Closed-Form Solution and Reliability Analysis of Deep Tunnel Supported by a Concrete Liner and a Covered Compressible Layer Within the Viscoelastic Burger Rock

Abstract

Time-dependent rock behavior can significantly affect the design and the construction method of tunnel support lining. As an innovation, a concrete liner covered with a compressible layer is thus of great interest to reduce the underground space project costs. Indeed, the compressible material usually exhibits very high deformability under oedometric loading (30–60% volumetric deformation under a weak variation in axial stress). This helps to absorb the convergence of the excavated wall of the squeezing rock, and thus limit the transmission of stress to the inner concrete lining. This study firstly aims at deriving a closed-form solution for a deep-circular tunnel excavated within the viscoelastic Burger rock and supported by a double-layer concrete/compressible material. The inner concrete layer is assumed to be linear elastic, while the outer compressible layer is described by a tri-linear elastic model. The analytical solution is derived under the integral equation form. Secondly, the effects of uncertainty of time-dependent mechanical properties of host rock on the failure probability of tunnel for a period of 100 years are studied by introducing the closed-form solution into the well-known Kriging-based reliability analysis AK-MCS (Active learning reliability method combining Kriging and Monte Carlo Simulation) method. Notably, the benefit of the compressible layer is highlighted from the reliability analysis.

Appendices

Appendix A: Determination of the Supporting Pressures of the Outer Tri-Linear Compressible Liner and Inner Elastic Liner

The supporting pressures of two liners in which the first liner owes a tri-linear elastic behavior are mathematically conducted in more detail. The derivation process bases on the solution of the integral equation and is derived from the compatibility conditions of displacement at the interface between the rock mass and the compressible liner and at the interface of two liners.

Determination of Supporting Pressures for the Period $t_{0} \le t \le t_{1}$.

Corresponding to this period, the elastic behavior of the compressible liner is in the first slope (i.e., elasticity E_c = E_c1). From the compatibility conditions of displacement at the interface between the rock mass and the first support, one obtain the following equation:

$$ \begin{gathered} \frac{{R_{1} }}{2}\left\{ {\int_{0}^{{t_{0} }} {p_{0}^{h} } \chi (\tau )H(t_{0} - \tau )d\tau - \int_{0}^{t} {p_{0}^{h} } \chi (\tau )H(t - \tau )d\tau + \int_{{t_{0} }}^{t} {p_{11} (\tau )} H(t - \tau )d\tau } \right\}\,\, \hfill \\ = a_{00,1} p_{11} (t) + a_{01,1} p_{21} (t), \hfill \\ \end{gathered} $$

(38)

with

$$ a_{00,1} = - \frac{1}{{2G_{L1,1}^{{}} }}\frac{{R_{1}^{{}} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,1}^{{}} }}\frac{{R_{1}^{3} }}{{R_{1}^{2} - R_{2}^{2} }}\,{,}\,\,\,\,a_{01,1} = \frac{1}{{2G_{L1,1}^{{}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,1}^{{}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }}\,, $$

(39)

The other compatibility conditions of displacement at the interface between two liners (Eq. (22)) yields

$$ p_{21} (t) = - \frac{{a_{10,1} p_{11} (t)}}{{a_{11,1} }} $$

(40)

with

$$ \begin{gathered} a_{10,1} = - \frac{1}{{2G_{L1,1}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,1}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }}\,{,}\, \hfill \\ \,a_{10,1}^{*} = \frac{1}{{2G_{L1,1}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,1}^{{}} }}\frac{{R_{2}^{3} }}{{R_{1}^{2} - R_{2}^{2} }}\,, \hfill \\ a_{11,1} = a_{10,1}^{*} + \frac{1}{{2G_{L2}^{{}} }}\frac{{R_{3}^{2} R_{2}^{{}} }}{{R_{2}^{2} - R_{3}^{2} }} + \frac{{1 + \nu_{L2}^{{}} }}{{K_{L2}^{{}} }}\frac{{R_{2}^{3} }}{{R_{2}^{2} - R_{3}^{2} }}, \hfill \\ \end{gathered} $$

(41)

Substituting Eq. (40) in Eq. (38) and after some developments, we deduce

$$ \begin{gathered} p_{11} \left( t \right) = \frac{{R_{1} a_{11,1} G_{M} }}{{2G_{M} (a_{00,1} a_{11,1} - a_{01,1} a_{10,1} ) - R_{1} a_{11,1} }}\int_{{t_{0} }}^{t} {(\frac{1}{{\eta_{M} }} + \frac{{\exp ( - G_{K} (t - \tau )/\eta_{K} )}}{{\eta_{K} }})p_{11} (\tau )d\tau } \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} + \frac{{2G_{M} (a_{00,1} a_{11,1} - a_{01,1} a_{10,1} )}}{{2G_{M} (a_{00,1} a_{11,1} - a_{01,1} a_{10,1} ) - R_{1} a_{11,1} }}f_{1}^{*} (t), \hfill \\ \end{gathered} $$

