Convergence-confinement analysis for tunnels with combined bolt–cable system considering the effects of intermediate principal stress

Abstract

The mechanical responses of tunnels reinforced by combined bolt–cable system are investigated in this paper. An analytical model of the convergence-confinement type is proposed that accounts for the sequential installation of finite-length fully grouted bolts and cables, as well as the effects of intermediate principal stress. The rock mass is assumed to be strain-softening material, obeying the unified strength theory or the Drucker–Prager criterion. The analytical model is divided into three stages including before bolt installation, bolt–rock interaction and after cable installation, of which the latter two stages are the focus of this study. According to the distribution and extent of the plastic zone and the relative bolt–cable lengths, six forms during the bolt–rock interaction and ten forms after cable installation are categorized and solved. Using the critical support pressures for the transitions of different forms, the modified whole-process ground reaction curve is obtained and validated by 2D numerical simulations and an existing model. Furthermore, the fictitious pressure is introduced to quantify the three-dimensional space effect of the tunnel face, whose solution is given by combining the proposed ground reaction curve and the longitudinal displacement profile proposed by Hoek. By using the strain compatibility conditions of bolts, cables and rock, the longitudinal tunnel displacement, stresses and plastic radii considering the sequential installation of bolts and cables are obtained. The results of the proposed model agree well with the 3D numerical simulations. Parametric analyses are conducted to investigate the effect of intermediate principal stress on the tunnel responses by different criteria. The proposed model provides a convenient alternative tool for the combined bolt–cable design of tunnels at the preliminary stage.

Appendices

Appendix A. Solutions for the GRCs of bolt-supported tunnels

The elastic modulus of bolt-reinforced zone is set as $E_{g}$. The circumferential space, longitudinal space and uniform length of bolts are set as $S_{cb}$, $S_{lb}$ and $L_{b}$. The elastic modulus, cross-sectional area and pretension load of a single bolt are set as $E_{b}$, $A_{b}$ and $F_{b}$, respectively.

A.1 The solution for Form A1

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(36)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{1 + \mu }{{\alpha E_{g} }}r_{0} \left[ {\left( {1 - 2\mu } \right)\left( {p_{{r_{b} }} r_{b}^{2} - p_{i} r_{0}^{2} } \right) - \left( {p_{i} - p_{{r_{b} }} } \right)r_{b}^{2} } \right] $$

(37)

where $p_{{r_{b} }}$ is the radial stress at the interface of the bolt-reinforced zone and the natural zone ($r = r_{b}$), which can be obtained by

$$ p_{{r_{b} }} = \frac{{2\left[ {E\left( {1 - \mu } \right)p_{i} r_{0}^{2} + \alpha E_{g} \left( {1 - \mu } \right)p_{0} } \right]}}{{\alpha E_{g} + E\left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{0}^{2} } \right]}} $$

(38)

where $\alpha = r_{b}^{2} - r_{0}^{2}$.

A.2 The solution for Form A2

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(39)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{Dr_{0} }}{h + 1}\left[ {\left( {h - 1} \right) + 2\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right] $$

(40)

where $D = \frac{{\left( {1 + \mu } \right)\left[ {p_{0} \left( {\xi_{0} - 1} \right) + \sigma_{p} } \right]}}{{E\left( {1 + \xi_{0} } \right)}}$.

The softening zone radius $r_{1}$ required in Eq. (40) can be obtained from the equation:

$$ \begin{gathered} \frac{1}{{1 - \xi_{0} }}\left( {B - \sigma_{p} - \frac{{2DE^{\prime}}}{h + 1}} \right) + \left[ {p_{i} - \frac{B}{{1 - \xi_{0} }} + \frac{A}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} \hfill \\ = \frac{2M}{{1 - \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{{1 - \xi_{0} }} + \frac{1}{{h + \xi_{0} }}\left( {A + \frac{{2DE^{\prime}}}{h + 1}} \right)\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} \hfill \\ \end{gathered} $$

(41)

where $A = \frac{{2Dh\xi_{0} E_{b} A_{b} - 2S_{bc} S_{lc} DE^{\prime}}}{{\left( {1 + h} \right)S_{bc} S_{lc} }}$, $B = \sigma_{p} - A - \frac{{\xi_{0} }}{{S_{bc} S_{lc} }}\left( {DE_{b} A_{b} + F_{b} } \right)$,.$M = \frac{{\left( {1 - \xi_{0} } \right)p_{0} - \sigma_{p} }}{{1 + \xi_{0} }} - \frac{{DE^{\prime}}}{{h + \xi_{0} }}$.

