij
k 1
k 1
mij k
(t ) z
k
m 1, 2; with i 1, 2; j 1, 2, 3, 4
(4)
(5)
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where υijR ( z, t ) and ψijR ( z, t ) denote the potentials of the rock subjected to the supporting forces induced by the rock–liner interaction in the first liner stage (i=1) or the second liner stage (i=2) in
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Case (j); υ1Sij and ψ1Sij denote the potentials of the first liner in Case (j) during the first (i=1) or the
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second (i=2) liner stage; υ2Sij and ψ 2Sij denote the potentials of the second liner. akij (t ) , bkij (t ) , ckmij (t ) , d kmij (t ) , ekmij (t ) , and
f kmij (t ) are the timedependent coefficients which will be determined by
boundary and interface compatibility conditions. By using the complex variable method introduced by Muskhelishvili [24], displacements and stresses of the elastic liners can be expressed as follows (each quantity with subscript mij indicates that it is for the mth liner in the ith liner stage in Case (j)):
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Displacements: u ρS ( mij ) ( z, t ) iuθS ( mij ) ( z, t )
S υmij ( z, t ) Re 2 z σ θS ( mij ) ( z , t )
σ ρS ( mij ) ( z , t )
Stresses:
S υmij ( z, t ) eiθ S S κ Sm υmij ( z, t ) z ψmij ( z, t ) (6) 2GSm z
z
S 2 υmij ( z, t )
z
2
S ψmij ( z, t )
z
e2iθ (7)
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S S 2 υmij ( z , t ) ψmij ( z, t ) 2iθ S ( mij ) σ ρθ ( z , t ) Im z e 2 z z
where κSm 3 4νSm ; νSm and GSm are the Poisson’s ratio and shear modulus, respectively (first liner: m=1 and the second one: m=2).
The tractions, whose values are the same as the initial stresses but the directions are opposite, are
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exerted on tunnel boundary to address the displacements and stresses occurring after tunnel excavation induced by farfield initial stresses. According to the initial stress state, the expressions of tra tractions, σ tra respectively along radial and tangential directions, can be determined as ρ and σ ρθ
( λ 1) p0 ( λ 1) p0 2iθ (1 λ) p0 2iθ Re[ e ], σ tra e ] ρθ Im[ 2 2 2
(8)
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σ tra ρ
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follows:
For the rock (see Figure 2b), only tractions are applied to the tunnel boundary during excavation
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stage, however, both tractions and supporting forces between the rock and the first liner are exerted on applied to the tunnel boundary during the first and second liner stages. The radial and
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circumferential displacements ( u ρR and uθR ), as well as normal ( σ ρR and σ θR ) and shear ( σ ρθR ) stresses
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of the rock occurring from t=0 during the first and second liner stages, can be expressed as follows: u ( z, t ) iu ( z, t ) R ρ
R θ
u ρR ( ex ) ( z, t ) u ρR (ij ) ( z, t ) i uθR ( ex ) ( z, t ) uθR (ij ) ( z, t )
;
First liner stage : i 1 Second liner stage : i 2
σ ρR ( z, t ) σ ρR ( ex ) ( z, t ) σ ρR (ij ) ( z, t ) First liner stage : σ θR ( z, t ) σ θR ( ex ) ( z, t ) σ θR (ij ) ( z, t ) ; σ ( z, t ) σ R ρθ
R ( ex ) ρθ
( z, t ) σ
R ( ij ) ρθ
( z, t )
(9)
i 1
Second liner stage : i 2
(10)
where the superscript ―R(ex)‖ represents that the quantity is induced by the tractions (its exact 10 / 56
ACCEPTED MANUSCRIPT expressions will be provided in the next subsection); and the quantities with superscript ―R(ij)‖ are induced by supporting forces in the first (i=1) or second (i=2) liner stages in Case (j), which can then be obtained by substituting Eq.(4) into Eqs. (1) and (3). Note that the displacements and stresses fields of the rock and liners are addressed so long as the coefficients in potentials (Eqs. (4) and (5)) have been determined. The boundary and
determination of these coefficients. 3.3 Boundary and compatibility conditions in the four cases
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compatibility conditions are satisfied in the next subsection to build a set of equations for
As described in Section 2, Cases (1)(4) with various interface conditions are considered in this
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paper. The various interface conditions considered in this study include (1) noslip condition, which indicates full continuity of displacements and stresses along the interface; (2) fullslip condition, which implies that the displacements and stresses are only continuous in radial direction, and the
stresses are presented as follows:
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shear stress at the interface should be zero. The compatibility conditions for displacements and
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For noslip conditions at the rock–liner interface ( ρ R1 , t t1 ):
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u ρR ( z, t ) iuθR ( z, t ) u ρR (ex) ( z, t1 ) iuθR (ex) ( z, t1 )
u ρS (1ij ) ( z, t ) iuθS (1ij ) ( z, t )
z R1eiθ
σ ρR (ij ) ( z, t ) σ ρS (1ij ) ( z, t ) R ( ij ) S (1ij ) σ ρθ ( z, t ) σ ρθ ( z, t )
z R1eiθ z R1eiθ
z R1eiθ
(11)
0
(12)
0
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For fullslip conditions at the rock–liner interface ( ρ R1 , t t1 ): Re u ρR ( z, t ) iuθR ( z, t ) u ρR ( ex ) ( z, t1 ) iuθR ( ex ) ( z, t1 ) R ( ij ) σ ρθ ( z, t ) S (1ij ) σ ρθ ( z, t )
z R1eiθ z R1eiθ
z R1eiθ
Re u ρS (1ij ) ( z, t ) iuθS (1ij ) ( z, t )
(13)
0 0
σ ρR (ij ) ( z, t ) σ ρS (1ij ) ( z, t )
(14) z R1eiθ
For noslip conditions at the liner–liner interface ( ρ R2 , t t2 ): 11 / 56
z R1eiθ
0
ACCEPTED MANUSCRIPT u ρS (12 j ) ( z, t ) iuθS (12 j ) ( z, t ) u ρS (12 j ) ( z, t2 ) iuθS (12 j ) ( z, t2 )
u ρS (22 j ) ( z, t ) iuθS (22 j ) ( z, t )
z R2 eiθ
σ ρS (12 j ) ( z, t ) σ ρS (22 j ) ( z, t )
z R2 eiθ
S (12 j ) S (22 j ) σ ρθ ( z, t ) σ ρθ ( z, t )
z R2 eiθ
z R2 eiθ
(15)
0
(16)
0
For fullslip conditions at the liner–liner interface ( ρ R2 , t t2 ):
S (12 j ) σ ρθ ( z, t ) S (22 j ) σ ρθ ( z, t )
z R2 eiθ z R2 eiθ
z R2 eiθ
u ρS (22 j ) ( z, t )
0 0
z R2 eiθ
(17)
(18)
0
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σ ρS (12 j ) ( z, t ) σ ρS ( 22 j ) ( z, t )
z R2eiθ
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u ρS (12 j ) ( z, t ) u ρS (11 j ) ( z, t2 )
During the first liner stage (t1≤t＜t2), the stress boundary condition along inner boundary of the first liner are σ ρS (11 j ) ( z, t )
z R2 eiθ
0;
S (11 j ) σ ρθ ( z, t )
z R2 eiθ
0
(19)
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During the second liner stage (t≥t2), the stress boundary condition along inner boundary of the first
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liner are
z R3eiθ
qwat (t );
S (22 j ) σ ρθ ( z, t )
z R3eiθ
0
(20)
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σ ρS (22 j ) ( z, t )
Table 3 provides the boundary and compatibility conditions that should be satisfied in Cases
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(1)(4) during the first and second liner stages. According to these conditions, the equations with respect to the undetermined coefficients in potentials (see Eqs. (4) and (5)) can be obtained
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accordingly.
3.4 Solutions at excavation and the first liner stages tra During the excavation stage (0≤t＜t1), only tractions ( σ tra ρ and σ ρθ in Eq.(8)) are applied to the
internal boundary of the tunnel, and the incremental displacements occurring from t=0 in the polar coordinate reference system can be obtained [22]:
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u ρR ( ex ) ( z, t ) iuθR ( ex ) ( z, t )
eiθ p0 z 1 1 1 [ A1 (t )(1 λ) 2 A1 (t )(1 λ) A2 (t )(1 λ) A3 (t )(1 λ) 3 ] (21) 4 z z z z
where z ρeiθ ; A1 (t ) R12 0t H (t τ )dτ ; A2 (t ) R12 0t I (t τ )dτ ; and A3 (t ) R14 0t H (t τ )dτ . The total stresses of the rock in the excavation stage are as follows:
σ
R ( ex ) ρθ
( z, t ) ( λ 1) p0 2 ( z, t )
(1 λ) R12 ( λ 1) p0 2iθ z2 Re[ e ] p0 Re (1 λ) R12 1 λ R12 3(1 λ) R14 2iθ 2 [ z ]e z3 2 z2 2 z4
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σ σ
R ( ex ) ρ R ( ex ) θ
(22)
( λ 1) p0 2iθ (1 λ) R12 3(1 λ) R14 2iθ Im[ e ] p0 Im [ z ]e 2 z3 2 z4
During the first liner stage (t1≤t＜t2), tractions and supporting forces induced by rock–liner
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interaction are both applied to the tunnel boundary, whereas only supporting forces are applied to the first liner. With regard to a circular tunnel supported by singleliner, Wang et al. [25] presented detailed derivation of analytical solutions for the first liner stage.