(42)

with

$$ \begin{gathered} f_{1}^{*} \left( t \right) = \frac{{R_{1} a_{11,1} }}{{2(a_{00,1} a_{11,1} - a_{01,1} a_{10,1} )}}\left( {\frac{{p_{0}^{h} }}{{G_{M} }}\left( {\chi (t_{0} ) - \chi (t)} \right) - \frac{{p_{0}^{h} }}{{\eta_{M} }}\int\limits_{{t_{0} }}^{t} {\chi (\tau )d\tau } } \right) \hfill \\ \begin{array}{*{20}c} {} & {} & + \\ \end{array} \frac{{R_{1} a_{11,1} }}{{2(a_{00,1} a_{11,1} - a_{01,1} a_{10,1} )}}\frac{{p_{0}^{h} }}{{\eta_{K} }}\left( {\int\limits_{0}^{{t_{0} }} {\chi (\tau )\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t_{0} - \tau )} \right)d\tau - \int\limits_{0}^{t} {\chi (\tau )\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)d\tau } } } \right), \hfill \\ \end{gathered} $$

(43)

By noting

$$ e_{1,1} = \frac{{R_{1} a_{11,1} G_{M} }}{{2G_{M} (a_{00,1} a_{11,1} - a_{01,1} a_{10,1} ) - R_{1} a_{11,1} }} $$

(44)

Eq. (42) becomes

$$ p_{11} \left( t \right) = f_{1} (t) + \int_{{t_{0} }}^{t} {\left( {\frac{{e_{1,1} }}{{\eta_{M} }} + \frac{{e_{1,1} }}{{\eta_{K} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)p_{11} \left( \tau \right)d\tau } , $$

(45)

with

$$ f_{1} (t) = \frac{{2G_{M} (a_{00,1} a_{11,1} - a_{01,1} a_{10,1} )}}{{2G_{M} (a_{00,1} a_{11,1} - a_{01,1} a_{10,1} ) - R_{1} a_{11,1} }}f_{1}^{*} (t) $$

(46)

Equation (Eq. (43)) takes the form of the standard integral equation (see Eq. (23)) with the free term $f_{1}^{{}} (t)$ and the Kernel $\left( {\frac{{e_{1,1} }}{{\eta_{M} }} + \frac{{e_{1,1} }}{{\eta_{K} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)$, whereas the corresponding parameters $E{}_{1},E{}_{2},\lambda_{1} ,\lambda_{2}$ are respectively equal to $E_{1} = - \frac{{e_{1,1} }}{{\eta_{M} }},\,\,\,\,\,E_{2} = - \frac{{e_{1,1} }}{{\eta_{K} }},$$\lambda_{1} = 0,$ $\lambda_{2} = - \frac{{G_{K} }}{{\eta_{K} }}$.

Since $\eta_{K} > 0,\,\,\eta_{M} \, > 0$, the discriminant of the quadratic Eq. (27) is positive, the solution $p_{11}^{{}} (t)$ takes the same form of Eq. (24), which means

$$ p_{11}^{{}} (t) = f_{1}^{{}} (t) + \int_{{t_{0} }}^{t} {\left( {F_{1,1} e^{{\mu_{1,1} (t - \tau )}} + F_{2,1} e^{{\mu_{2,1} (t - \tau )}} } \right)} f_{1}^{{}} (\tau )d\tau , $$

(47)

with

$$ \begin{gathered} \begin{array}{*{20}c} {F_{1,1} = E_{1} \frac{{\mu_{1,1} - \lambda_{2} }}{{\mu_{2,1} - \mu_{1,1} }} + E_{2} \frac{{\mu_{1,1} - \lambda_{1} }}{{\mu_{2,1} - \mu_{1,1} }},} & {F_{2,1} = E_{1} \frac{{\mu_{2,1} - \lambda_{2} }}{{\mu_{1,1} - \mu_{2,1} }} + E_{2} \frac{{\mu_{2,1} - \lambda_{1} }}{{\mu_{1,1} - \mu_{2,1} }},} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mu_{1,1} = \frac{{\lambda_{1} + \lambda_{2} - E_{1} - E_{2} + \sqrt {D_{a} } }}{2},} & {\mu_{2,1} = \frac{{\lambda_{1} + \lambda_{2} - E_{1} - E_{2} - \sqrt {D_{a} } }}{2},} \\ \end{array} \hfill \\ D_{a} = \left( {E_{1} E_{2} - \lambda_{1} - \lambda_{2} } \right)^{2} - 4\left( {\lambda_{1} \lambda_{2} - E_{1} \lambda_{2} - E_{2} \lambda_{1} } \right) \hfill \\ \end{gathered} $$

(48)

Determination of Supporting Pressures for the Period $t_{1} \le t \le t_{2}$.

