Time-Dependent Reliability Analysis of Deep Tunnel in the Viscoelastic Burger Rock with Sequential Installation of Liners

Abstract

The behavior of deep tunnel is usually time dependent due to the sequential construction process as well as the rheological characteristic of the rock mass. Moreover, the uncertainty of different parameters such as the viscoelastic properties can play an essential role on the stability of tunnel and its supports. The reliability analysis based on the probabilistic approach, which has been largely applied in the last 2 decades to treat this kind of problems, presents an efficient tool but its application is usually limited in case of time-independent behavior of tunnel supported by a single liner. This study aims at assessment of time-dependent reliability of deep circular tunnel counting for rheological characteristic of rock mass and sequential construction of liners. Since the reliability analysis, using direct Monte Carlo Simulation and a FEM software could be very time-consuming, we establish firstly a closed form solution for the response of a deep tunnel excavated in the viscoelastic Burger rock accounting for the sequential installation of two liners. The validation of this analytical solution is performed by comparing the results of this solution with some previous existing analytical solutions as well as with numerical results obtained from the finite element simulation in some case studies. Once validated, this analytical solution is used as a fast tool for the time-dependent reliability analysis by direct Monte Carlo Simulation. The influence of different parameters on the probability of failure of a double liners tunnel in respect with its life service, by taking into account of the sequential construction, was discussed in the last part of the paper.

Appendix 1: Determination of the supporting pressures of two liners

In this appendix, we detail the process to determine the supporting pressures of two liners by using the compatibility conditions of displacement at the interface between the rock mass and the first liner (Eq. 16) and at the interface of two liners (Eq. 17).

1.1 1.1. Determination of Supporting Pressures after the Installation of the First Liner

By substituting the Eq. (11) and Eq. (14a) in Eq. (16), the radial incremental displacement of rock from time t₁ to the generic time t > t₁ can be written in the integral form:

$$ \begin{aligned} & \frac{1}{2R_{1}} \left\{ \int_{0}^{t_{1}} p_{0}^{{\text{h}}} \chi (\tau )R^{2} (\tau )H(t_{1} - \tau ){\text{d}}\tau \right. \\ & \left. \quad - \int_{0}^{t} p_{ 0}^{{\text{h}}} \chi (\tau )R^{2} (\tau )H(t - \tau ){\text{d}}\tau + R_{1}^{2} \int_{t_{1}}^{t} {p_{1} (\tau )} H(t - \tau ){\text{d}}\tau \right\} \\ & \quad = a_{00} p_{1} (t) + a_{01} p_{2} (t), \end{aligned} $$

(46)

with:

$$ \begin{aligned} & a_{00} = - \frac{1}{{2G_{\text{L1}} }}\frac{{R_{1}^{{}} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + v_{\text{L1}} }}{{K_{\text{L1}} }}\frac{{R_{1}^{3} }}{{R_{1}^{2} - R_{2}^{2} }} ,\\ & a_{01} = \frac{1}{{2G_{\text{L1}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + v_{\text{L1}} }}{{K_{\text{L1}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }}. \end{aligned}$$

(47)

Regarding with the conditions expressed in Eq. (4), the Eq. (46) written for the instant $t$ with $t_{1} \le t < t_{2}$ is:

$$ \frac{1}{{2R_{1} }}\left( {\int_{0}^{{t_{1} }} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t_{1} - \tau ){\text{d}}\tau - } \int_{0}^{t} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t - \tau ){\text{d}}\tau + R_{1}^{2} \int_{{t_{1} }}^{t} {p_{11} (\tau )H(t - \tau ){\text{d}}\tau } } } \right) = a_{00} p_{11} (t). $$

(48)