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3.5 Solutions at the second liner stage
During the second liner stage (t≥t2), the compatibility conditions at the rock–liner and liner–liner
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interfaces, as well as the boundary condition at the inner boundary of the second liner, should be satisfied, as shown in Table 3. In the following paragraph, the boundary and compatibility
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conditions are first expressed by potentials, and then the equations with respect to the undetermined
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coefficients in potentials can be obtained. Substituting Eqs.(6), (7), (9) and (10) into Eqs. (11) and (12), yields the specific expression of
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noslip conditions at the rockliner interface by potentials: Re D( z , t ) D( z , t1 ) e iθ 2
e iθ 2
υ1Rj ( z , τ ) t2 t2 R R dτ I ( t τ ) υ ( z , τ ) dτ H ( t τ ) z ψ ( z , τ ) t1 t1 1j 1j z
υ2Rj ( z , τ ) t t t2 I (t τ )υ2Rj ( z, τ )dτ t2 H (t τ ) z ψ2Rj ( z, τ ) dτ z
Re
υ12S j ( z , t ) e iθ S κ S 1υ12 j ( z , t ) z ψ12S j ( z, t ) 2GS 1 z 13 / 56
z R1eiθ
(23) z R1eiθ
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υ2Rj ( z, t ) z
υ12S j ( z, t ) ψ2Rj ( z, t ) υ12S j ( z, t ) z ψ12S j ( z, t ) z
υ2Rj ( z, t ) z
0
(24)
z R1eiθ
z 1 1 1 1 where D( z, t ) eiθ A1 (t )(1 λ) 2 A1 (t )(1 λ) A2 (t )(1 λ) A3 (t )(1 λ) 3 . 4 z z z z
Similarly, other conditions are expressed by potentials as follows:
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The fullslip compatibility conditions at the rockliner interface are
υ1Rj ( z, τ ) t2 t2 R R dτ I ( t τ ) υ ( z , τ ) dτ H ( t τ ) z ψ ( z , τ ) t1 t1 1j 1j z R iθ υ2 j ( z, τ ) e t t t2 I (t τ )υ2Rj ( z, τ )dτ t2 H (t τ ) z ψ2Rj ( z, τ ) dτ (25) z R1eiθ 2 z S υ ( z , t ) eiθ κ S 1υ12S j ( z, t ) z 12 j Re ψ12S j ( z, t ) z R1eiθ 2GS 1 z
Im e 2iθ [ z
eiθ 2
2 υ2Rj ( z, t ) z 2
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Re D( z , t ) D( z, t1 )
ψ2Rj ( z, t ) z
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2 υ12S j ( z , t ) ψ12S j ( z, t ) Im e2iθ [ z ] z 2 z υ2Rj ( z , t )
0,
]
2 υ2Rj ( z, t )
z R1e
iθ
0,
(26)
z R1e R 2j iθ
ψ ( z, t )
e2iθ [ z ] z R eiθ 1 z z 2 z S 2 S S υ12 j ( z , t ) 2iθ υ12 j ( z, t ) ψ12 j ( z, t ) Re 2 e [z ] z z 2 z
z R1e
PT
ED
Re 2
0 iθ
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The compatibility conditions at the linerliner interface are
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υS ( z , t ) υS ( z, t2 ) e iθ e iθ κ S1υ12S j ( z, t ) z 12 j κ S1υ11S j ( z, t2 ) z 11 j ψ12S j ( z, t ) ψ11S j ( z, t2 ) 2GS 1 z 2GS1 z z R2 eiθ z R eiθ 2
υ2S2 j ( z , t ) e S S κ S 2 υ22 j ( z, t ) z ψ22 j ( z, t ) 2GS 2 z z R2 eiθ
(27)
iθ
υ ( z, t ) z S 12 j
υ12S j ( z, t ) z
ψ
S 12 j
S υ22 j ( z, t ) S S ( z, t ) υ22 j ( z, t ) z ψ22 j ( z, t ) z
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0 z R2 e
iθ
(28)
ACCEPTED MANUSCRIPT υS ( z , t ) e iθ κ S 1υ12S j ( z , t ) z 12 j ψ12S j ( z, t ) 2GS 1 z Re
υS ( z, t2 ) e iθ κ S 1υ11S j ( z , t2 ) z 11 j ψ11S j ( z, t2 ) 2GS 1 z S υ22 e iθ j ( z, t ) S S κ S 2 υ22 ψ22 j ( z, t ) z j ( z, t ) 2GS 2 z
Im e 2iθ [ z
2 υ12S j ( z, t ) z 2
ψ12S j ( z, t ) z
]
S S 2 υ22 ψ22 j ( z, t ) j ( z, t ) Im e2iθ [ z ] 2 z z
υ12S j ( z , t )
2 υ12S j ( z, t )
(29) z R2 eiθ
, z R2 eiθ
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Re
0,
z R2 eiθ
0,
z R2 eiθ S 12 j
ψ
(30)
( z, t )
e [z ] z R eiθ 2 z z 2 z S 2 S S υ22 j ( z, t ) 2iθ υ22 j ( z, t ) ψ22 j ( z , t ) Re 2 e [z ] z z 2 z
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Re 2
2 iθ
z R2 e
0 iθ
The boundary condition of the inner boundary of the first liner (t1≤t＜t2) is 0
2 iθ
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z
2 υ11S j ( z, t ) ψ11S j ( z, t ) ] e [z ] z 2 z
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2Re[
υ11S j ( z, t )
(31)
z R2 eiθ
The boundary condition of the inner boundary of the second liner (t≥t2) is S υ22 j ( z, t )
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2Re[
z
] e2iθ [ z
S S 2 υ22 ψ22 j ( z, t ) j ( z, t ) ] 2 z z
qwat (t ) .
(32)
z R3e
iθ
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Substituting the Eqs. (4) and (5) into Eqs. (23)–(32), and by setting the coefficients in the term
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eikθ (k 0， 1， 2， ) to zero, we can build a set of linear algebraic equations with regard to the undetermined coefficients in the potentials. These linear algebraic equations can be easily solved, and the displacements and stresses are then obtained for the rock and liners accordingly. Owing to the length limit of the paper, the detailed derivations are provided only for Case (1) and (2) in Appendix A. Both cases are more commonly encountered than the other two in the geoengineering. Solutions for Case (3) and (4) can be obtained in a similar solving process.
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4. Verification of the Analytical Solutions and comparison with other solutions 4.1 Verification of the Analytical Solutions In this section, an example under the boundary conditions of Case (1), i.e. noslip at both rock–liner and liner–liner interfaces, is carried out, and the analytical and numerical results are compared to validate the proposed method and solutions in previous sections. Then both the
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boundary and compatibility conditions are checked for other cases to validate the correctness of derivation. The Finite Element Method (FEM) code ANSYS is employed (version 13.0).
The numerical model is consistent with the analytical one, and both of them are calculated under
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planestrain conditions with small deformation. Only a quarter of the tunnel structure is analyzed in FEM (see Figure 3) because of the double symmetry of x and y axis of geometry and boundary conditions. Moreover, the vertical displacements along the bottom boundary and the horizontal displacements along left boundary are restrained in the numerical model. Figure 4 shows the mesh of
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vicinity of the tunnel in FEM simulations. The parameters employed in the numerical simulations are listed in Table 4.
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In numerical simulation, the initial pressures were first applied to the boundary of ground without
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holes for a sufficiently long time (100 days in this study), and then the tunnel (part I in Figure 4) was instantaneously excavated (t=0) by removing the corresponding elements in this area. After that, the
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first and second liners were installed at t=2nd day and 5th day respectively. Five days later, the uniform internal pressure (water pressure) was applied along the inner boundary of the second liner. The
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comparison between analytical and numerical results in terms of incremental displacements and total stresses of the rock at the rock–liner interface is illustrated in Figure 5, where the incremental displacements were obtained by subtracting the displacements occurring before excavation from the total ones. A good agreement between both results is evident, which validates the proposed analytical method in derivation. Since the derivation procedure in Case (1), which has been validated by comparison between 16 / 56
ACCEPTED MANUSCRIPT analytical and numerical results, is the same as that in Cases (2)(4), only the boundary and compatibility conditions are checked for other cases in the following to verify the correctness of the derivation. The displacements and stresses of the rock and the first liner at the rock–liner interface versus the angle θ ( θ is polar angle of the coordinate) in Case (2) are plotted in Figure 6. Three specific time points are set: (1) t 9th day (before the internal pressure was applied); (2) t 11st day
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(shortly after the internal pressure was applied); and (3) t 50th day (when the final state was achieved). Noted that the displacements/stresses of the rock are all consistent with those of the first liner, indicating that the compatibility conditions of both displacement and stress at the rockliner
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interface are satisfied. Figures 7(a) and 7 (b) show the distribution of incremental displacements of the first liner occurring after installation of the second liner, as well as the stresses along the liner–liner interface at three specific time points. The quantities for the second liner along the liner–liner interface are also plotted in these figures. It is demonstrated that the displacements/stresses
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of the first and second liners agree well along the linerliner interface. Figure 7(c) shows stresses
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along the inner boundary of the second liner at three specific time points. Noted that the radial normal stresses are equal to uniform water pressure at the t=11st and 50th day, but are zero at the t=9th day.