The behavior of the compressible liner is in the second elastic slope (i.e., E_c = E_c2) during this period. The compatibility conditions of displacement at the interface between the rock mass leads to

$$ \begin{gathered} \frac{{R_{1} }}{2}\left\{ {\int_{0}^{{t_{0} }} {p_{0}^{h} } \chi (\tau )H(t_{0} - \tau )d\tau - \int_{0}^{t} {p_{0}^{h} } \chi (\tau )H(t - \tau )d\tau + \int_{{t_{0} }}^{{t_{1} }} {p_{11} (\tau )} H(t - \tau )d\tau + \int_{{t_{1} }}^{t} {p_{12} (\tau )} H(t - \tau )d\tau } \right\}\,\, \hfill \\ = a_{00,1} p_{11} (t_{1} ) + a_{01,1} p_{21} (t_{1} ) + a_{00,2} \left( {p_{12} (t) - p_{11} (t_{1} )} \right) + a_{01,2} \left( {p_{22} (t) - p_{21} (t_{1} )} \right), \hfill \\ \end{gathered} $$

(49)

with

$$ \begin{gathered} a_{00,2} = - \frac{1}{{2G_{L1,2}^{{}} }}\frac{{R_{1}^{{}} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,2}^{{}} }}\frac{{R_{1}^{3} }}{{R_{1}^{2} - R_{2}^{2} }}\,{,}\,\,\,\, \hfill \\ a_{01,2} = \frac{1}{{2G_{L1,2}^{{}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,2}^{{}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }}\,, \hfill \\ \end{gathered} $$

(50)

We develop the compatibility conditions of displacement at the interface between two liners in form

$$ a_{10,1} p_{11} (t_{1} ) + a_{10,1}^{*} p_{21} (t_{1} ) + a_{10,2} \left( {p_{12} (t) - p_{11} (t_{1} )} \right) + a_{10,2}^{*} \left( {p_{22} (t) - p_{21} (t_{1} )} \right) = - (a_{11,2} - a_{10,2}^{*} )p_{22} (t) $$

(51)

which is expressed in a more compact form such as

$$ p_{22} (t) = - \frac{{a_{10,2} p_{12} (t)}}{{a_{11,2} }} + \frac{{(a_{10,2} - a_{10,1} )}}{{a_{11,2} }}p_{11} (t_{1} ) + \frac{{(a_{11,2} - a_{11,1} )}}{{a_{11,2} }}p_{21} (t_{1} ) $$

(52)

with

$$ \begin{gathered} a_{10,2} = - \frac{1}{{2G_{L1,2}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,2}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }}\,{,}\,\,\,\, \hfill \\ a_{10,2}^{*} = \frac{1}{{2G_{L1,2}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,2}^{{}} }}\frac{{R_{2}^{3} }}{{R_{1}^{2} - R_{2}^{2} }}\,, \hfill \\ a_{11,2} = a_{10,2}^{*} + \frac{1}{{2G_{L2}^{{}} }}\frac{{R_{3}^{2} R_{2}^{{}} }}{{R_{2}^{2} - R_{3}^{2} }} + \frac{{1 + \nu_{L2}^{{}} }}{{K_{L2}^{{}} }}\frac{{R_{2}^{3} }}{{R_{2}^{2} - R_{3}^{2} }}, \hfill \\ \end{gathered} $$

(53)

Substituting Eq. (52) in Eq. (49) and after some developments, the supporting pressure of the outer liner can be written as

$$ \begin{gathered} p_{12} \left( t \right) = \frac{{R_{1} a_{11,2} G_{M} }}{{2G_{M} (a_{00,2} a_{11,2} - a_{01,2} a_{10,2} ) - R_{1} a_{11,2} }}\int_{{t_{1} }}^{t} {(\frac{1}{{\eta_{M} }} + \frac{{\exp ( - G_{K} (t - \tau )/\eta_{K} )}}{{\eta_{K} }})p_{12} (\tau )d\tau } \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} + \frac{{2G_{M} (a_{00,2} a_{11,2} - a_{01,2} a_{10,2} )}}{{2G_{M} (a_{00,2} a_{11,2} - a_{01,2} a_{10,2} ) - R_{1} a_{11,2} }}f_{2}^{*} (t), \hfill \\ \end{gathered} $$

(54)

with

$$ \begin{gathered} f_{2}^{*} \left( t \right) = \frac{{R_{1} a_{11,2} }}{{2(a_{00,2} a_{11,2} - a_{01,2} a_{10,2} )}}\left( {p_{0}^{h} \int\limits_{0}^{{t_{0} }} {\chi (\tau )} H(t_{0} - \tau )d\tau - p_{0}^{h} \int\limits_{0}^{t} {\chi (\tau )} H(t - \tau )d\tau + \int\limits_{{t_{0} }}^{{t_{1} }} {p_{11} (\tau )H(t - \tau )d\tau } } \right) \hfill \\ \begin{array}{*{20}c} {} & {} & { - \frac{{\left( {a_{00,1} - a_{00,2} } \right)a_{11,2} + \left( {a_{10,2} - a_{10,1} } \right)a_{01,2} }}{{\left( {a_{00,2} a_{11,2} - a_{01,2} a_{10,2} } \right)}}} \\ \end{array} p_{11} (t_{1} ) - \frac{{\left( {a_{01,1} - a_{01,2} } \right)a_{11,2} + \left( {a_{11,2} - a_{11,1} } \right)a_{01,2} }}{{\left( {a_{00,2} a_{11,2} - a_{01,2} a_{10,2} } \right)}}p_{21} (t_{1} ), \hfill \\ \end{gathered} $$

(55)

By noting

$$ e_{1,2} = \frac{{R_{1} a_{11,2} G_{M} }}{{2G_{M} \left( {a_{00,2} a_{11,2} - a_{01,2} a_{10,2} } \right) - R_{1} a_{11,2} }} $$