This Eq. (48) results the second type Volterra equation (Polyanin and Manzhirov 2008) for $p_{11} (t)$:

$$ p_{11} \left( t \right) = \frac{{R_{1} }}{{2a_{00} }}\int_{{t_{1} }}^{t} {p_{11} (\tau )H(t - \tau ){\text{d}}\tau } + \frac{1}{{2a_{00} R_{1} }}\left( {\int_{0}^{{t_{1} }} {p_{0}^{h} \chi (\tau )R^{2} (\tau )H(t_{1} - \tau ){\text{d}}\tau } } \right.\left. { - \int_{0}^{t} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t - \tau ){\text{d}}\tau } } \right). $$

(49)

Substituting Eq. (13) into Eq. (49) yields this following equation:

$$ \begin{aligned} p_{{11}} \left( t \right) & = \frac{{R_{1} }}{{2a_{{00}} }}\int_{{t_{1} }}^{t} {p_{{11}} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta _{{\text{K}}} }}\exp \left( { - \frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}(t - \tau )} \right) + \frac{1}{{\eta _{H} }}} \right){\text{d}}\tau } \\ & \quad + \frac{1}{{2a_{{00}} R_{1} }}\left\{ {\int_{0}^{{t_{1} }} {p_{0}^{{\text{h}}} \chi (\tau )R^{2} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t_{1} - \tau ) + \frac{1}{{\eta _{{\text{K}}} }}\exp \left( { - \frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}(t_{1} - \tau )} \right) + \frac{1}{{\eta _{H} }}} \right){\text{d}}\tau } } \right. \\ & \quad - \left. {\int_{0}^{t} {p_{0}^{{\text{h}}} \chi (\tau )R^{2} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta _{{\text{K}}} }}\exp \left( { - \frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}(t - \tau )} \right) + \frac{1}{{\eta _{H} }}} \right){\text{d}}\tau } } \right\} \\ \end{aligned} $$

(50)

which can be expanded in the form:

$$ \begin{aligned} p_{11} \left( t \right) = & \frac{{R_{1} }}{{2a_{00} }}\frac{1}{{G_{H} }}\int_{{t_{1} }}^{t} {p_{11} (\tau )\delta (t - \tau ){\text{d}}\tau } + \frac{{R_{1} }}{{2a_{00} }}\frac{1}{{\eta_{\text{K}} }}\int_{{t_{1} }}^{t} {p_{11} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right){\text{d}}\tau } \\ & + \frac{{R_{1} }}{{2a_{00} }}\frac{1}{{\eta_{H} }}\int_{{t_{1} }}^{t} {p_{11} (\tau ){\text{d}}\tau } + \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{G_{H} }}\left( {\int_{0}^{{t_{1} }} {\chi (\tau )\delta (t_{1} - \tau )R^{2} (\tau ){\text{d}}\tau } - \int_{0}^{t} {\chi (\tau )\delta (t - \tau )R^{2} (\tau ){\text{d}}\tau } } \right) \\ & + \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{\eta_{\text{K}} }}\left( {\int_{0}^{{t_{1} }} {\chi (\tau )R^{2} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t_{1} - \tau )} \right){\text{d}}\tau } - \int_{0}^{t} {\chi (\tau )R^{2} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right){\text{d}}\tau } } \right) \\ & + \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{\eta_{H} }}\left( {\int_{0}^{{t_{1} }} {\chi (\tau )R^{2} (\tau ){\text{d}}\tau } - \int_{0}^{t} {\chi (\tau )R^{2} (\tau ){\text{d}}\tau } } \right). \\ \end{aligned} $$

(51)