A.3 The solution for Form A3

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(42)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{\left( {1 + \mu } \right)r_{1} }}{{E_{g} \beta }}\left\{ {\left[ {\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} + \left( {p_{{r_{b} }} r_{b}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)\left( {1 - 2\mu } \right)} \right] + \frac{{2\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} }}{h + 1}\left[ {\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h} - 1} \right]} \right\} $$

(43)

where $\beta = r_{b}^{2} - r_{1}^{2}$.

The softening zone radius $r_{1}$, the radial stresses $p_{{r_{1} }}$ at $r = r_{1}$ and $p_{{r_{b} }}$ at $r = r_{b}$ can be obtained from the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{1} }} = \frac{B}{{1 - \xi_{0} }} - \frac{A}{{h + \xi_{0} }} + \left[ {p_{i} - \frac{B}{{1 - \xi_{0} }} + \frac{A}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} \hfill \\ p_{{r_{b} }} = \frac{{2\left[ {E\left( {1 - \mu } \right)p_{{r_{1} }} r_{1}^{2} + \alpha E_{g} \left( {1 - \mu } \right)p_{0} } \right]}}{{\beta E_{g} + E\left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{1}^{2} } \right]}} \hfill \\ \left[ {p_{i} - \frac{B}{{1 - \xi_{0} }} + \frac{A}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} + \frac{B}{{1 - \xi_{0} }} = \frac{1}{{1 + \xi_{0} }}\left[ {\frac{{2\left( {p_{{r_{b} }} r_{b}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)}}{\beta } - A - B} \right] + \frac{A}{{h + \xi_{0} }} \hfill \\ \end{gathered} \right. $$

(44)

A.4 The solution for Form A4

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{2} + dr}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(45)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{2Dr_{0} }}{f + 1}t^{h + 1} \left[ {\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} + \frac{{D\left( {f - h} \right)}}{{\left( {f + 1} \right)\left( {h + 1} \right)}}} \right] + \frac{{Dr_{0} \left( {h - 1} \right)}}{h + 1} $$

(46)

where $t = {{r_{1} } \mathord{\left/ {\vphantom {{r_{1} } {r_{2} }}} \right. \kern-\nulldelimiterspace} {r_{2} }}$.

The softening zone radius $r_{1}$ and the residual zone radius $r_{2}$ required in Eq. (46) can be obtained from the following equation set:

$$ \left\{ \begin{gathered} \left[ {p_{i} - \frac{{B^{\prime}}}{{1 - \xi_{r} }} + \frac{{A^{\prime}}}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} = \frac{{\sigma_{pr} - B^{\prime}}}{{1 - \xi_{r} }} + \left( {p_{{r_{2} }} - \frac{{\sigma_{pr} }}{{1 - \xi_{r} }}} \right)\left( {\frac{{r_{b} }}{{r_{2} }}} \right)^{{\xi_{r} - 1}} + \frac{{A^{\prime}}}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{b} }}} \right)^{f + 1} \hfill \\ p_{{r_{2} }} = \frac{2M}{{1 - \xi_{0} }}t^{{1 - \xi_{0} }} + \frac{{\sigma_{p} }}{{1 - \xi_{0} }} + \frac{{2DE^{\prime}}}{h + 1}\left[ {\frac{1}{{h + \xi_{0} }}t^{h + 1} + \frac{1}{{1 - \xi_{0} }}} \right] \hfill \\ r_{1} = r_{2} \left[ {\left( {\sigma_{p} - \sigma_{pr} } \right)\frac{h + 1}{{2DE^{\prime}}} + 1} \right]^{{\frac{1}{h + 1}}} \hfill \\ \end{gathered} \right. $$