(56)

Eq. (54) becomes

$$ p_{12} \left( t \right) = f_{2} (t) + \int_{{t_{1} }}^{t} {\left( {\frac{{e_{1,2} }}{{\eta_{M} }} + \frac{{e_{1,2} }}{{\eta_{K} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)p_{12} \left( \tau \right)d\tau } , $$

(57)

with

$$ f_{2} (t) = \frac{{2G_{M} (a_{00,2} a_{11,2} - a_{01,2} a_{10,2} )}}{{2G_{M} (a_{00,2} a_{11,2} - a_{01,2} a_{10,2} ) - R_{1} a_{11,2} }}f_{2}^{*} (t) $$

(58)

The standard integral equation (in Eq. (57)) has the free term $f_{2}^{{}} (t)$ and the Kernel $\left( {\frac{{e_{1,2} }}{{\eta_{M} }} + \frac{{e_{1,2} }}{{\eta_{K} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)$, whereas the corresponding parameters $E{}_{1},E{}_{2},\lambda_{1} ,\lambda_{2}$ are respectively equal to $E_{1} = - \frac{{e_{1,2} }}{{\eta_{M} }},\,\,\,\,\,E_{2} = - \frac{{e_{1,2} }}{{\eta_{K} }},$$\lambda_{1} = 0,$ $\lambda_{2} = - \frac{{G_{K} }}{{\eta_{K} }}$.

By using the integral equation (24), the supporting pressure $p_{12}^{{}} (t)$ takes the following form

$$ p_{12}^{{}} (t) = f_{2}^{{}} (t) + \int_{{t_{1} }}^{t} {\left( {F_{1,2} e^{{\mu_{1,2} (t - \tau )}} + F_{2,2} e^{{\mu_{2,2} (t - \tau )}} } \right)} f_{2}^{{}} (\tau )d\tau , $$

(59)

with

$$ \begin{gathered} \begin{array}{*{20}c} {F_{1,2} = E_{1} \frac{{\mu_{1,2} - \lambda_{2} }}{{\mu_{2,2} - \mu_{1,2} }} + E_{2} \frac{{\mu_{1,2} - \lambda_{1} }}{{\mu_{2,2} - \mu_{1,2} }},} & {F_{2,2} = E_{1} \frac{{\mu_{2,2} - \lambda_{2} }}{{\mu_{1,2} - \mu_{2,2} }} + E_{2} \frac{{\mu_{2,2} - \lambda_{1} }}{{\mu_{1,2} - \mu_{2,2} }},} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mu_{1,2} = \frac{{\lambda_{1} + \lambda_{2} - E_{1} - E_{2} + \sqrt {D_{a} } }}{2},} & {\mu_{2,2} = \frac{{\lambda_{1} + \lambda_{2} - E_{1} - E_{2} - \sqrt {D_{a} } }}{2},} \\ \end{array} \hfill \\ D_{a} = \left( {E_{1} E_{2} - \lambda_{1} - \lambda_{2} } \right)^{2} - 4\left( {\lambda_{1} \lambda_{2} - E_{1} \lambda_{2} - E_{2} \lambda_{1} } \right) \hfill \\ \end{gathered} $$

(60)

Determination of supporting pressures for the period $t_{2} \le t$.

The compressible liner behaves in the third elastic slope (i.e., E_c = E_c3) in this period. The compatibility conditions of displacement at the interface between the rock mass reads

$$ \begin{gathered} \frac{{R_{1} }}{2}\left\{ {\int_{0}^{{t_{0} }} {p_{0}^{h} } \chi (\tau )H(t_{0} - \tau )d\tau - \int_{0}^{t} {p_{0}^{h} } \chi (\tau )H(t - \tau )d\tau + \int_{{t_{0} }}^{{t_{1} }} {p_{11} (\tau )} H(t - \tau )d\tau + \int_{{t_{1} }}^{{t_{2} }} {p_{12} (\tau )} H(t - \tau )d\tau } \right\}\,\, \hfill \\ + \frac{{R_{1} }}{2}\int_{{t_{2} }}^{t} {p_{13} (\tau )} H(t - \tau )d\tau = a_{00,1} p_{11} (t_{1} ) + a_{01,1} p_{21} (t_{1} ) + a_{00,2} \left( {p_{12} (t_{2} ) - p_{11} (t_{1} )} \right) + a_{01,2} \left( {p_{22} (t_{2} ) - p_{21} (t_{1} )} \right) \hfill \\ + a_{00,3} \left( {p_{13} (t) - p_{12} (t_{2} )} \right) + a_{01,3} \left( {p_{23} (t) - p_{22} (t_{2} )} \right), \hfill \\ \end{gathered} $$

(61)

with

$$ \begin{gathered} a_{00,3} = - \frac{1}{{2G_{L1,3}^{{}} }}\frac{{R_{1}^{{}} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,3}^{{}} }}\frac{{R_{1}^{3} }}{{R_{1}^{2} - R_{2}^{2} }}\,{,}\,\,\,\, \hfill \\ a_{01,3} = \frac{1}{{2G_{L1,3}^{{}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,3}^{{}} }}\frac{{R_{1} R_{2}^{2} }}{{R_{1}^{2} - R_{2}^{2} }}\,, \hfill \\ \end{gathered} $$