Supposing now $\varphi_{1}^{B} (t) = p_{11} \left( t \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)$, the following integral equation can be obtained:

$$ \begin{aligned} \varphi_{1}^{B} (t) = \; & \frac{{R_{1} }}{{2a_{00} }}\frac{1}{{G_{H} }}\varphi_{1}^{B} (t) + \frac{{R_{1} }}{{2a_{00} }}\frac{1}{{\eta_{\text{K}} }}\int_{{t_{1} }}^{t} {\varphi_{1}^{B} (\tau ){\text{d}}\tau } + \frac{{R_{1} }}{{2a_{00} }}\frac{1}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\varphi_{1}^{B} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } \\ & + \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{G_{H} }}R_{1}^{2} \left( {\chi (t_{1} ) - \chi (t)} \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right) + \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{\eta_{\text{K}} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - t_{1} )} \right)\int_{0}^{{t_{1} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } \\ & - \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{\eta_{\text{K}} }}\int_{0}^{t} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } + \frac{1}{{2a_{00} R_{1} }}\frac{{p_{0}^{\text{h}} }}{{\eta_{H} }}R_{1}^{2} \exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\chi (\tau ){\text{d}}\tau } . \\ \end{aligned} $$

(52)

Then by defining $e_{1} = (R_{1} G_{H} )/(2G_{H} a_{00} - R_{1} )$, we can rewrite Eq. (52) in form:

$$ \begin{aligned} \varphi_{1}^{B} (t) & = \frac{{e_{1} }}{{\eta_{\text{K}} }}\int_{{t_{1} }}^{t} {\varphi_{1}^{B} (\tau ){\text{d}}\tau } + \frac{{e_{1} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\varphi_{1}^{B} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } + \frac{{e_{1} p_{0}^{\text{h}} }}{{G_{H} }}\left( {\chi (t_{1} ) - \chi (t)} \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right) \\ & \quad + \frac{{e_{1} p_{0}^{\text{h}} }}{{\eta_{\text{K}} R_{1}^{2} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - t_{1} )} \right)\int_{0}^{{t_{1} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } - \frac{{e_{1} p_{0}^{\text{h}} }}{{\eta_{K} R_{1}^{2} }}\int_{0}^{t} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } \\ & \quad + \frac{{e_{1} p_{0}^{\text{h}} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\chi (\tau ){\text{d}}\tau } . \\ \end{aligned} $$

(53)

By defining:

$$ \begin{aligned} f_{1}^{B} (t) & & = \frac{{e_{1} p_{0}^{{\text{h}}} }}{{G_{H} }}\left( {\chi (t_{1} ) - \chi (t)} \right)\exp \left( {\frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}t} \right) + \frac{{e_{1} p_{0}^{{\text{h}}} }}{{\eta _{{\text{K}}} R_{1} ^{2} }}\exp \left( {\frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}(t - t_{1} )} \right)\int_{0}^{{t_{1} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}\tau } \right){\text{d}}\tau } \\ & \quad - \frac{{e_{1} p_{0}^{{\text{h}}} }}{{\eta _{{\text{K}}} R_{1} ^{2} }}\int_{0}^{t} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}\tau } \right)d\tau } + \frac{{e_{1} p_{0}^{{\text{h}}} }}{{\eta _{H} }}\exp \left( {\frac{{G_{{\text{K}}} }}{{\eta _{{\text{K}}} }}t} \right)\int_{{t_{1} }}^{t} {\chi (\tau ){\text{d}}\tau } , \\ \end{aligned} $$

(54)

with $\lambda_{1}^{B} = e_{1} /\eta_{\text{K}}$ and posing:

$$ A_{1} = \int_{0}^{{t_{0} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } = \int_{0}^{{t_{0} }} {\left( {1 - m_{1} \exp \left( { - \frac{{m_{2} v_{\text{l}} \tau }}{{\left( {R_{\text{ini}} + v_{\text{r}} \tau } \right)}}} \right)} \right)\left( {R_{\text{ini}} + v_{\text{r}} \tau } \right)^{2} \exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } , $$

(55a)

$$ A_{2} = \int_{{t_{0} }}^{{t_{1} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } = \int_{{t_{0} }}^{{t_{1} }} {\left( {1 - m_{1} \exp \left( { - \frac{{m_{2} v_{\text{l}} \tau }}{{R_{\text{fin}} }}} \right)} \right)R_{\text{fin}}^{2} \exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } , $$