(47)

where $A^{\prime} = \frac{{2Df\xi_{r} E_{b} A_{b} }}{{\left( {f + 1} \right)S_{cb} S_{lb} }}t^{h + 1}$, $B^{\prime} = \sigma_{pr} - \frac{D}{h + 1}\left[ {3h - 1 + \frac{2f - 2hf - 4h}{{f + 1}}t^{h + 1} } \right] - \xi_{r} \frac{{E_{b} A_{b} }}{{S_{cb} S_{lb} }}$.

A.5 The solution for Form A5

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{2} + dr}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - dr}} } \right. = u_{r} \left| {_{{r = r_{2} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(48)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{2Dr_{0} }}{f + 1}t^{h + 1} \left[ {\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} + \frac{{D\left( {f - h} \right)}}{{\left( {f + 1} \right)\left( {h + 1} \right)}}} \right] + \frac{{Dr_{0} \left( {h - 1} \right)}}{h + 1} $$

(49)

The variables $r_{1}$ and $r_{2}$ required in Eq. (49) and the radial stresses $p_{{r_{b} }}$ at $r = r_{b}$ and $p_{{r_{2} }}$ at $r = r_{2}$ can be obtained from the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{2} }} = \frac{{B^{\prime}}}{{1 - \xi_{r} }} - \frac{{A^{\prime}}}{{f + \xi_{0} }} + \left[ {m^{\prime} + \frac{{A^{\prime}}}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ t^{h + 1} = \frac{1}{A}\left[ {p_{{r_{2} }} \left( {\xi_{r} - \xi_{0} } \right) + B^{\prime} - B + A^{\prime}} \right] \hfill \\ p_{{r_{b} }} = \frac{B}{{1 - \xi_{0} }} - \frac{A}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} + \left[ {p_{{r_{2} }} - \frac{B}{{1 - \xi_{0} }} + \frac{A}{{h + \xi_{0} }}t^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{2} }}} \right)^{{\xi_{0} - 1}} \hfill \\ \frac{1}{{1 - \xi_{0} }}\left( {B - \sigma_{p} - \frac{{2DE^{\prime}}}{h + 1}} \right) + \left[ {p_{{r_{2} }} - \frac{B}{{1 - \xi_{0} }} + \frac{A}{{h + \xi_{0} }}t^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{2} }}} \right)^{{\xi_{0} - 1}} \hfill \\ = \frac{2M}{{1 - \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{{1 - \xi_{0} }} + \frac{1}{{h + \xi_{0} }}\left( {\frac{{2DE^{\prime}}}{h + 1} + A} \right)\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} \hfill \\ \end{gathered} \right. $$

(50)

A.6 The solution for Form A6

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - {\text{d}}r}} } \right. = \sigma_{r} \left| {_{{r = r_{b} {\text{ + d}}r}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - {\text{d}}r}} } \right. = \sigma_{r} \left| {_{{r = r_{2} {\text{ + d}}r}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - {\text{d}}r}} } \right. = u_{r} \left| {_{{r = r_{1} {\text{ + d}}r}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - {\text{d}}r}} } \right. = u_{r} \left| {_{{r = r_{2} {\text{ + d}}r}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - {\text{d}}r}} } \right. = u_{r} \left| {_{{r = r_{b} {\text{ + d}}r}} } \right. \hfill \\ \end{gathered} \right. $$

(51)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{u_{{r_{1} }} f}}{f + 1} - M\left[ {\left( {h - f} \right)t^{h + 1} + 1} \right] + \left\{ {\frac{{u_{{r_{1} }} }}{f + 1} + M\left[ {\left( {2f - h + 1} \right)t^{h + 1} + 1} \right] - \left( {f + 1} \right)r_{1} - r_{2} } \right\}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f} $$

(52)

where $M = \frac{{2\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} r_{2} }}{{E_{g} \beta \left( {h + 1} \right)\left( {f + 1} \right)}}$.