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The shear stresses are zero all the time. It can be noted that the solutions perfectly satisfied the compatibility conditions of the rock–liner and liner–liner interfaces, as well as the stress boundary
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condition of the second liner. Therefore, we can confirm that the solutions for the case with fullslip
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condition (Case 2) are correct. 4.2 Comparison between the new solution and those in the references In this Subsection, the Case (1) solution in this study (hereafter referred to as the ―new solution‖)
is compared with the elastic and viscoelastic solutions for lined tunnels, respectively described by Naggar et al. [15] and Wang et al. [3]. The elastic solution [15] was proposed for deep buried circular tunnel supported by two elastic liners, and considered the delayed installation of liners, as well as the full/noslip conditions at rockliner and linerliner interfaces. However, the time interval 17 / 56
ACCEPTED MANUSCRIPT between the installation of the first and second liners was not considered, i.e. the two liners were installed at the same time. Figure 8(a) plots the comparison of the radial displacements between the new solution and the results in reference [15], under noslip conditions at the rockliner and linerliner interfaces. The parameters employed in the two solutions are provided in Table 5, in which the initial Young’s
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modulus of the generalized Kelvin viscoelastic models (EM in Figure 1c) in the new solution is the same as that in the elastic solution. Owing to the assumption of elasticity, the elastic displacements are the constant values against time. It is also shown in Figure 8(a) that the final displacement obtained by the new solution is larger than that obtained by the elastic solution, when the first and
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second liners are installed immediately after the tunnel excavation, since the additional supporting pressure due to the rock rheology is generated in the viscoelastic case. However, in the case where the two liners were both installed with a delay at the third day, the final displacement obtained by
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the new solution is smaller than the elastic one, because the displacement has partly occurred prior to the liner installation in the viscoelastic case (new solution). It is shown that, by employing the
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new solution, the timedependent displacement and stress can be predicted, thus accounting for the
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realistic timedependency of the rock mass and rockliner interaction. The viscoelastic solution in reference [3] was presented for a circular tunnel supported by two
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elastic liners, which were sequentially excavated in a viscoelastic rock mass. The timedependent tunnel excavation and liner installation times were both considered in this solution, but the initial
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stresses were assumed to be hydrostatic ones (a particular case in this study, =1). By employing the parameters shown in Table 6, the radial displacements versus time, which was provided in reference [3] and in this study, are plotted in Figure 8(b). It may be noted that the new solution for cases under =1 is consistent with the previous solution [3]. In addition, the timedependent displacements for cases in which nonhydrostatic initial stresses are considered ( 1 ), can be predicted by the new solution, the results of which show quite different values and variation forms 18 / 56
ACCEPTED MANUSCRIPT from the results of the previous solution.
5. Analysis and Discussion For the rock with good mechanical properties or subject to low stresses, the exhibited mechanical behavior shows limited viscosity, which can be simulated by generalized Kelvin viscoelastic models (see Figure 1c). Regarding this type of rheological behavior commonly encountered in rock
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engineering, a parametric analysis for effects of the installation time of the second liner on the displacements and stresses of the rock is carried out for Cases (1) and (2), and a method for design of the second liner is provided by means of the analytical model.
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In order to simplify the analysis, all the variables are expressed in dimensionless forms. The shear moduli, geometry parameters and stresses are normalized by GK , R1 , p0 , respectively. Time spans are normalized by TK which is the retardation time of viscoelastic model ( TK ηK / GK for generalized Kelvin model). Tunnel convergence is normalized by u ρe , which is the
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excavationinduced elastic convergence of the unlined circular tunnel subjected to hydrostatic initial
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stress p0 :
p0 R1 (GM GK ) 2GM GK
(33)
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u ρe
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According to relevant literatures, the displacements versus time for different types of viscoelastic models have been presented to address axisymmetric problems [4]. Therefore, only a commonly
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used viscoelastic model, i.e. The constitutive parameters of generalized Kelvin model adopted in the analysis are the following: GM / GK 2.0 , ηK / GK 10 , and the horizontal to vertical stress ratio of initial stresses is taken as λ 0.65 . As a preliminary support, the first liner should be installed as soon as the tunnel excavation is completed, hence t1 TK =0 in the following analysis. The other parameters of the first liner are assumed within their commonly used values in this section: thickness d1 R1 0.02 , shear modulus GS1 GK 10.0 and Poisson’s ratio νS1 0.2 . Internal
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pressure (water pressure) is 0.01p0 , which is applied at the time t3 , and
t3 t2 0.5 . TK
5.1 Effect of installation time of the second liner Installation time of the second liner is a crucial parameter in tunnel design. Early installation of the second liner will significantly benefit the rock stability since the large pressure to the rock are
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induced, however, the second liner is probably yielded if it is installed too early, for it bears excessive stresses due to the rock rheology. On the other hand, later installation of the second liner will induce larger pressure on the first liner and result in a more unstable tunnel, though it is economical to design the second liner. In order to investigate the influence of the installation time of
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the second liner on the resulting displacements and stresses of the surrounding rock, examples of Case (1) and (2) are presented herein. Three options are considered for installation time of the second liner:
(a) t2 / TK 0 , i.e. both liners are immediately installed after completion of the excavation;
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(b) t2 / TK 1 ; and
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(c) t2 / TK 2 .
In all the cases, the shear modulus and thickness of the second liner are set to commonly used
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values, GS 2 / GK 13 and d2 R1 0.0383 , respectively, and Poisson’s ratio of the second liner is
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νS 2 0.2 .
The radial and circumferential final displacements of the rock at the rock–liner interface versus
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the angle θ (the polar angle in coordinate) are plotted in Figure 9 for different installation time of the second liner. The results show that radial displacements increase gradually with the increase of angle θ , while circumferential displacements first increase to the peak value at θ 45 and then decrease. It can be seen that the shape of displacements versus the angle is very similar at different installation time of the second liner, however, earlier installation of the second liner leads to smaller radial and circumferential displacements in Case (1). As a result of comparison between Case (1) 20 / 56
ACCEPTED MANUSCRIPT and (2), no significant differences of radial displacement are found, while the circumferential displacements are almost the same at different installation time of the second liner in Case (2), which is quite different from the change of circumferential displacement in Case (1). The maximum difference between the maximum and minimum values of final radial displacements at different installation time of the second liner in Case (1), accounts for 23.2% of the maximum displacement,
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while that between the final circumferential displacements is only 13.5% of the maximum displacement.
Figure 10 shows the normal and shear stresses of the rock in Cases (1) and (2) along the rock–liner interface versus at various installation time of the second liner. Note that the variation
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patterns of stress against the angle are very similar at different installation time of the second liner in both Case (1) and (2): the radial and hoop stresses decrease gradually with the increase of angle θ . However, the shear stresses first increase to a peak value at θ 45 and then decrease. Earlier
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installation leads to larger radial stresses and smaller hoop stresses in both Case (1) and (2), which is beneficial to the tunnel stability. Comparison between the quantities in Case (1) and (2) shows
ED
that, the values of radial and hoop stress have small difference, but shear stresses are larger when the
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second liner is installed earlier in Case (1), however, in Case (2), shear stress is independent of installation time of the second liner and becomes the minimum one among all the cases. The
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maximum differences of radial, hoop and shear stresses at the three specific installation time of the second liner, account for 47.2%, 10.3% and 40.76% of the corresponding largest values in Case (1),
AC
respectively.
Displacements and stresses of the rock and liners are timedependent, mainly because of
rheological properties of the rock. The displacements and stresses in Case (1) versus time are plotted in Figures 11 and 12 respectively, where the quantities are at the tunnel boundary with θ 45 . The displacements, radial and shear stresses increase, while the hoop stress decreases with time, and finally reach the stable values. Earlier installation of the second liner leads to slower increase of 21 / 56
ACCEPTED MANUSCRIPT displacements after installation of the second liner, resulting in smaller final displacement values, as shown in Figure 11. However, it can be seen from Figure 12 that, the earlier the second liner is installed, the more quickly the radial and shear stresses increase with time, the larger the final stress values are reached, and the more quickly the hoop stress decreases with time. 5.2 Analyticsbased method for design of the second liner
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Owing to the rock rheology, the final pressure applied to the outer boundary of the second liner is significantly affected by installation time of the second liner. Early installation of the second liner leads to larger final pressure on a support system, probably resulting in failure of the second liner. In this subsection, the final stress fields of the second liner determined by the proposed analytical
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model, as well as the MohrCoulomb failure criterion, are employed to predict the initial failure of the second liner for different liner installation time.
According to the MohrCoulomb failure criterion, an equivalent stress of the second liner, σ ES 2 , is
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defined as follows:
(34)
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σ ES 2 σ1S 2 ασ3S 2 σt
where α σtS 2 / σcS 2 , with σ tS 2 and σ cS 2 being tensile and compressive strengths of the liner material,
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respectively; σ1S 2 and σ 3S 2 are the major and minor principal stresses, respectively. We assume that a
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point will fail in case σ ES 2 ≥0, so as to approximately predict the initial failure in the second liner. It is also noted that the larger the positive equivalent stress of a point is, the earlier this point will fail.
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Through a number of calculations, the point (Point A in Figure 4) at inner boundary of the second liner with θ 0 is the first one to fail if λ＜1 , while the first failure point (Point B in Figure 4) is at the inner boundary with θ 90 , if λ＞1 . In the tunnel engineering, in order to ensure the tunnel stability, the second liner is not expected to fail, i.e., σ ES 2 of the first failure point should be negative. Figure 13 shows the flow chart for design of the second liner, following the principle that no failure points appear in the second liner. 22 / 56
ACCEPTED MANUSCRIPT Based on the flow chart, an example below shows how the optimal liner installation time or thickness was determined. In the application example, the second liner is made of concrete reinforced by steel. The parameters employed in the example are presented in Table 7. By using these parameters in the analytical models proposed for Cases (1) and (2), we could calculate equivalent stresses. Figures
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14(a) and 14(b) show the equivalent stresses at Point A (i.e., the first failure point in the second liner) as a function of installation time and the thickness of the second liner respectively, where the dash line without bars represents the zero equivalent stress. For the cases with different liner thicknesses (curves with bars) in Figure 14(a), the earliest installation time of the second liner (the economic
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installation time) is determined by the intersection of the zero equivalent stress line and the equivalent stress curves. It is also noted that, the bigger the liner thickness is, the earlier the economic installation time is. Given the same liner thicknesses in Case (1) and (2), the economic
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installation time in Case (1) is later than that in Case (2). In Figure 14(b), the intersections of the zero equivalent stress line and the equivalent stress curves for variable liner installation times relate
ED
to the smallest liner thickness of the second liner (the economic liner thickness). The economic liner
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thickness in Case (1) would be greater than that in Case (2), in case the liner installation time in both cases are the same. With the proposed analytical models in this paper, the optimization design
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of the second liner can be conducted conveniently, and the recommended parameters can be used in preliminary design of tunnel construction.