(55b)

the Eq. (54) can be rewritten as:

$$ \begin{aligned} f_{1}^{B} (t) & = \frac{{e_{1} p_{0}^{\text{h}} }}{{G_{H} }}\left( {\chi (t_{1} ) - \chi (t)} \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right) \\ & \quad + \frac{{e_{1} p_{0}^{h} }}{{\eta_{\text{K}} R_{1}^{2} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - t_{1} )} \right).\left( {A_{1} + A_{2} } \right) \hfill \\ & \quad - \frac{{e_{1} p_{0}^{\text{h}} }}{{\eta_{\text{K}} R_{1}^{2} }}\left( {A_{1} + A_{2} + R_{\text{fin}}^{2} \int_{{t_{1} }}^{t} {\chi (\tau )\exp \left( {\frac{{G_{K} }}{{\eta_{K} }}\tau } \right)d\tau } } \right) \\ & \quad + \frac{{e_{1} p_{0}^{\text{h}} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right). \hfill \\ \end{aligned} $$

(56)

Thus, the function $\varphi_{1}^{B} (t)$ in Eq. (53) can be simplified as:

$$ \begin{aligned} \varphi_{1}^{B} (t) & = \frac{{e_{1} }}{{\eta_{\text{K}} }}\int_{{t_{1} }}^{t} {\varphi_{1}^{B} (\tau ){\text{d}}\tau } + \frac{{e_{1} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\varphi_{1}^{B} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } + f_{1}^{B} (t) \\ & = \int_{{t_{1} }}^{t} {\left( {\frac{{e_{1} }}{{\eta_{\text{K}} }} + \frac{{e_{1} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right)} \right)\varphi_{1}^{B} (\tau )d\tau } + f_{1}^{B} (t). \\ \end{aligned} $$

(57)

This last equation presents in effect the standard integral equation as shown in (Eq. 18) with the free term $f_{1}^{B} (t)$ and the Kernel $\left( {\frac{{e_{1} }}{{\eta_{\text{K}} }} + \frac{{e_{1} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right)} \right)$ whereas the corresponding parameters $E{}_{1},E{}_{2},\lambda_{1} ,\lambda_{2}$ are respectively equal to $E_{1} = - \frac{{e_{1} }}{{\eta_{\text{K}} }},\;E_{2} = - \frac{{e_{1} }}{{\eta_{H} }},\;\lambda_{1} = 0,\;\lambda_{2} = \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}$.

Due to the fact that $\eta_{K} > 0,\,\,\eta_{H} \, > 0$, the discriminant of the quadratic equation (Eq. 19) is positive ($D_{a} > 0$), thus the solution $\varphi_{1}^{B} (t)$ takes the same form of Eq. (21) which means that:

$$ \varphi_{1}^{B} (t) = f_{1}^{B} (t) + \int_{{t_{1} }}^{t} {\left( {F_{1} {\text{e}}^{{\mu_{1} (t - \tau )}} + F_{2} {\text{e}}^{{\mu_{2} (t - \tau )}} } \right)} f_{1}^{B} (\tau ){\text{d}}\tau , $$

(58)

Correspondingly, the solution of $p_{11} \left( t \right)$ can be calculated:

$$ p_{11} \left( t \right) = \varphi_{1}^{B} (t)\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right), $$

(59)

1.2 1.2. Determination of Supporting Pressure after the Installation of the Second Liner

The determination of the supporting pressure in the liners after the second installation stage consists of calculating the pressures $p_{12} (t)$ and $p_{22} (t)$ with $t \ge t_{2}$.