The radial displacement $u_{{r_{1} }}$ at $r = r_{1}$ can be obtained by

$$ u_{{r_{1} }} = \frac{1 + \mu }{{\beta E_{g} }}r_{1} \left[ {\left( {1 - 2\mu } \right)\left( {p_{{r_{b} }} r_{b}^{2} - p_{{r_{1} }} r_{1}^{2} } \right) - \left( {p_{{r_{1} }} - p_{{r_{b} }} } \right)r_{b}^{2} } \right] $$

(53)

The variables $r_{1}$, $r_{2}$, $p_{{r_{b} }}$ and $p_{{r_{1} }}$ required in Eqs. (52) and (53) and the radial stress $p_{{r_{2} }}$ at $r = r_{2}$ can be obtained from the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{b} }} = \frac{{2\left[ {E\left( {1 - \mu } \right)p_{{r_{1} }} r_{1}^{2} + \alpha E_{g} \left( {1 - \mu } \right)p_{0} } \right]}}{{\beta E_{g} + E\left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{1}^{2} } \right]}} \hfill \\ p_{{r_{2} }} = \frac{{B^{\prime}}}{{1 - \xi_{r} }} - \frac{{A^{\prime}}}{{f + \xi_{r} }} + \left[ {p_{i} - \frac{{B^{\prime}}}{{1 - \xi_{r} }} + \frac{{A^{\prime}}}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ t^{h + 1} = \frac{1}{A}\left[ {p_{{r_{2} }} \left( {\xi_{r} - \xi_{0} } \right) + B^{\prime} - B + A^{\prime}} \right] \hfill \\ p_{{r_{1} }} = \frac{B}{{1 - \xi_{0} }}\left( {1 - t^{{\xi_{0} - 1}} } \right) + \frac{A}{{h + \xi_{0} }}\left( {t^{h + 1} - 1} \right) + p_{{r_{2} }} t^{{\xi_{0} - 1}} \hfill \\ r_{1} = \left\{ {\frac{{p_{{r_{b} }} r_{b}^{2} }}{{p_{{r_{1} }} }} - \frac{1}{2}\left[ {\left( {1 + \xi_{0} } \right) + \frac{A + B}{{p_{{r_{1} }} }}} \right]\beta } \right\}^{0.5} \hfill \\ \end{gathered} \right. $$

(54)

Appendix B. Solutions for the GRCs of tunnels supported by combined bolt–cable system

The elastic modulus of bolt–cable-reinforced zone and the cable-only-reinforced zone are set as $E_{s}$ and $E_{d}$, respectively. The circumferential space, longitudinal space and uniform length of cables are set as $S_{cc}$, $S_{lc}$ and $L_{c}$. The elastic modulus, cross-sectional area and pretension load of a single cable are set as $E_{c}$, $A_{c}$ and $F_{c}$, respectively. The other symbols are consistent with Appendix A.

B.1 The solution for Form B1

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{c} - dr}} } \right. = u_{r} \left| {_{{r = r_{c} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(55)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{1 + \mu }{{\alpha E_{s} }}r_{0} \left[ {\left( {1 - 2\mu } \right)\left( {p_{{r_{b} }} r_{b}^{2} - p_{i} r_{0}^{2} } \right) - \left( {p_{i} - p_{{r_{b} }} } \right)r_{b}^{2} } \right] $$

(56)

where $p_{{r_{b} }}$ and $p_{{r_{c} }}$ are the radial stress at $r = r_{b}$ and $r = r_{c}$, respectively, which can be obtained by the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{b} }} = \frac{{2\left( {1 - \mu } \right)\left[ {\alpha E_{s} p_{{r_{b} }} r_{c}^{2} + \eta E_{d} p_{i} r_{0}^{2} } \right]}}{{\alpha E_{s} \left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{c}^{2} } \right] + \eta E_{d} \left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{0}^{2} } \right]}} \hfill \\ p_{{r_{c} }} = \frac{{2\left[ {E\left( {1 - \mu } \right)p_{{r_{b} }} r_{b}^{2} + \eta E_{d} \left( {1 - \mu } \right)p_{0} } \right]}}{{\eta E_{d} + E\left[ {\left( {1 - 2\mu } \right)r_{c}^{2} + r_{b}^{2} } \right]}} \hfill \\ \end{gathered} \right. $$

(57)

where $\eta = r_{c}^{2} - r_{b}^{2}$.