(62)

The compatibility conditions of displacement at the interface between two liners is developed as

$$ \begin{gathered} a_{10,1} p_{11} (t_{1} ) + a_{10,1}^{*} p_{21} (t_{1} ) + a_{10,2} \left( {p_{12} (t_{2} ) - p_{11} (t_{1} )} \right) + a_{10,2}^{*} \left( {p_{22} (t_{2} ) - p_{21} (t_{1} )} \right) + a_{10,3} \left( {p_{13} (t) - p_{12} (t_{2} )} \right) \hfill \\ + a_{10,3}^{*} \left( {p_{23} (t) - p_{22} (t_{2} )} \right) = - (a_{11,3} - a_{10,3}^{*} )p_{23} (t) \hfill \\ \end{gathered} $$

(63)

which can be rewritten in a more compact form as follows

$$ \begin{gathered} p_{23} (t) = - \frac{{a_{10,3} p_{13} (t)}}{{a_{11,3} }} + \frac{{\left( {a_{10,2} - a_{10,1} } \right)p_{11} (t_{1} )}}{{a_{11,3} }} + \frac{{\left( {a_{11,2} - a_{11,1} } \right)p_{21} (t_{1} )}}{{a_{11,3} }} \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} + \frac{{\left( {a_{10,3} - a_{10,2} } \right)p_{12} (t_{2} )}}{{a_{11,3} }} + \frac{{\left( {a_{11,3} - a_{11,2} } \right)p_{22} (t_{2} )}}{{a_{11,3} }}, \hfill \\ \end{gathered} $$

(64)

with

$$ \begin{gathered} a_{10,3} = - \frac{1}{{2G_{L1,3}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} - \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,3}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }}\,{,}\,\,\, \hfill \\ \,a_{10,3}^{*} = \frac{1}{{2G_{L1,3}^{{}} }}\frac{{R_{1}^{2} R_{2}^{{}} }}{{R_{1}^{2} - R_{2}^{2} }} + \frac{{1 + \nu_{L1}^{{}} }}{{K_{L1,3}^{{}} }}\frac{{R_{2}^{3} }}{{R_{1}^{2} - R_{2}^{2} }}\,, \hfill \\ a_{11,3} = a_{10,3}^{*} + \frac{1}{{2G_{L2}^{{}} }}\frac{{R_{3}^{2} R_{2}^{{}} }}{{R_{2}^{2} - R_{3}^{2} }} + \frac{{1 + \nu_{L2}^{{}} }}{{K_{L2}^{{}} }}\frac{{R_{2}^{3} }}{{R_{2}^{2} - R_{3}^{2} }}, \hfill \\ \end{gathered} $$

(65)

Substituting Eq. (64) in Eq. (61) and after some developments, one deduces the following supporting pressure of the first liner

$$ \begin{gathered} p_{13} \left( t \right) = \frac{{R_{1} a_{11,3} G_{M} }}{{2G_{M} (a_{00,3} a_{11,3} - a_{01,3} a_{10,3} ) - R_{1} a_{11,3} }}\int_{{t_{2} }}^{t} {(\frac{1}{{\eta_{M} }} + \frac{{\exp ( - G_{K} (t - \tau )/\eta_{K} )}}{{\eta_{K} }})p_{13} (\tau )d\tau } \hfill \\ \begin{array}{*{20}c} {} & {} & {} \\ \end{array} + \frac{{2G_{M} (a_{00,3} a_{11,3} - a_{01,3} a_{10,3} )}}{{2G_{M} (a_{00,3} a_{11,3} - a_{01,3} a_{10,3} ) - R_{1} a_{11,3} }}f_{3}^{*} (t), \hfill \\ \end{gathered} $$

(66)

with

$$ \begin{gathered} \begin{array}{*{20}c} {} & {f_{3}^{*} \left( t \right) = } & { - \frac{{\left( {a_{00,1} - a_{00,2} } \right)a_{11,3} + \left( {a_{10,2} - a_{10,1} } \right)a_{01,3} }}{{\left( {a_{00,3} a_{11,3} - a_{01,3} a_{10,3} } \right)}}} \\ \end{array} p_{11} (t_{1} ) \hfill \\ - \frac{{\left( {a_{01,1} - a_{01,2} } \right)a_{11,3} + \left( {a_{11,2} - a_{11,1} } \right)a_{01,3} }}{{\left( {a_{00,3} a_{11,3} - a_{01,3} a_{10,3} } \right)}}p_{21} (t_{1} ) \hfill \\ - \frac{{\left( {a_{00,2} - a_{00,3} } \right)a_{11,3} + \left( {a_{10,3} - a_{10,2} } \right)a_{01,3} }}{{\left( {a_{00,3} a_{11,3} - a_{01,3} a_{10,3} } \right)}}p_{12} (t_{2} ) \hfill \\ - \frac{{\left( {a_{01,2} - a_{01,3} } \right)a_{11,3} + \left( {a_{11,3} - a_{11,2} } \right)a_{01,3} }}{{\left( {a_{00,3} a_{11,3} - a_{01,3} a_{10,3} } \right)}}p_{22} (t_{2} ) \hfill \\ + \frac{{R_{1} a_{11,3} }}{{2(a_{00,3} a_{11,3} - a_{01,3} a_{10,3} )}}\left( {p_{0}^{h} \int\limits_{0}^{{t_{0} }} {\chi (\tau )} H(t_{0} - \tau )d\tau - p_{0}^{h} \int\limits_{0}^{t} {\chi (\tau )} H(t - \tau )d\tau } \right) \hfill \\ + \frac{{R_{1} a_{11,3} }}{{2(a_{00,3} a_{11,3} - a_{01,3} a_{10,3} )}}\left( {\int\limits_{{t_{0} }}^{{t_{1} }} {p_{11} (\tau )H(t - \tau )d\tau } + \int\limits_{{t_{1} }}^{{t_{2} }} {p_{12} (\tau )H(t - \tau )d\tau } } \right) \hfill \\ \end{gathered} $$