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6. Conclusions
Timedependent analytical stress and displacement fields around deeply buried lined circular
tunnels were proposed, where the sequential installation of two liners, viscoelastic properties of the host rock and nonhydrostatic initial stress field were all taken into account. Various interface conditions, such as noslip and fullslip conditions of rock–liner and liner–liner interfaces, represented by Cases (1)(4) in the derivation, were considered taken into account accordingly. The 23 / 56
ACCEPTED MANUSCRIPT complex variable method, Laplace transformation technique, and extension of correspondence principle were employed in the derivation of the analytical solutions. As a verification step, a good agreement between analytical solutions and results from finite element analyses was obtained eventually. Parametric investigation was then carried out, while the generalized Kelvin viscoelastic model
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for the rock mass was assumed under interface conditions represented by Cases (1) and (2). It was demonstrated that there was no significant difference between Case (1) and (2) in terms of the radial displacements and normal stresses along the rock–liner interface; the circumferential displacements and shear stresses were independent of installation time of the second liner in Case (2).
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The analyticsbased method was used in design of the second liner, and the economic installation time or thicknesses of the second liner was recommended in various liner situations. The proposed analytical solutions provide a quick and contingent alternative method for
M
preliminary tunnel design, with consideration of various viscoelastic rock mass and installation times of the first and second liners. Analytical solutions combined with failure criterion can be
ED
employed to predict the initial failure zone of the tunnel. However, it should be point out that the
PT
solution is under the assumption of small deformation and without the consideration of plasticity, due to the limitation of current theory of mathematics and mechanics if complex conditions are
CE
considered. In the future, the formulas will be applied in practical projects, and the research in
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consideration of rock failure will be further performed to explore other possibilities.
Acknowledgements This study was financially supported by the National Natural Science Foundation of China (Grant Nos. 11572228, 51639008); the National Basic Research Program of China (Grant No. 2014CB046901); the State Key Laboratory of Disaster Reduction in Civil Engineering (Grant No. 24 / 56
ACCEPTED MANUSCRIPT SLDRCE14B11); and the Fundamental Research Funds for the Central Universities. These supports
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were greatly appreciated.
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Appendix A: Derivation of the potentials in Cases (1) and (2) A.1 Potentials in Case (1)Equation Chapter (Next) Section 1 In Subsection 4.3, the equations expressed by potentials (see Eqs. (23), (24), (27), (28), (32)) have been provided by satisfying the boundary and compatibility conditions in Case (1) during the second liner stage. Substituting Eqs. (4) and (5) into the above equations, and replacing the variable
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t by t '' t2 , the Laplace transformation with respect to t '' can be introduced to get the equations about the coefficients in the potentials as follows: 1 L 4
A1 (t ) A1 (t1 ) (1 λ)
1 1 R1 4
1 2iθ 1 e L R1 4
A3 (t ) A3 (t1 ) (1 λ)
1 2iθ e R13
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A2 (t ) A2 (t1 ) (1 λ)
L
1 2iθ 1 e L R1 4
A1 (t ) A1 (t1 ) (1 λ)
t2 t2 1 1 1 L I (t τ )a111 ( τ ) R11 dτ e 2iθ L H (t τ ) a111 ( τ ) R11 dτ e 2iθ 2 2 2 t1 t1
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t2 1 t2 1 L H (t τ )b111 ( τ ) R11 dτ L H (t τ )b311 ( τ ) R13 dτ e2iθ Iˆ( s) aˆk21 ( s) 2 k 1 t1 2 t1 1 1 1 R1 k e( k 1)iθ kHˆ ( s )aˆk21 ( s )R1 k e ( k 1)iθ Hˆ ( s )bˆk21 ( s )R1 k e ( k 1)iθ [κS1 2 k 1 2 k 1 2GS 1
k 1
k 1
(A.1)
k 1
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121 k ( k 1) iθ k ( k 1) iθ κ S 1 dˆk121 ( s )R1k e( k 1)iθ kcˆ121 kdˆk121 ( s ) cˆk ( s )R1 e k ( s )R1 e k 1
k 1
k 1
eˆ ( s )R e k 1
121 k
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R e
( k 1) iθ
( k 1) iθ
fˆ k 1
121 k
k 1
( s )R e
( k 1) iθ
]
k 1
CE
21 k kiθ 21 k ( k 2) iθ k kiθ bk21 ( s ) R1 k e kiθ c121 ak ( s) R1 e (k )ak ( s ) R1 e k ( s ) R1 e
k 1
k 1
k 1
k 1
k 1
k 1
k 1
k ( k 2) iθ k d k121 ( s) R1k e kiθ (k )c121 kd k121 ( s ) R1k e ( k 2)iθ e121 k ( s ) R1 e k ( s ) R1
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ekiθ f k121 ( s) R1k e kiθ k 1
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(A.2)
ACCEPTED MANUSCRIPT 1 k ( k 1) iθ k ( k 1) iθ [ κ S 1cˆ121 κ S 1 dˆk121 ( s ) R2k e ( k 1)iθ kcˆ121 k ( s ) R2 e k ( s ) R2 e k 1 k 1 k 1 2GS 1 1 121 k ( k 1) iθ k ( k 1) iθ eˆ121 fˆk121 ( s ) R2k e ( k 1)iθ ] [κS1 kdˆk ( s ) R2 e k ( s ) R2 e k 1 k 1 k 1 2GS 1 cˆ111 R 1e 2iθ κ dˆ111 R κ dˆ111 R 3 e 2iθ cˆ111 R 1e 2iθ dˆ111 R 3dˆ111 R 3 e 2iθ 1
2
S1 1
eˆ1111 R21 eˆ3111 R23 e 2iθ
2
S1 3
2
1
2
1
2
3
2
1 fˆ1111 R2 e 2iθ ] [ κ S 2 cˆk221 ( s ) R2 k e( k 1) iθ κ S 2 dˆk221 ( s) k 1 2GS 2 k 1
k 1
k 1
k 1
(A.3)
R2k e( k 1)iθ kcˆk221 ( s ) R2 k e ( k 1) iθ kdˆk221 ( s ) R2k e( k 1)iθ eˆk221 ( s ) R2 k e ( k 1)iθ
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221 k ( k 1) iθ ] fˆk ( s ) R2 e
k 1
k 1
k 1
121 k kiθ k ( k 2) iθ dˆk121 ( s ) R2k e kiθ kcˆ121 kdˆk121 ( s ) R2k e ( k 2)iθ cˆk ( s) R2 e k ( s ) R2 e
k 1
k 1
k 1
k 1
k 1
k 1
k kiθ eˆ121 fˆk121 ( s ) R2k e kiθ cˆk221 ( s ) R2 k e kiθ dˆk221 (s ) R2k e kiθ k ( s ) R2 e
k 1
k 1
k 1
(A.4)
221 k ( k 2) iθ kdˆk221 ( s ) R2k e( k 2) iθ eˆk221 ( s ) R2 k e kiθ fˆk221 (s ) R2k e kiθ kcˆk ( s) R2 e
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k 1
221 k 1 ( k 1) iθ (k )cˆk221 ( s) R3 k 1e( k 1)iθ kdˆk221 ( s) R3k 1e ( k 1) iθ (k )cˆk ( s) R3 e