Using the radial displacement in each liner expressed in Eqs. (14a) and (15a), the compatibility condition of the displacement at the interface of two liners (see Eq. 17) can be rewritten in the form:

$$ a_{10} p_{12} (t) + a_{11} p_{22} (t) = a_{10} p_{12} (t_{2} ),\quad (t \ge t_{2} ) $$

(60)

with:

$$ \begin{aligned} a_{10} = & - \frac{1}{{2G_{\text{L1}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + v_{\text{L1}} }}{{K_{\text{L1}} }}\frac{{R_{1}^{2} R_{2} }}{{R_{1}^{2} - R_{2}^{2} }} ,\\ a_{11} = \; & \frac{1}{{2G_{\text{L1}} }}\frac{{R_{1}^{2} R_{2} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + v_{\text{L1}} }}{{K_{\text{L1}} }}\frac{{R_{2}^{3} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{1}{{2G_{\text{L2}} }}\frac{{R_{3}^{2} R_{2} }}{{R_{2}^{2} - R_{3}^{2} }} + \frac{{1 + \nu_{\text{L2}} }}{{K_{\text{L2}} }}\frac{{R_{2}^{3} }}{{R_{2}^{2} - R_{3}^{2} }}. \\ \end{aligned} $$

(61)

The Eq. (46) written for the instant $t$ (with $t \ge t_{2}$) has the following form:

$$ \begin{aligned} \frac{1}{{2R_{1} }}\left( {\int_{0}^{{t_{1} }} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t_{1} - \tau ){\text{d}}\tau - } \int_{0}^{t} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t - \tau ){\text{d}}\tau + R_{1}^{2} \int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau )H(t - \tau ){\text{d}}\tau } } } \right. \hfill \\ \left. {\quad + \; R_{1}^{2} \int_{{t_{2} }}^{{t_{{}} }} {p_{12} (\tau )H(t - \tau ){\text{d}}\tau } } \right) = a_{00} p_{12} (t) + a_{01} p_{22} (t). \hfill \\ \end{aligned} $$

(62)

Using the relationship between $p_{22} (t)$ and $p_{12} (t)$ as shown in Eq. (60), one can deduce the following integral equation for $p_{12} (t)$:

$$ \begin{aligned} p_{12} (t) = & \frac{{R_{1} a_{11} }}{{2(a_{00} a_{11} - a_{01} a_{10} )}}\int_{{t_{2} }}^{t} {p_{12} (\tau )H(t - \tau ){\text{d}}\tau } - \frac{{a_{01} a_{10} }}{{a_{00} a_{11} - a_{01} a_{10} }}p_{11} (t_{2} ) \\ & \quad + \frac{{a_{11} }}{{2R_{1} (a_{00} a_{11} - a_{01} a_{10} )}}\left( {\int_{0}^{{t_{1} }} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t_{1} - \tau ){\text{d}}\tau - } \int_{0}^{t} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )H(t - \tau ){\text{d}}\tau + R_{1}^{2} \int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau )H(t - \tau ){\text{d}}\tau } } } \right). \\ \end{aligned} $$

(63)

Substituting $H(t)$ from Eq. (13) into Eq. (63), we have:

$$ \begin{aligned} p_{12} (t) = \frac{{R_{1} a_{11} }}{{2(a_{00} a_{11} - a_{01} a_{10} )}}\int_{{t_{2} }}^{t} {p_{12} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right){\text{d}}\tau } - \frac{{a_{01} a_{10} }}{{a_{00} a_{11} - a_{01} a_{10} }}p_{11} (t_{2} ) \hfill \\ \quad \quad \quad + \frac{{a_{11} }}{{2R_{1} (a_{00} a_{11} - a_{01} a_{10} )}}\left\{ {\int_{0}^{{t_{1} }} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t_{1} - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t_{1} - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right){\text{d}}\tau } } \right. \hfill \\ \quad \quad \quad - \int_{0}^{t} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right){\text{d}}\tau } \hfill \\ \left. {\quad \quad \quad + R_{1}^{2} \int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right){\text{d}}\tau } } \right\}. \hfill \\ \end{aligned} $$

(64)