(67)

Noting

$$ e_{1,3} = \frac{{R_{1} a_{11,3} G_{M} }}{{2G_{M} \left( {a_{00,3} a_{11,3} - a_{01,3} a_{10,3} } \right) - R_{1} a_{11,3} }} $$

(68)

Thus Eq. (66) becomes

$$ p_{13} \left( t \right) = f_{3} (t) + \int_{{t_{1} }}^{t} {\left( {\frac{{e_{1,3} }}{{\eta_{M} }} + \frac{{e_{1,3} }}{{\eta_{K} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)p_{13} \left( \tau \right)d\tau } , $$

(69)

with

$$ f_{3} (t) = \frac{{2G_{M} (a_{00,3} a_{11,3} - a_{01,3} a_{10,3} )}}{{2G_{M} (a_{00,3} a_{11,3} - a_{01,3} a_{10,3} ) - R_{1} a_{11,3} }}f_{3}^{*} (t) $$

(70)

We has now the standard integral equation (69) with the free term $f_{3}^{{}} (t)$ and the Kernel $\left( {\frac{{e_{1,3} }}{{\eta_{M} }} + \frac{{e_{1,3} }}{{\eta_{K} }}\exp \left( { - \frac{{G_{K} }}{{\eta_{K} }}(t - \tau )} \right)} \right)$, whereas the corresponding parameters $E{}_{1},E{}_{2},\lambda_{1} ,\lambda_{2}$ are respectively equal to $E_{1} = - \frac{{e_{1,3} }}{{\eta_{M} }},\,\,\,\,\,E_{2} = - \frac{{e_{1,3} }}{{\eta_{K} }},$$\lambda_{1} = 0,$ $\lambda_{2} = - \frac{{G_{K} }}{{\eta_{K} }}$.

The solution $p_{13}^{{}} (t)$ takes the same form of Eq. (24), i.e.

$$ p_{13}^{{}} (t) = f_{3}^{{}} (t) + \int_{{t_{2} }}^{t} {\left( {F_{1,3} e^{{\mu_{1,3} (t - \tau )}} + F_{2,3} e^{{\mu_{2,3} (t - \tau )}} } \right)} f_{3}^{{}} (\tau )d\tau , $$

(71)

with

$$ \begin{gathered} \begin{array}{*{20}c} {F_{1,3} = E_{1} \frac{{\mu_{1,3} - \lambda_{2} }}{{\mu_{2,3} - \mu_{1,3} }} + E_{2} \frac{{\mu_{1,3} - \lambda_{1} }}{{\mu_{2,3} - \mu_{1,3} }},} & {F_{2,3} = E_{1} \frac{{\mu_{2,3} - \lambda_{2} }}{{\mu_{1,3} - \mu_{2,3} }} + E_{2} \frac{{\mu_{2,3} - \lambda_{1} }}{{\mu_{1,3} - \mu_{2,3} }},} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\mu_{1,3} = \frac{{\lambda_{1} + \lambda_{2} - E_{1} - E_{2} + \sqrt {D_{a} } }}{2},} & {\mu_{2,3} = \frac{{\lambda_{1} + \lambda_{2} - E_{1} - E_{2} - \sqrt {D_{a} } }}{2},} \\ \end{array} \hfill \\ D_{a} = \left( {E_{1} E_{2} - \lambda_{1} - \lambda_{2} } \right)^{2} - 4\left( {\lambda_{1} \lambda_{2} - E_{1} \lambda_{2} - E_{2} \lambda_{1} } \right) \hfill \\ \end{gathered} $$

(72)

1.1 Determination of the unknown time instants t ₁ and t ₂

Following the definition of the tri-linear elastic behavior of the compressible liner (Eq. (3)), the two unknown time parameters t₁ and t₂ correspond to the instant at which the elastic behavior of the first liner changes from the first to the second slope and from the second to the third slope, respectively. Correspondingly, these two-time instants are calculated from the following condition

$$ \varepsilon_{v} (t) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{\lim 1} } & {if} & {t = t_{1} ,} \\ {\varepsilon_{\lim 2} } & {if} & {t = t_{2} ,} \\ \end{array} } \right. $$

(73)