k 1
k 1
kdˆ k 1
221 k
k 1 ( k 1) iθ 3
( s) R e
k 1
k (k 1)dˆ
221 k
k 1
k 1 ( k 1) iθ 3
(s) R
k 1
k 1
e
k (k 1)dˆk221 ( s ) R3k 1 k 1
(A.5)
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e( k 1)iθ (k )eˆk221 ( s ) R3 k 1e( k 1)iθ kfˆk221 ( s ) R3k 1e( k 1)iθ qˆwat ( s)
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The linear system of algebraic equations with regard to the coefficients in the Laplace domain is then obtained by setting the coefficients of the term eikθ (k 0, 1, 2, ) on both sides of Eqs. (A.1)
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e0 :
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eikθ in Eq. (A.1):
eiθ :
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(A.5) being equality, as shown in the following:
t2 1 1 1 L A1 (t ) A1 (t1 ) (1 λ) L [ H (t τ )b111 ( τ ) R11 dτ ] 4 R1 2 t1 1 ˆ 1 H ( s)bˆ121 ( s) R11 [(κ S1 1)dˆ1121 ( s) R1 eˆ1121 ( s) R11 ] 2 2GS 1
(A.6)
1 1 Hˆ ( s)bˆ221 ( s) R12 [κS1 dˆ2121 ( s) R12 eˆ2121 ( s) R12 ] 2 2GS1
(A.7)
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1 1 R13 2 t2 t2 1 1 L [ H (t τ )a111 ( τ ) R11 dτ ] L [ H (t τ )b311 ( τ ) R13 dτ ] Hˆ ( s) aˆ121 ( s) R11 2 2 t1 t1 1 1 Hˆ ( s )bˆ321 ( s ) R13 [κ S 1 dˆ3121 ( s) R13 cˆ1121 ( s) R11 eˆ3121 ( s) R13 ] 2 2GS1
(A.8)
e( k 1)iθ : (k =2,3, )
1 ˆ 1 kH ( s)aˆk21 ( s ) R1 k Hˆ ( s )bˆk21 2 ( s ) R1 k 2 2 2 1 k 2 k k 2 [κ S 1 dˆk121 kcˆ121 eˆ121 ] 2 ( s ) R1 k ( s ) R1 k 2 ( s ) R1 2GS1
(A.9)
A1 (t ) A1 (t1 ) (1 λ)
eiθ :
0
e2iθ :
A3 (t ) A3 (t1 ) (1 λ)
1 [2dˆ2121 ( s) R12 ] 2GS1
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t2 p0 1 1 1 L A2 (t ) A2 (t1 ) (1 λ) L [ I (t2 τ )a111 ( τ ) R11 dτ ] Iˆ( s) aˆ121 ( s) R11 4 R1 2 2 t1 1 [κ S 1cˆ1121 ( s) R11 2dˆ3121 ( s) R13 fˆ1121 ( s) R1 ] 2GS 1
1ˆ 1 k k 2 I ( s)aˆk21 R1 k [κS1cˆ121 (k 2)dˆk121 fˆk121 ( s) R1k ] k ( s ) R1 2 ( s ) R1 2 2GS1
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eikθ in Eq. (A.2):
bˆ121 (s) R11 2dˆ1121 (s) R1 eˆ1121 (s) R11
(A.14)
2 bˆ221 (s) R12 dˆ2121 (s) R12 eˆ121 2 ( s) R1
(A.15)
k 2 k ( k 2) ˆ121 kaˆk21 (s) R1 k bˆk212 (s) R1( k 2) dˆk121 kcˆ121 2 ( s) R1 k (s ) R1 ek 2 (s ) R1
(A.16)
k k 2 ˆ121 aˆk21 (s) R1 k cˆ121 fˆk121 (s) R1k k ( s) R1 (k 2)d k 2 ( s) R1
(A.17)
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e kiθ : (k 1, 2, )
(A.12)
(A.13)
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AC
e( k 2)iθ : (k 1,2, )
(A.11)
2dˆ2121 (s) R12 0
e0 :
eiθ :
(A.10)
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e ( k 1)iθ : (k = 2, 3, )
e2iθ :
1 1 L R1 4
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1 L 4
eikθ in Eq. (A.3): e0 :
1 [(κ S 1 1)dˆ1121 ( s) R2 eˆ1121 ( s) R21 ( κ S1 1) dˆ1111 ( s) R2 eˆ1111 ( s) R21 ] 2GS 1 1 [(κ S 2 1)dˆ1221 ( s) R2 eˆ1221 ( s) R21 ] 2GS 2
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(A.18)
ACCEPTED MANUSCRIPT 1 1 2 [κS1 dˆ2121 ( s) R22 eˆ121 [κS 2 dˆ2221 ( s) R22 eˆ2221 ( s) R22 ] 2 ( s ) R2 ] 2GS1 2GS 2
(A.19)
e2iθ :
1 [κ S 1 dˆ3121 ( s ) R23 cˆ1(2) ( s ) R21 eˆ3121 (s ) R23 κ S1 dˆ3111 ( s) R23 cˆ1111 ( s) R21 2GS 1 1 eˆ3111 ( s) R23 ] [κ S 2 dˆ3221 ( s) R23 cˆ1221 ( s) R21 eˆ3221 ( s) R23 ] 2GS 2
(A.20)
e( k 1)iθ : (k =2,3, )
1 k 2 k ( k 2) ˆ121 [κ S 1 dˆk121 kcˆ121 ] 2 ( s ) R2 k ( s ) R2 ek 2 ( s ) R2 2GS 1 1 k 2 ( k 2) [κ S 2 dˆk221 kcˆk221 ( s) R2 k eˆk221 ] 2 ( s ) R2 2 ( s ) R2 2GS 2
(A.21)
eiθ :
1 1 [2dˆ2121 ( s) R22 ] [2dˆ2221 ( s) R22 ] 2GS1 2GS 2
(A.22)
e2iθ :
1 [κ S 1cˆ1121 ( s) R21 3dˆ3121 ( s) R23 fˆ1121 ( s) R2 κ S1 cˆ1111 ( s) R21 3dˆ3111 ( s) R23 2GS 1 1 fˆ1111 ( s) R2 ] [κ S 2 cˆ1221 ( s) R21 3dˆ3221 ( s) R23 fˆ1221 ( s) R2 ] 2GS 2
(A.23)
e( k 1)iθ : (k = 2, 3, )
1 k k 2 ˆ121 [κ S 1cˆ121 fˆk121 ( s) R2k ] k ( s ) R2 ( k 2) d k 2 ( s ) R2 2GS 1 1 k 2 [κ S 2 cˆk221 ( s) R2 k (k 2)dˆk221 fˆk221 ( s) R2k ] 2 ( s ) R2 2GS 2
(A.24)
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eiθ :
eikθ in Eq. (A.4):
2dˆ2121 (s) R 22 2dˆ2221 (s) R 22
(A.25)
2dˆ1121 (s) R2 eˆ1121 (s) R21 2dˆ1221 (s) R2 fˆ1221 (s) R21
(A.26)
2 2 2 ˆ 221 ˆ221 dˆ2121 (s) R22 eˆ121 2 ( s) R2 d 2 ( s) R2 e2 ( s) R2
(A.27)
e( k 2)iθ : (k 1,2, )
k 2 k ( k 2) ˆ121 dˆk121 kcˆ121 2 ( s ) R2 k ( s ) R2 ek 2 ( s ) R2 221 k 2 221 k 221 ( k 2) dˆk 2 ( s) R2 kcˆk ( s) R2 eˆk 2 (s) R2
(A.28)
e kiθ : (k 1, 2, )
k k 2 ˆ121 cˆ121 fˆk121 ( s) R2k k ( s ) R2 ( k 2) d k 2 ( s ) R2
(A.29)
PT
e0 :
AC
e2iθ :
CE
eiθ :
k 2 cˆk221 ( s) R2 k (k 2)dˆk221 fˆk221 ( s ) R2k 2 ( s ) R2
eikθ in Eq. (A.5):
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2dˆ1221 ( s)
eˆ1221 ( s) qˆwat ( s) R32
(A.30)
eiθ :
2dˆ2221 (s) R3 2dˆ2221 (s) R3 0
(A.31)
e( k 1)iθ : (k 1,2, )
k 1 cˆk221 (s) R3 k 1 (k 2)dˆk221 fˆk221 (s) R3k 1 0 2 ( s) R3
(A.32)
e iθ :
2dˆ2221 (s) R3 2eˆ2221 (s) R33 0
(A.33)
e( k 1)iθ : (k 1, 2, )
k 1 k 3 kcˆk221 (s) R3 k 1 dˆk221 eˆk221 0 2 ( s) R3 2 ( s) R3
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e0 :
(A.34)
Therefore, a system of linear algebraic equations is obtained with regard to the coefficients in the Laplace domain. It can be noted that the equations provided in Eqs. (A.9), (A.12), (A.16), (A.17),
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(A.21), (A.24), (A.28), (A.29), (A.32), (A.34) are singular linearly with respect to aˆk21 ( s) , bˆk21 2 (s) , cˆkm 21 (s) , dˆkm212 ( s) , eˆkm212 ( s) and fˆkm 21 ( s) ( k 2 and m 1, 2 ), and therefore these coefficients should
be zero. All the coefficients in Laplace domain can be calculated by solving the equations (A.6)
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(A.34), and then the coefficients in time domain can be obtained by inverse Laplace transform.
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A.2 Potentials in Case (2)
The equations expressed by potentials (see Eqs. (23), (24), (29), (30), (32)) have been provided by
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satisfying the boundary and compatibility conditions in Case (2) during the second liner stage. Only
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the new equations (29) and (30), obtained from the fullslip conditions along linerliner interface, are introduced in the following derivation to give the additional equations in Case (2).