B.2 The solution for Form B2

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(58)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{Dr_{0} }}{h + 1}\left[ {\left( {h - 1} \right) + 2\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right] $$

(59)

The softening zone radius $r_{1}$ required in Eq. (59) and the radial stress $p_{{r_{b} }}$ at $r = r_{b}$ can be obtained from the equation:

$$ \left\{ \begin{gathered} p_{{r_{b} }} = \frac{{B_{2} }}{{1 - \xi_{0} }} - \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} + \left[ {p_{i} - \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} \hfill \\ \frac{1}{{1 - \xi_{0} }}\left( {B_{1} - \sigma_{p} - \frac{{2DE^{\prime}}}{h + 1}} \right) + \left[ {p_{{r_{b} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{c} }}{{r_{b} }}} \right)^{{\xi_{0} - 1}} \hfill \\ = \frac{2M}{{1 - \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{c} }}} \right)^{{1 - \xi_{0} }} + \frac{1}{{h + \xi_{0} }}\left( {A_{1} + \frac{{2DE^{\prime}}}{h + 1}} \right)\left( {\frac{{r_{1} }}{{r_{c} }}} \right)^{h + 1} \hfill \\ \end{gathered} \right. $$

(60)

where $A_{1} = \frac{{2D\left( {h\xi_{0} E_{c} A_{c} - S_{cc} S_{lc} E^{\prime}} \right)}}{{\left( {1 + h} \right)S_{cc} S_{lc} }}$, $B_{1} = \sigma_{p} - A_{1} - \frac{{\xi_{0} }}{{S_{cc} S_{lc} }}\left( {DE_{c} A_{c} + F_{c} } \right)$, $A_{2} = \frac{2D}{{1 + h}}\left[ {h\xi_{0} \left( {\frac{{E_{b} A_{b} }}{{S_{cb} S_{lb} }} + \frac{{E_{c} A_{c} }}{{S_{cc} S_{lc} }}} \right) - E^{\prime}} \right]$, $B_{2} = \sigma_{p} - A_{2} - \xi_{0} \left[ {D\left( {\frac{{E_{b} A_{b} }}{{S_{cb} S_{lb} }} + \frac{{E_{c} A_{c} }}{{S_{cc} S_{lc} }}} \right) + \frac{{F_{b} }}{{S_{cb} S_{lb} }} + \frac{{F_{c} }}{{S_{cc} S_{lc} }}} \right]$.

B.3 The solution for Form B3

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(61)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{\left( {1 + \mu } \right)r_{1} }}{{E_{d} \left( {\beta + \eta } \right)}}\left\{ {\left[ {\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} + \left( {p_{{r_{b} }} r_{b}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)\left( {1 - 2\mu } \right)} \right] + \frac{{2\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} }}{h + 1}\left[ {\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h} - 1} \right]} \right\} $$

(62)