Defining $\varphi_{2}^{B} (t) = p_{12} \left( t \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)$, after some developments, the simplified form of the integral equation $\varphi_{2}^{B} (t)$ can be obtained:

$$ \varphi_{2}^{B} (t) = \frac{{R_{1} a_{11} }}{{2(a_{00} a_{11} - a_{01} a_{10} )}}\left\{ {\frac{1}{{G_{H} }}\varphi_{2}^{B} (t) + \frac{1}{{\eta_{\text{K}} }}\int_{{t_{2} }}^{t} {\varphi_{2}^{B} (\tau ){\text{d}}\tau + \frac{1}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{2} }}^{t} {\varphi_{2}^{B} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right)d\tau } } } \right\} + f_{2}^{B*} (t), $$

(65)

where

$$ \begin{aligned} f_{2}^{B*} (t) = \left\{ {\frac{{a_{11} }}{{2R_{1} (a_{00} a_{11} - a_{01} a_{10} )}}\left\{ {\int_{0}^{{t_{1} }} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t_{1} - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t_{1} - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right){\text{d}}\tau } } \right.} \right. \hfill \\ \quad \quad \quad \quad - \int_{0}^{t} {p_{0}^{\text{h}} \chi (\tau )R^{2} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right){\text{d}}\tau } \hfill \\ \quad \quad \quad \quad \left. { + R_{1}^{2} \int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau )\left( {\frac{1}{{G_{H} }}\delta (t - \tau ) + \frac{1}{{\eta_{\text{K}} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right) + \frac{1}{{\eta_{H} }}} \right)d\tau } } \right\}\left. { - \frac{{a_{01} a_{10} }}{{a_{00} a_{11} - a_{01} a_{10} }}p_{11} (t_{2} )} \right\}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right). \hfill \\ \end{aligned} $$

(66)

Defining $e_{2} = \frac{{a_{11} G_{H} R_{1} }}{{2G_{H} \left( {a_{00} a_{11} - a_{01} a_{10} } \right) - R_{1} a_{11} }}$ and $\lambda_{2}^{B} = e_{2} /\eta_{K}$, Eq. (65) can be rewritten in a more compact form:

$$ \varphi_{2}^{B} (t) = \frac{{e_{2} }}{{\eta_{\text{K}} }}\int_{{t_{2} }}^{t} {\varphi_{2}^{B} (\tau ){\text{d}}\tau } + \frac{{e_{2} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{2} }}^{t} {\varphi_{2}^{B} (\tau )\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } + f_{2}^{B} (t), $$

(67)

where:

$$ \begin{aligned} f_{2}^{B} (t) = & e_{2} f_{2}^{B*} (t) = \frac{{e_{2} p_{0}^{\text{h}} }}{{G_{H} }}\left( {\chi (t_{1} ) - \chi (t)} \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right) + \frac{{e_{2} p_{0}^{\text{h}} }}{{\eta_{\text{K}} R_{1}^{2} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - t_{1} )} \right)\int_{0}^{{t_{1} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{K} }}{{\eta_{K} }}\tau } \right){\text{d}}\tau } \\ & - \frac{{e_{2} p_{0}^{\text{h}} }}{{\eta_{K} R_{1}^{2} }}\int_{0}^{t} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{E} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } + \frac{{e_{2} }}{{\eta_{\text{K}} }}\int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau )\exp \left( {\frac{{G_{e} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } - \frac{{2e_{2} a_{01} a_{10} }}{{a_{11} R_{1} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)p_{11} (t_{2} ) \\ & - \frac{{e_{2} p_{0}^{\text{h}} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\chi (\tau ){\text{d}}\tau } + \frac{{e_{2} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau ){\text{d}}\tau } . \\ \end{aligned} $$

(68)

Defining $A_{3} = \int_{{t_{1} }}^{{t_{2} }} {\chi (\tau )R^{2} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau }$ with $R(t) = R_{fin} \,\,\,{\text{if}}\,\,\,\,t_{1} \le t$ and:

$$ B_{1} = \int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } ,\quad B_{2} = \frac{{2e_{2} a_{01} a_{10} }}{{a_{11} R_{1} }}p_{11} (t_{2} ),\quad B_{3} = \int_{{t_{1} }}^{{t_{2} }} {p_{11} (\tau ){\text{d}}\tau } . $$