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Substituting Eqs. (4) and (5) into Eqs. (29) and (30), and introducing Laplace transformation with respect to t '' , yields the equations about the coefficients in the potentials:
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Re
1 k ( k 1) iθ [ κ S 1cˆ122 κ S 1 dˆk122 ( s) R2k e( k 1)iθ kcˆk122 ( s ) R2 k e ( k 1)iθ k ( s ) R2 e k 1 k 1 2GS 1 k 1
k 1
k 1
k 1
k ( k 1) iθ kdˆk122 ( s ) R2k e( k 1)iθ eˆ122 fˆk122 ( s ) R2k e ( k 1)iθ ] k ( s ) R2 e
1 [κ S1 2GS 1
cˆ1112 R21e 2iθ κ S 1 dˆ1112 R2 κ S1 dˆ3112 R23 e 2iθ cˆ1112 R21e 2iθ dˆ1112 R2 3dˆ3112 R23 e 2iθ eˆ1112 R21
(A.35) fˆ1112 R2 e 2iθ ] Re
1 [ κ S 2 cˆk222 ( s ) R2 k e( k 1)iθ κ S 2 dˆk222 ( s) k 1 2GS 2 k 1
k 1
k 1
k 1
3 2 iθ 2
eˆ R e 112 3
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R2k e( k 1)iθ kcˆk222 ( s ) R2 k e( k 1) iθ kdˆk222 ( s ) R2k e( k 1) iθ eˆk222 ( s ) R2 k e( k 1)iθ
222 k ( k 1) iθ ] fˆk ( s ) R2 e
k 1
k 1
k 1
k 1
Re
k (k 1)dˆ
122 k
k 1
2 cˆ Re
k 1
222 k
k 1 ( k 1) iθ 2
(s) R e
( s ) (k ) R
 k 1 ( k 1) iθ 2
e
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 k 1 ( k 1) iθ  k 1 ( k 1) iθ 2 cˆ122 2 kdˆk122 ( s) R2k 1e( k 1) iθ k (k 1)cˆ122 k ( s ) ( k ) R2 e k ( s ) R2 e
 k 1 ( k 1) iθ 2
keˆ (s ) R 122 k
k 1
2 kdˆ k 1
k 1
k 1
222 k
e
k 1 ( k 1) iθ 2
( s) R e
kfˆ k 1
122 k
(s) R e
k (k 1)cˆ
222 k
k 1
k 1 ( k 1) iθ 2
(A.36)  k 1 ( k 1) iθ 2
(s) R
e
k (k 1)dˆk222 ( s ) R2k 1e( k 1)iθ keˆk222 ( s ) R2 k 1e( k 1) iθ kfˆk222 ( s ) R2k 1e( k 1) iθ
k 1
k 1
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k 1
Im
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122 k 1 ( k 1) iθ k (k 1)dˆk122 ( s ) R2k 1e( k 1) iθ k (k 1)cˆk ( s) R2 e
0
(A.37)
0
(A.38)
k 1 ( k 1) iθ keˆ122 e kfˆk122 ( s ) R2k 1e( k 1) iθ k ( s ) R2 k 1
PT
k 1
222 k 1 ( k 1) iθ k (k 1)dˆk222 ( s ) R2k 1e( k 1) iθ k (k 1)cˆk ( s) R2 e
Im
k 1
keˆ
k 1 ( k 1) iθ 2
(s) R
e
CE
k 1
222 k
k 1
kfˆ k 1
222 k
k 1 ( k 1) iθ 2
(s) R
e
AC
Similar to the process in Appendix A.1, the coefficients of term eikθ ( k 0， 1， 2， ) on both sides of Eqs. (A.35)(A.38) should be equal, yields: eikθ in Eq. (A.35):
0
e :
1 2κ S1 dˆ1122 R2 2dˆ1122 R2 2eˆ1122 R21 2κ S1 dˆ1112 R2 2dˆ1112 R2 2eˆ1112 R21 2GS1 1 2κ S 2 dˆ1222 R2 2dˆ1222 R2 2eˆ1222 R21 2GS 2
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(A.39)
ACCEPTED MANUSCRIPT 1 ˆ122 2 κS1 d2 R2 2dˆ2122 R22 eˆ2122 R22 2GS1
e iθ :
1 κS 2 dˆ2222 R22 2dˆ2222 R22 eˆ2222 R22 2GS 2
1 κ S 1cˆ1122 ( s ) R21 κ S 1 dˆ3122 ( s) R23 cˆ1122 ( s ) R21 3dˆ3122 ( s ) R23 eˆ3122 ( s ) R23 2GS1 fˆ 122 ( s ) R κ cˆ112 ( s) R 1 κ dˆ112 ( s) R 3 cˆ112 ( s) R 1 3dˆ112 ( s) R 3 eˆ112 ( s ) 1
e2iθ :
2
S1 1
2
S1 3
2
1
2
3
2
3
1 R23 fˆ1112 ( s ) R2 κ S 2 cˆ1222 ( s) R21 κ S 2 dˆ3222 ( s ) R23 cˆ1222 ( s ) R21 3 2GS 2 dˆ 222 ( s ) R 3 eˆ222 ( s ) R 3 fˆ 222 ( s ) R 2
3
2
1
1 k k 2 k k 2 ˆ122 ˆ122 κ S 1cˆ122 cˆ121 k ( s ) R2 κ S 1 d k 2 ( s ) R2 k ( s ) kR2 ( k 2) d k 2 ( s ) R2 2GS1 1 k 2 k 2 eˆ122 fˆk122 ( s ) R2k κ S 2 cˆk222 ( s) R2 k κ S 2 dˆk222 cˆk222 ( s) k 2 ( s ) R2 2 ( s ) R2 2GS 2 k 222 k 2 2 kR (k 2)dˆ ( s) R eˆ 22 ( s) R k 2 fˆ 222 ( s ) R k
e ( k 1)iθ : (k = 2, 3, )
k 2
2
k 2
2
k
2
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2
(A.41)
2
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3
(A.40)
eikθ in Eq.(A.36):
(A.42)
2dˆ1122 (s) eˆ1122 (s) R22 2dˆ1222 (s) eˆ1222 (s) R22
(A.43)
e iθ :
3 3 ˆ 222 ˆ222 dˆ2122 (s) R2 eˆ122 2 ( s) R2 d 2 ( s) R2 e2 ( s) R2
(A.44)
k 1 k 1 k 1 2kcˆ122 2(k 2)dˆk122 k (k 1)cˆ122 (k 2)(k 1) k ( s ) R2 2 ( s ) R2 k ( s ) R2 dˆ122 ( s ) R k 1 (k 2)eˆ122 ( s) R k 3 kfˆ 122 ( s) R k 1 2kcˆ 222 ( s) R k 1 k 2
k 2
2
2
k
2
k
2
k 1 k 1 2(k 2)dˆk222 k (k 1)cˆk222 ( s ) R2 k 1 (k 2)(k 1)dˆk222 2 ( s ) R2 2 ( s ) R2
ED
e ( k 1)iθ : (k 1, 2, )
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e0 :
(A.45)
e0 :
AC
e iθ :
CE
eikθ in Eq.(A.37):
PT
k 3 (k 2)eˆk222 kfˆk222 ( s) R2k 1 2 ( s ) R2
e ( k 1)iθ : (k 1, 2, )
eˆ1122 (s) R22 eˆ1122 (s) R22 0
(A.46)
3 2dˆ2122 (s) R2 2eˆ122 2 ( s) R2 0
(A.47)
k 1 k 1 k 3 k (k 1)cˆ122 (k 2)(k 1)dˆk122 (k 2)eˆk122 k ( s ) R2 2 ( s ) R2 2 ( s ) R2 kfˆ 122 ( s) R k 1 0
(A.48)
k
2
eikθ in Eq.(A.38): e0 :
eˆ1222 (s) R22 eˆ1222 (s) R22 0
(A.49)
e iθ :
2dˆ2222 (s) R2 2eˆ2222 (s) R23 0
(A.50)
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e ( k 1)iθ : (k 1, 2, )
k 1  k 3 k (k 1)cˆk222 (s) R2 k 1 (k 2)(k 1)dˆk222 (k 2)eˆk222 2 ( s ) R2 2 ( s ) R2 222 k 1 kfˆ ( s) R 0 k
(A.51)
2
Eqs. (A.39)(A.51), along with Eqs. (A.6)(A.17) and (A.30)(A.34), provide a system of linear algebraic equations with regard to the coefficients in the Laplace domain in Case (2). It can be noted that the equations (A.9), (A.12), (A.16), (A.17), (A.32), (A.34), (A.42), (A.45), (A.48), (A.51), are
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singular linearly with respect to aˆk21 ( s) , bˆk21 2 (s) , cˆkm 21 (s) , dˆkm212 (s) , eˆkm212 (s) and fˆkm 21 ( s) ( k 2 and m 1, 2 ), and therefore these coefficients are zero. The expression of other nonzero coefficients in
AC
CE
PT
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the potentials can be determined by solving the other equations.
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References: [1] P.P. Oreste, Analysis of structural interaction in tunnels using the covergence–confinement approach, Tunnelling and Underground Space Technology. 18 (4) (2003) 347363. [2] D.F. Malan, Manuel rocha medal recipient simulating the timedependent behaviour of
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excavations in hard rock, Rock Mechanics and Rock Engineering. 35 (4) (2002) 225254. [3] H.N. Wang, Y. Li, Q. Ni, S. Utili, M.J. Jiang, F. Liu, Analytical solutions for the construction of deeply buried circular tunnels with two liners in rheological rock, Rock Mechanics and Rock
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Engineering. 46 (6) (2013) 14811498.
[4] H.N. Wang, S. Utili, M.J. Jiang, An analytical approach for the sequential excavation of axisymmetric lined tunnels in viscoelastic rock, International Journal of Rock Mechanics and
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Mining Sciences. 68 (6) (2014) 85106.
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[5] H.N. Wang, G.S. Zeng, S. Utili, M.J. Jiang, L. Wu, Analytical solutions of stresses and displacements for deeply buried twin tunnels in viscoelastic rock, International Journal of Rock
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Mechanics and Mining Sciences. 93 (2017) 1329.
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[6] M.Y. Fattah, Boundary element analysis of a lined tunnel problem, International Journal of
AC
Engineering. 25 (2) (2012) 8997. [7] M.J. Jiang, Z.F. Shen, S. Utili, DEM modeling of cantilever retaining excavations: Implications for lunar constructions, Engineering Computations. 33 (2) (2016) 366394.
[8] C.W. Boon, G.T. Houlsby, S. Utili, Designing tunnel support in jointed rock masses via the DEM, Rock Mechanics and Rock Engineering. 48 (2) (2015) 603632. [9] M.J. Jiang, Z.Y. Yin, Influence of soil conditioning on ground deformation during longitudinal 34 / 56
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tunneling, Comptes Rendus Mecanique. 342 (3) (2014) 189197. [10] H.M. Tian, W.Z. Chen, D.S. Yang, G.J. Wu, X.J. Tan, Numerical analysis on the interaction of shotcrete liner with rock for yielding supports, Tunnelling and Underground Space Technology. 54 (2016) 2028.
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[11] M.J. Jiang, H. Chen, G.B. Crosta, Numerical modeling of rock mechanical behavior and fracture propagation by a new bond contact model, International Journal of Rock Mechanics and Mining Sciences. 78 (2015) 175189.
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[12] C. CarranzaTorres, C. Fairhurst, The elastoplastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion, International Journal of Rock Mechanics and Mining Sciences. 36 (6) (1999) 777809.