Using Eq. (18), the first time instant t₁ can be determined from the following equation

$$ \varepsilon_{v1} (t_{1} ) = \frac{{2\left( {1 + \nu_{c} } \right)\left( {R_{2}^{2} p_{21} (t_{1} ) - R_{1}^{2} p_{11} (t_{1} )} \right)}}{{K_{c1} (R_{1}^{2} - R_{2}^{2} )}} = \varepsilon_{\lim 1} , $$

(74)

whilst Eq. (19) allows deriving the second time instant t₂ from the following equation

$$ \varepsilon_{v2} (t_{2} ) = \varepsilon_{\lim 1} + \frac{{2\left( {1 + \nu_{c} } \right)\left( {R_{2}^{2} \left( {p_{22} (t_{2} ) - p_{21} (t_{1} )} \right) - R_{1}^{2} \left( {p_{12} (t_{2} ) - p_{11} (t_{1} )} \right)} \right)}}{{K_{c2} (R_{1}^{2} - R_{2}^{2} )}} = \varepsilon_{\lim 2} , $$

(75)

Regarding the formula of p₁₁(t), p₂₁(t), one can state that the equation to determine t₁ (Eq. (74)) is nonlinear. It is also the case for the determination of the second time instant t₂ by substituting the expressions of p₁₁(t), p₂₁(t), p₁₂(t), p₂₂(t) in Eqs. ((45), (40), (57), (52)) into Eq. (75). The numerical solution of these nonlinear equations can be conducted without difficulty by using one of the common solvers (Matlab, Mathematical, Excel, etc.).

Figure 12 summarizes the principal steps to derive the tunnel convergence and stress state in the concrete liner.

Fig. 12

Flowchart of the closed-form solution of tunnel convergence and stress state in the concrete liner

Appendix B: Kriging metamodeling technique

The principal idea of the Kriging metamodeling technique lies on the approximation of the LSF by a surrogate (a Gaussian process) (Matheron 1973; Echard et al. 2011; Bichon et al. 2011) written in the form:

$$ G({\text{X}}) \approx \overline{G} ({\text{X}}) = {\text{k}}({\text{X}})^{T} {\upbeta } + Z({\text{X}}) = \sum\limits_{i}^{n} {\beta_{i} k_{i} ({\text{X}}) + Z({\text{X}})} $$

(76)

In Eq. (76), β and k(X) are the vectors of the regression coefficient and the vector of basis-functions of n elements while the first term k(X)^Tβ represents the mean value of the Gaussian process (i.e., the trend of the process). The second term Z(X) is assumed to have the zero-mean stationary Gaussian process and variance $C_{{ZZ}} ({\mathbf{X}},{\mathbf{X}}') = \sigma _{Z}^{2} R({\mathbf{\theta }},{\mathbf{X}},{\mathbf{X}}^\prime )$, with $\sigma_{Z}^{2}$ constant process variance, and the kernel function R(θ, X, X’) presents the prescribed auto-correlation function with respect to the hyperparameter vector ${{\varvec{\uptheta}}}$. Among the different functions proposed in the literature, the following squared exponential auto-correlation (Gaussian) function is selected:

$$ R\left( {{{\varvec{\uptheta}}};{\text{X}},{\text{ X}}^{\prime}} \right) = \prod\limits_{i = 1}^{d} {\exp \left[ { - \theta_{i} \left( {X_{i}^{{}} - X_{i}^{^{\prime}} } \right)^{2} } \right]} $$

(77)

where d is the dimension of the random variable vector X, and X_i, θ_i (i = 1, …, d) is the i-th component of X and ${{\varvec{\uptheta}}}$.

The necessary unknown parameters $\sigma_{Z}^{2} ,{{\varvec{\upbeta}}},{{\varvec{\uptheta}}}$ to construct the Kriging metamodel $\overline{G} ({\text{X}})$ in Eq. (76) can be calibrated from an optimization process by using the exact results (gathered in a vector y) of the performance function $G({\text{X}})$ that are evaluated at different observation points (called the training points). Initially, the data of these training points (gathered in a matrix S) can be generated from the Latin Hypercube Sampling (LHS) technique. Then, the construction step of these data (or the matrix S), also known as the Design of Experiment (DoE), is iteratively updated by adding the new training points by using a so-called learning function. Meanwhile, the Kriging metamodel $\overline{G} ({\text{X}})$ is built in an iterative manner until a convergence criterion is satisfied. Note that the calibration of the hyperparameters $\sigma_{Z}^{2} ,{{\varvec{\upbeta}}},{{\varvec{\uptheta}}}$ (i.e., the optimization process) was successfully implemented in many software. In our study here, the well-known MATLAB toolbox DACE developed by (Lophaven et al. 2002) is chosen.