(69)

Therefore, $f_{2}^{B} (t)$ can be expressed in form:

$$ \begin{aligned} f_{2}^{B} (t) = & \frac{{e_{2} p_{0}^{\text{h}} }}{{G_{H} }}\left( {\chi (t_{1} ) - \chi (t)} \right)\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right) + \frac{{e_{2} p_{0}^{h} }}{{\eta_{\text{K}} R_{1}^{2} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - t_{1} )} \right) \cdot \left( {A_{1} + A_{2} } \right) \\ & - \frac{{e_{2} p_{0}^{\text{h}} }}{{\eta_{E} R_{1}^{2} }}\left( {A_{1} + A_{2} + A_{3} + R_{E}^{2} \int_{{t_{2} }}^{t} {\chi (\tau )\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}\tau } \right){\text{d}}\tau } } \right) + \frac{{e_{2} }}{{\eta_{\text{K}} }}B_{1} \\ & - B_{2} \exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right) - \frac{{e_{2} p_{0}^{\text{h}} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right)\int_{{t_{1} }}^{t} {\chi (\tau ){\text{d}}\tau } + \frac{{e_{2} }}{{\eta_{H} }}B_{3} \exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right). \\ \end{aligned} $$

(70)

We obtain also the following standard integral equation:

$$ \varphi_{2}^{B} (t) = \int_{{t_{2} }}^{t} {\left( {\frac{{e_{2} }}{{\eta_{\text{K}} }} + \frac{{e_{2} }}{{\eta_{H} }}\exp \left( {\frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}(t - \tau )} \right)} \right)\varphi_{2}^{B} (\tau ){\text{d}}\tau } + f_{2}^{B} (t), $$

(71)

which has the Kernel $\left( {\frac{{e_{2} }}{{\eta_{K} }} + \frac{{e_{2} }}{{\eta_{H} }}\exp \left( {\frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)$ and the free term $f_{2}^{B} (t)$.

This integral equation with the corresponding parameters $E_{1} = - \frac{{e_{2} }}{{\eta_{\text{K}} }},\quad E_{2} = - \frac{{e_{2} }}{{\eta_{H} }},\quad \lambda_{1} = 0,\quad \lambda_{2} = \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}$ results a positive discriminant ($D_{a} > 0$) of the characteristic equation (Eq. 19). Thus, the solution of Eq. (71) has the same form as one presented in Eq. (22) meaning that:

$$ \varphi_{2}^{B} (t) = f_{2}^{B} (t) + \int_{{t_{2} }}^{t} {\left( {F_{1} {\text{e}}^{{\mu_{1} (t - \tau )}} + F_{2} {\text{e}}^{{\mu_{2} (t - \tau )}} } \right)} f_{2}^{B} (\tau ){\text{d}}\tau . $$

(72)

From this last solution of $\varphi_{2}^{B} \left( t \right)$, one can deduce the solution of $p_{12} \left( t \right)$:

$$ p_{12} \left( t \right) = \varphi_{2}^{B} (t)\exp \left( { - \frac{{G_{\text{K}} }}{{\eta_{\text{K}} }}t} \right). $$

(73)

Through which the solution of $p_{22} \left( t \right)$ can be determined using the Eq. (60) as shown in Eq. (25).

To simplify the presentation, the expressions of the supporting pressures as well as the displacements in the rock mass are written in the integral form but it is worth to note that their explicitly analytical expressions can be found without difficulty. The numerical integration is only needed for the functions containing the term $\chi (t)$ determined in the range $t < t_{0}$(i.e. during the excavation stage) such as ones expressed in Eqs. (28 and 29) due to the present of time in both the numerator and denominator of the exponential function (see also Eq. 55a).