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[13] M.B. Wang, S.C. Li, A complex variable solution for stress and displacement field around a
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lined circular tunnel at great depth, International Journal for Numerical and Analytical Methods in Geomechanics. 33 (7) (2009) 939951.
PT
[14] S.C. Li, M.B. Wang, Elastic analysis of stress–displacement field for a lined circular tunnel at
CE
great depth due to ground loads and internal pressure, Tunnelling and Underground Space
AC
Technology. 23 (6) (2008) 609617. [15] H. El Naggar, S.D. Hinchberger, K.Y. Lo, A closedform solution for composite tunnel linings in a homogeneous infinite isotropic elastic medium, Canadian Geotechnical Journal. 45 (2) (2008) 266287. [16] A.Z. Lu, L.Q. Zhang, N. Zhang, Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay, International Journal of Rock Mechanics and 35 / 56
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Mining Sciences, 48 (3) (2011) 514519. [17] C. CarranzaTorres, B. Rysdahl, M. Kasim, On the elastic analysis of a circular lined tunnel considering the delayed installation of the support, International Journal of Rock Mechanics and Mining Sciences. 61 (10) (2013) 5785.
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[18] D.P. Mason, H. Abelman, Support provided to rock excavations by a system of two liners, International Journal of Rock Mechanics and Mining Sciences. 46 (7) (2009) 11971205. [19] A. Fahimifar, F.M. Tehrani, A. Hedayat, A. Vakilzadeh, Analytical solution for the
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excavation of circular tunnels in a viscoelastic Burger’s material under hydrostatic stress field, Tunnelling and Underground Space Technology. 25 (4) (2010) 297304. [20] P. Nomikos, R. Rahmannejad, A. Sofianos, Supported axisymmetric tunnels within linear
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viscoelastic burgers rocks, Rock Mechanics and Rock Engineering. 44 (5) (2011) 553564.
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[21] K.Y. Lo, C.M.K. Yuen, Design of tunnel lining in rock for long term time effects, Canadian Geotechnical Journal. 18 (1) (1981) 2439.
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[22] H.N. Wang, S. Utili, M.J. Jiang, P. He, Analytical solutions for tunnels of elliptical
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crosssection in rheological rock accounting for sequential excavation, Rock Mechanics and
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Rock Engineering. 48 (5) (2014) 19972029. [23] R.M. Christensen, L.B. Freund, Theory of viscoelasticity, Academic Press, 1982. [24] N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Noordhoff International Publishing, 1977. [25] H.N. Wang, F. Song, M.J. Jiang, Analytical solutions for the construction of circular tunnel accounting for time·dependent characteristic of the rheological rock, Journal of Tongji 36 / 56
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University：Natural Science. 44 (12) (2016) 18351844. (in Chinese)
M
EM (GM )
EK (GK )
K (a)
Maxwell
(b) Kelvin model
model EK (GK )
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EK (GK )
M
EM (GM )
K
K
EM (GM )
(d) Burgers model
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(c) Generalized Kelvin model
AC
CE
PT
ED
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Figure 1 Physical viscoelastic models.
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p0
R1
R2 R3
R ( ij )
p0
R ( ij )
S (2ij )
S (2ij )
S (2ij )
(b) Rock
(c) First liner
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(a) Rock and liners
qwat (t )
S (2ij )
tra
tra
qw (t )
S (1ij ) S (1ij )
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Infinite boundary
AC
CE
PT
ED
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Figure 2 Boundary conditions and reference coordinate systems.
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(d) Second liner
ACCEPTED MANUSCRIPT
p0
94m
qw (t )
5.5m 5.8m 6m
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94m
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p0
Figure 3 Constraint, Calculation domain and boundary conditions stresses in FEM numerical
AC
CE
PT
ED
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simulations.
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Farboundaries located at distances x=100 m and y=100 m with 8620 elements. Point B
Second liner
First liner
y Point A
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Part x
AC
CE
PT
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Figure 4 Mesh of the vicinity of the tunnel.
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Analytical FEM
ux at t=20th day
x at t=20th day
uy at t=6th day
y at t=6th day
uy at t=20th day 90 80 70 0.06 60 50 0.05
y at t=20th day 90 80 70 40.0 60 50 35.0 40 [] 30.0 25.0 30 20.0 20 15.0 10 10.0 0 5.0
0.04 0.03 0.02 0.01 0.00
40 [] 30 20 10 0
(a)
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x at t=6th day
Stresses [MPa]
Displacements [m]
Analytical FEM ux at t=6th day
(b)
Figure 5 Comparison between analytical solutions (noslip conditions) and FEM results for
AC
CE
PT
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(a) excavation induced displacements versus ; (b) total stresses versus .
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uR(22)
0.010
t=9th day t=11st day
0.005
t=50th day
uS(122)
2
uR(22) uS(122)
R(22)
S(122)
R(22) S(122)
th
t=9 day t=11st day
1
Stresses [MPa]
0.000 0.005 0.010
t=50th day
0 1 2
0.015 0.020
0
10
20
30
40
50
60
70
80
90 100
3
0
10
20
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Displacements [m]
0.015
30
40
50
60
70
80
90 100
(a)
(b)
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Figure 6 Displacements and stresses of the rock and first liner at the rockliner interface versus the angle in Case (2): (a) radial and circumferential displacements; (b) radial stress and shear stress. The
AC
CE
PT
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quantities are presented at the three specific times.
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0.010
0.4 t=9th day t=11st day
0.000
t=50th day
0.010 0.015
0
10
20
30
40
50
60
70
80
S(222)
(a)
S(222) S(222)
0.8 t=9th day
S(222) at t=50th day
0.0 S(222) th S(222) at t=11th day 0.2 at t=9 day S(222) th S(222) at t=11th day at t=9 day 0.4 0 10 20 30 40 50 60 70 80 90 100
20
30
40
50
60
70
80
90 100
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10
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t=50 day
S(222) at t=11th day
0
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Stresses [MPa]
0.2
0.4
(b)
S(222)
t=11st day
0.4
0.2
th
0.6
t=50th day
0.0
0.6
90 100
1.0
S(122) S(222)
t=9 day t=11st day
0.2
0.005
0.020
S(222)
th
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0.005
S(122)
uS(222)
Stresses [MPa]
Displacements [m]
uS(122)
PT
Figure 7 Boundary Displacements and stresses of first and second liners versus the angle in Case (2): (a) radial displacements （c） at linerliner interface; (b) stresses at linerliner interface; (c) stresses on
AC
CE
inner boundary of the second liner. The quantities are presented at the three specific times.
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0.004
0.075
t1=t2=3rd day
t1=t2=0 day New solution: d1=0.3m, d2=0.1m
0.045
u[m]
u(=0) [m]
0.060
New solution: d1=0.3m, d2=0.3m
0.003
0.002
0.030
New solution: New solution: New solution: Solution in reference [3]：
0.015
Solution in reference [15]: d1=0.3m,d2=0.1m
0.000
Solution in reference [15]: d1=0.3m,d2=0.3m
0
4
8
12
16
20
24
28
32
36
40
t [day]
0.000
0
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0.001
3
6
9
12
15
18
21
24
27
30
t [day]
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(a) (b) Figure 8 Comparison on radial displacements versus time between new solution and (a) the solution in the reference [15]; (b) the solution in the reference [3]. The displacements are for the point on inner
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CE
PT
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boundary of second liner ( =0 ).
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1.05
0.24
0.90
t2/TK=0
0.21
t2/TK=0
t2/TK=1.0
0.18
t2/TK=1.0
Case (2)
0.150
0.145
t2/TK=2.0
t2/TK=2.0
0.140
0.15
uu
0.75
u/u
Case (1)
Case (2)
uu
Case (1)
0.60
0.135 35
40
45
50
55
0.12 0.09 0.06
0.45
0
10
20
30
40
50
60
70
80
0.00
90 100
(a)
0
10
CR IP T
0.03
0.30
20
30
40
50
60
70
80
90
100
(b)
AN US
Figure 9 Normalized final displacements of the rock along rockliner interface in Cases (1) and (2), versus the angle for various installation times of second liner: (a) radial displacements; (b) circumferential displacements. Three specific installation times of the first liner are
AC
CE
PT
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M
considered.
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ACCEPTED MANUSCRIPT
0.5
2.4
t2/TK=0
2.1
t2/TK=2.0
1.8 0
1.8
3
6
9
12
15
1.5
1.0
p0
p0
0.3
2.0
1.9
t2/TK=1.0
0.4
p0
2.1
Case (2) p0
Case (1)
0.9
0.8
1.2
Case (1) Case (2)
0.7 75
78
t2/TK=0
0.2
81
84
87
90
0.1
0.6
0
10
20
30
40
50
60
70
80
90 100
0
10
(a)
0.054
t2/TK=1.0
0.052 0.050
t2/TK=2.0
0.048 35
40
45
50
0.06
0
10
20
30
40
50
60
60
70
80
90
100
70
55
80
90 100
ED
(c)
50
M
0.03 0.00
40
AN US
p0
0.09
0.056
Case (2) p0
0.12
30
(b)
0.15 Case (1)
20
t2/TK=0
CR IP T
0.9 t2/TK=1.0 t2/TK=2.0
PT
Figure 10 Normalized final stresses of the rock along the rockliner interface versus the angle
CE
for various installation times of second liner: (a) radial stress; (b) hoop stress; (c) shear
AC
stress. The quantities in both Cases (1) and (2) are plotted, and three specific installation times of the first liner are considered.