Based on the constructed metamodel $\overline{G} ({\text{X}})$, the performance function can be predicted for any realization of the random input vector X, which follows the Normal distribution. More precisely, Kriging predictor provides the following mean and variance values $\left( {\mu_{{\overline{G} }} ({\text{X}}),\sigma_{{\overline{G} }}^{2} ({\text{X}})} \right)$(Lophaven et al. 2002; Gaspar et al. 2015):

$$ \begin{gathered} \mu_{{\overline{G} }} ({\text{X}}) = {\text{k}}({\text{X}})^{T} \overline{{\upbeta }} {\text{ + r}}({\text{X}})^{T} {\mathbf{R}}^{ - 1} ({\text{y - }}{\mathbf{K}}\overline{{\upbeta }} ) \hfill \\ \sigma_{{\overline{G} }}^{2} ({\text{X}}) = \sigma_{Z}^{2} \left( {1 - {\text{r}}({\text{X}})^{T} {\mathbf{R}}^{ - 1} {\text{r}}({\text{X}}) + {\text{u}}({\text{X}})^{T} ({\mathbf{K}}^{T} {\mathbf{R}}^{ - 1} {\mathbf{K}})^{ - 1} {\text{u}}({\text{X}})} \right) \hfill \\ \end{gathered} $$

(78)

with:

$$ \begin{array}{*{20}c} {\overline{{\upbeta }} = ({\text{K}}^{T} {\mathbf{R}}^{ - 1} {\text{K}})^{ - 1} {\text{K}}^{T} {\mathbf{R}}^{ - 1} {\text{y,}}} & {{\text{u}}({\text{X}}) = } \\ \end{array} {\text{K}}^{T} {\mathbf{R}}^{ - 1} {\text{r}}({\text{X}}) - {\text{k}}({\text{X}}) $$

(79)

In Eqs. (78) and (79), R is the matrix of correlation between each pair of N_DoE training points of DoE while K and r(X) are the regression matrix and the vector of cross-correlation between the prediction point X and each training point of DoE:

$$ \begin{gathered} \begin{array}{*{20}c} {y_{i} = G({\mathbf{X}}^{(i)} ),} & {i = 1,...,N_{DoE} } \\ \end{array} \hfill \\ r_{i} ({\mathbf{X}}){\mathbf{ = }}R({{\varvec{\uptheta}}},{\mathbf{X}},{\mathbf{X}}^{(i)} ), \, i = 1,...,N_{DoE} \hfill \\ \begin{array}{*{20}c} {R_{ij} = R({{\varvec{\uptheta}}},{\mathbf{X}}^{(i)} ,{\mathbf{X}}^{(j)} ),} & {i,j = 1,...,N_{DoE} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {K_{ij} = k_{j} ({\mathbf{X}}^{(i)} ),} & {i = 1,...,N_{DoE} ,} & {j = 1,...,n} \\ \end{array} \hfill \\ \end{gathered} $$

(80)

The interpolation (by using Kriging predictor $\overline{G} ({\text{X}})$) with respect to N_MCS random samples generated from the sampling MCS technique allows to estimate the failure probability:

$$ \begin{array}{*{20}c} {P_{f} \approx \frac{1}{{N_{MCS} }}\sum\limits_{i = 1}^{{N_{MCS} }} {{\mathbf{I}}\left( {\overline{G} ({\text{X}}^{(i)} )} \right)} ,} & {{\mathbf{I}}\left( {\overline{G} ({\text{X}}^{(i)} )} \right) = \left\{ {\begin{array}{*{20}c} 1 & {if} & {\mu_{{\overline{G} }} ({\text{X}}^{(i)} ) \le 0} \\ 0 & {if} & {\mu_{{\overline{G} }} ({\text{X}}^{(i)} ) > 0} \\ \end{array} } \right.} \\ \end{array} $$

(81)

For the sake of clarity, Table

Table 3 Principal steps for the Kriging-based reliability analysis

3 is summarized the essential steps to conduct an iterative Kriging-based reliability analysis. Follow that, it is necessary to define a stopping criterion to verify the convergence of the process (step 6) and an appropriate learning function to find out the best new training points X^* to enrich the DoE and hence the Kriging metamodel (step 7). The following stopping criterion proposed by (Gaspar et al. 2015) is chosen:

$$ \begin{array}{*{20}c} {\frac{{\left| {P_{f}^{(i)} - P_{f}^{(1)} } \right|}}{{P_{f}^{(1)} }} \le \gamma ,} & {\forall i \in \left\{ {2,...,N_{\gamma } } \right\}} \\ \end{array} $$

(82)

where P_f⁽¹⁾ is the chosen reference value that is used to detect the stabilization of failure probability whilst P_f⁽ⁱ⁾ (i = 2, …, N_γ) are the failure probability in the next (N_γ−1) iterations. The parameters = γ0.01 and N_γ = 10 are sufficient to ensure the accuracy of the obtained probability of failure.

To enrich the DoE, we base on the U-learning function, initially proposed in the well-known AK-MCS method of (Echard et al. 2011), to choose the appropriate new training points. In fact, this function, defined as

$$ U({\text{X}}) = \frac{{\left| {\mu_{{\overline{G} }} ({\text{X}})} \right|}}{{\sigma_{{\overline{G} }} ({\text{X}})}} $$

(83)

That measures the probability of making a mistake on the sign of $G({\text{X}})$, by substituting $G({\text{X}})$ with $\overline{G} ({\text{X}})$. Thus, according to (Echard et al. 2011), the best choice of new training points X^*, are the samples that give the lowest U-value among N_MCS samples (i.e., X^* = argmin{U(X)}).