The softening zone radius $r_{1}$, the radial stresses $p_{{r_{1} }}$, $p_{{r_{b} }}$ and $p_{{r_{c} }}$ can be obtained from the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{1} }} = \frac{{B_{1} }}{{1 - \xi_{0} }} - \frac{{A_{1} }}{{h + \xi_{0} }} + \left[ {p_{{r_{b} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{{\xi_{0} - 1}} \hfill \\ p_{{r_{c} }} = \frac{{2\left( {1 - \mu } \right)\left[ {Ep_{{r_{1} }} r_{1}^{2} + E_{d} \left( {\beta + \eta } \right)\left( {1 - \mu } \right)p_{0} } \right]}}{{\left( {\beta + \eta } \right)E_{d} + E\left[ {\left( {1 - 2\mu } \right)r_{c}^{2} + r_{1}^{2} } \right]}} \hfill \\ p_{{r_{b} }} = \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} + \left[ {p_{i} - \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} \hfill \\ \left[ {p_{{r_{b} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{{\xi_{0} - 1}} + \frac{{B_{1} }}{{1 - \xi_{0} }} = \frac{1}{{1 + \xi_{0} }}\left[ {\frac{{2\left( {p_{{r_{c} }} r_{c}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)}}{{r_{c}^{2} - r_{1}^{2} }} - A_{1} - B_{1} } \right] + \frac{{A_{1} }}{{h + \xi_{0} }} \hfill \\ \end{gathered} \right. $$

(63)

B.4 The solution for Form B4

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(64)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{\left( {1 + \mu } \right)r_{1} }}{{E_{s} \left( {\beta + \eta } \right)}}\left\{ {\left[ {\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} + \left( {p_{{r_{b} }} r_{b}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)\left( {1 - 2\mu } \right)} \right] + \frac{{2\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} }}{h + 1}\left[ {\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h} - 1} \right]} \right\} $$

(65)

The softening zone radius $r_{1}$, the radial stresses $p_{{r_{1} }}$, $p_{{r_{b} }}$ and $p_{{r_{c} }}$ can be obtained from the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{b} }} = \frac{{\beta p_{{r_{c} }} r_{c}^{2} + \eta p_{{r_{1} }} r_{1}^{2} }}{{\left( {\eta + \beta } \right)r_{b}^{2} }} \hfill \\ p_{{r_{1} }} = \frac{{B_{2} }}{{1 - \xi_{0} }} - \frac{{A_{2} }}{{h + \xi_{0} }} + \left[ {p_{i} - \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} \hfill \\ p_{{r_{c} }} = \frac{{2\left( {1 - \mu } \right)\left[ {Ep_{{r_{1} }} r_{1}^{2} + E_{d} \left( {\eta + \beta } \right)p_{0} } \right]}}{{\left( {\eta + \beta } \right)E_{d} + E\left[ {\left( {1 - 2\mu } \right)r_{c}^{2} + r_{1}^{2} } \right]}} \hfill \\ \left[ {p_{i} - \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{0} }}} \right)^{{\xi_{0} - 1}} + \frac{{B_{2} }}{{1 - \xi_{0} }} = \frac{1}{{1 + \xi_{0} }}\left[ {\frac{{2\left( {p_{{r_{c} }} r_{c}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)}}{\eta + \beta } - A_{2} - B_{2} } \right] + \frac{{A_{2} }}{{h + \xi_{0} }} \hfill \\ \end{gathered} \right. $$

(66)

B.5 The solution for Form B5

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{2} + dr}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(67)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{2Dr_{0} }}{f + 1}t^{h + 1} \left[ {\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} + \frac{{D\left( {f - h} \right)}}{{\left( {f + 1} \right)\left( {h + 1} \right)}}} \right] + \frac{{Dr_{0} \left( {h - 1} \right)}}{h + 1} $$

(68)

The variables $r_{1}$ and $r_{2}$ required in Eq. (68) can be obtained from the equation set:

$$ \left\{ \begin{gathered} \left[ {p_{{r_{b} }} - \frac{{B^{\prime}_{1} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{1} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{b} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{c} }}{{r_{b} }}} \right)^{{\xi_{r} - 1}} = \frac{{\sigma_{pr} - B^{\prime}_{1} }}{{1 - \xi_{r} }} + \left( {p_{{r_{2} }} - \frac{{\sigma_{pr} }}{{1 - \xi_{r} }}} \right)\left( {\frac{{r_{c} }}{{r_{2} }}} \right)^{{\xi_{r} - 1}} + \frac{{A^{\prime}_{1} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{c} }}} \right)^{f + 1} \hfill \\ p_{{r_{2} }} = \frac{2M}{{1 - \xi_{0} }}t^{{1 - \xi_{0} }} + \frac{{\sigma_{p} }}{{1 - \xi_{0} }} + \frac{{2DE^{\prime}}}{h + 1}\left[ {\frac{1}{{h + \xi_{0} }}t^{h + 1} + \frac{1}{{1 - \xi_{0} }}} \right] \hfill \\ r_{1} = r_{2} \left[ {\left( {\sigma_{p} - \sigma_{pr} } \right)\frac{h + 1}{{2DE^{\prime}}} + 1} \right]^{{\frac{1}{h + 1}}} \hfill \\ p_{{r_{b} }} = \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{b} }}} \right)^{f + 1} + \left[ {p_{i} - \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ \end{gathered} \right. $$