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ACCEPTED MANUSCRIPT
0.16
0.75
0.14
0.45 Case (1) t2/TK=0 t2/TK=1.0 t2/TK=2.0
0.10 Case (1)
0.08
Case (2)
t2/TK=0 t2/TK=1.0
0.06
t2/TK=2.0
0.150 0.148 0.146 0.144 0.142 0.140 4.0
4.2
4.4
4.6
4.8
5.0
t/TK
0.04 0.0
0.15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.5
CR IP T
0.30
0.12
u/u（ ）
u/u（ ）
u/u（ ）
0.60
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
t/TK
t/TK
(a)
(b)
AN US
Figure 11 Normalized displacements of the rock on the tunnel boundary (
) in Case (1) and Case
(2), versus normalized time for various installation times of the second liner: (a) radial
AC
CE
PT
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displacements; (b) circumferential displacements.
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0.6
1.8 Case (1)
Case (2)
Case (1)
t2/TK=0
0.5
1.7
t2/TK=2.0
0.4
p（ ）
p（ ）
t2/TK=1.0
0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.060
0.050 4.0
4.2
4.4
4.6
4.8
t/TK
0.06 0.03
0.5
1.0
1.5
2.0
2.5
3.0
(a)
4.0
4.5
5.0
(b)
AC
CE
PT
(c)
5.0
ED
t/TK
3.5
AN US
0.055
0.09
0.00 0.0
1.4
CR IP T
0.12 t /T =1.0 2 K t2/TK=2.0
1.5
M
p（ ）
t2/TK=0
p（ ）
Case (2)
t2/TK=2.0
1.6
t/TK
0.065
Case (1)
t2/TK=1.0
1.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
t/TK
0.15
Case (2)
t2/TK=0
Figure 12 Normalized stresses of the rock on the tunnel boundary (
) in Cases (1) versus
normalized time for various installation times of the second liner: (a) radial stresses; (b) 48 / 56
hoop stresses; and (c) shear stresses.
ACCEPTED MANUSCRIPT
Input the parameters of (a) geometry, initial stresses, water pressure: R1 , , p0 , qw (b) material of rock mass: GM , GK ,K (c) first liner: GS1 , d1 , t1
Design for installation time of second liner
Input the parameters, GS 2 , d2 , into analytical model
CR IP T
Design for thickness of second liner
Input the parameters, GS 2 , t2 , into analytical model
Obtain
Obtain
Line of zero equivalent stress
Curve of equivalent stress of first failure point versus thickness of second liner
Economic thickness of second liner
M
Economic installation time of second liner
Intersection
AN US
Curve of equivalent stress of first Intersection failure point versus installation time of second liner
AC
CE
PT
ED
Figure 13 Flow chart in design of installation time/thickness of second liner.
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Case (2)
d2/R1=0.0250
0.5
d2/R1=0.0417
0.4
d2/R1=0.0583
0.3 0.2
0.12 0.08 0.04 0.00
0.04
0.1
0.08 0.8
0.9
0.2 0.00
1.1
1.2
t2/TK
0.0 0.1
1.0
Zero equivalent stress line
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
t2/TK
0.12 Case (1)
Case (2)
t2/TK=0.9
0.08
t2/TK=1.1
0.04
Zero equivalent stress line
0.00 0.04 0.08 0.02
0.03
CR IP T
Case (1)
Normalized effective stresses
0.6
Normalized effective stresses
Normalized effective stresses
ACCEPTED MANUSCRIPT
0.04
0.05
0.06
0.07
0.08
0.09
d2/R1
(a)
(b)
AN US
Figure 14 Normalized equivalent stresses of the second liner versus (a) installation time of second liner for various liner thicknesses; (b) thickness of second liner for various installation times of second liner. Both quantities in Case (1) and Case (2) are plotted.
ED
M
Table 1 Shear relaxation moduli for the four viscoelastic models
Viscoelastic
Maxwell
Kelvin
model
model
model
PT
Shear relaxation modulus G(t)
GM e
GM
M
t
GK K (t )
b2GM (M K )  b1GKM (b2  b1 ) A2
CE AC
a1
Generalized Kelvin model
GM 2 e GM GK
GM GK
K
t
GM GK GM GK
, a2 b2GKM b1GM (M K ) , b1 (b2  b1 ) A2
b2

t
A2 A2 2 4 A1 A3
A2 A2 2 4 A1 A3
A1 GM GK , A2 GMM GMK GKM , A3 MK

t
GM [a1e b1 a2e b2 ]
2 A3
2 A3
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Burgers model
,
ACCEPTED MANUSCRIPT
CR IP T
Table 2 The two functions defined in Eq. (2) for the four viscoelastic models
H(t)
I(t)
6 1 1 (t ) M 3K e GM GM 6GM 2
M 3K e GM
G exp K t K K
6
t
PT
ED
Kelvin model
1
M
G exp K K K 1
1 t t GM
AC
CE
Burgers model
6 1 6GM2 (t ) K (3K e GM ) 2 3K e GM GM
M
G 1 1 t exp K GM K K
Generalized
2
3K GK 1 G exp e t exp K t K K K K
1
Kelvin model
3K eGM exp t (3 K G ) e M M
AN US
1 1 (t ) GM M
Maxwell model
3K G GM (3K e GK ) 1 G exp e K t exp K (3 K G ) K e M K K
6 1 M (3K e GM ) GM 1
G 1 exp K t K K
t
3GM2 t 2 M K (3K e GM )
N N ( N1 2 ) t ( N1 2 ) t 2 2 ( M K M 1 )e (M K M 1 )e
M1 [GKM (3Ke GM )(M K ) 3KeGM (M K )2 ] / M 2 ,
2 M 2 [GM GK 3(GM GK ) Ke ]2M 6GM KeMK (GM GK 3GM Ke 3GK Ke ) 9GM2 Ke2K2
N1
M2 2(3Ke GM )MK
, N2 3GKM Ke GM GKM 3KeGM (M K )
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(3Ke GM )MK
ACCEPTED MANUSCRIPT
Table 3 Boundary and compatibility conditions in the four cases Stages
Boundary and compatibility conditions
Conditions expressed by potentials
First liner stage
Eqs. (11), (12) and (19)
—
Second liner stage
Eqs. (11), (12), (15), (16) and (20)
Eqs. (23), (24), (27), (28) and (32)
First liner stage
Eqs. (11), (12) and (19)
—
Second liner stage
Eqs. (11), (12), (17), (18) and (20)
First liner stage
Eqs. (13), (14) and (19)
Second liner stage
Eqs. (13), (14), (15), (16) and (20)
First liner stage
Eqs. (13), (14) and (19)
Second liner stage
Eqs. (13), (14), (17), (18) and (20)
Cases
Case (2)
Case (3)
Eqs. (23), (24), (29), (30) and (32) —
Eqs. (25), (26), (27), (28) and (32) —
AC
CE
PT
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Case (4)
CR IP T
Case (1)
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Eqs. (25), (26), (29), (30) and (32)
ACCEPTED MANUSCRIPT
Table 4 Inputting parameters of loading, geometrical and mechanical properties for rock and liners in FEM numerical simulations Initial
Ratio of
Geometrical parameters of tunnel and
stress
horizontal over
liners [m]
Liner installation Internal pressure of the liner
time
vertical stress
15
0.65
R1 6
d1
d2
0.05
0.1
Parameters for generalized Kelvin model GK [MPa]
[MPa.day]
2000
1000
10000
t1 [day]
t2 [day]
2.0
5.0
0 t 10
0 qwat (t ) 1.5
t 10
Elastic parameters of liners
Poisson’s ratio
GS1 [MPa]
GS2 [MPa]
AN US
GM [MPa]
qwat(t) [MPa]
CR IP T
p0 [MPa]
AC
CE
PT
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0.5
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10000
10000
S1
S2
0.2
0.2
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Table 5 Parameters employed in comparison between new solution and the one by Naggar et al. [15] Geometrical parameters of tunnel and liners [m]
p0 [MPa]
R1
d1
0.344
0.7
5.18
0.3
Parameters in elastic solution [15]
Elastic parameters of liners GS1 [MPa] GS2 [MPa]
S1
CR IP T
Initial stresses
12500
8333
S2
0.2
0.2
Parameters in new solution
Poisson’s ratio
GM [MPa]
GK [MPa]
[MPa.day]
Poisson’s ratio
90
0.4
32
32
300
0.5
AC
CE
PT
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Young’s modulus [MPa]
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Table 6 Parameters employed in comparison between new solution and the one by Wanag et al. [3].
p0 [MPa]
R1
d1
d2
6
0.1
0.2
Excavation process New solution Instantaneous
15
excavation Parameters for generalized Kelvin model GM [MPa]
GK [MPa]
2000
1000
Reference [3]
t1 [day]
t2 [day]
2m, 0 t 1 R(t ) t 1 6m,
1.0
8.0
Elastic parameters of liners
[MPa.day] Poisson’s ratio
GS1 [MPa]
GS2 [MPa]
S1
S2
10000
10000
0.2
0.2
0.5
AC
CE
PT
ED
M
AN US
10000
Liner installation time
CR IP T
Initial stress Geometrical parameters of tunnel and liners [m]
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Table 7 Inputting parameters in application example. Initial stress
Geometrical parameters of
Internal pressure of the liner
tunnel and liners [m]
15
0.65
R1
d1
6
0.12
qwat(t) [MPa]
0 t t2 5 0 qwat (t ) 0.15 t t2 5
Parameters for generalized Kelvin model GM [MPa]
1000
Poisson’s
[MPa.day]
ratio
10000
0.5
Compressiv
installation
strength of
e strength of
time
second liner
second liner
t1 [day]
σ tS 2 [MPa]
σ cS 2 [MPa]
2.0
2.39
26.8
Elastic parameters of liners GS1 [MPa]
GS2 [MPa]
S1
S2
10000
13541
0.2
0.2
AC
CE
PT
ED
M
AN US
2000
GK [MPa]
Tensile
CR IP T
p0 [MPa]
Liner
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