(69)

where $A^{\prime}_{1} = \frac{{2Df\xi_{r} E_{c} A_{c} t^{h + 1} }}{{\left( {1 + f} \right)S_{cc} S_{lc} }}$, $B^{\prime}_{1} = \sigma_{pr} - \frac{D}{h + 1}\left[ {3h - 1 + \frac{2f - 2hf - 4h}{{f + 1}}t^{h + 1} } \right] - \frac{{\xi_{r} E_{c} A_{c} }}{{S_{cc} S_{lc} }}$, $A^{\prime}_{2} = \frac{{2Df\xi_{r} t^{h + 1} }}{1 + f}\left( {\frac{{E_{b} A_{b} }}{{S_{cb} S_{lb} }} + \frac{{E_{c} A_{c} }}{{S_{cc} S_{lc} }}} \right)$, $B^{\prime}_{2} = \sigma_{pr} - \frac{D}{h + 1}\left[ {3h - 1 + \frac{2f - 2hf - 4h}{{f + 1}}t^{h + 1} } \right] - \xi_{r} \left( {\frac{{E_{b} A_{b} }}{{S_{cb} S_{lb} }} + \frac{{E_{c} A_{c} }}{{S_{cc} S_{lc} }}} \right)$.

B.6 The solution for Form B6

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - dr}} } \right. = u_{r} \left| {_{{r = r_{2} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(70)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{2Dr_{0} }}{f + 1}t^{h + 1} \left[ {\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} + \frac{{D\left( {f - h} \right)}}{{\left( {f + 1} \right)\left( {h + 1} \right)}}} \right] + \frac{{Dr_{0} \left( {h - 1} \right)}}{h + 1} $$

(71)

The variables $r_{1}$ and $r_{2}$ required in Eq. (71) can be obtained from the equation set:

$$ \left\{ \begin{gathered} p_{{r_{b} }} = \frac{{B_{2} }}{{1 - \xi_{0} }} - \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} + \left[ {p_{{r_{2} }} - \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}t^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{2} }}} \right)^{{\xi_{0} - 1}} \hfill \\ r_{2} = r_{1} \left[ {\frac{{p_{{r_{2} }} \left( {\xi_{r} - \xi } \right) - B_{2} + B^{\prime}_{2} }}{{A_{2} }}} \right]^{{\frac{1}{h + 1}}} \hfill \\ p_{{r_{2} }} = \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }} + \left[ {p_{i} - \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ \frac{1}{{1 - \xi_{0} }}\left( {B_{1} - \sigma_{p} - \frac{{2DE^{\prime}}}{h + 1}} \right) + \left[ {p_{{r_{b} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{c} }}{{r_{b} }}} \right)^{{\xi_{0} - 1}} \hfill \\ = \frac{2M}{{1 - \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{c} }}} \right)^{{1 - \xi_{0} }} + \frac{1}{{h + \xi_{0} }}\left( {\frac{{2DE^{\prime}}}{h + 1} + A_{1} } \right)\left( {\frac{{r_{1} }}{{r_{c} }}} \right)^{h + 1} \hfill \\ \end{gathered} \right. $$

(72)

B.7 The solution for Form B7

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - dr}} } \right. = u_{r} \left| {_{{r = r_{2} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(73)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{u_{{r_{1} }} f}}{f + 1} - M^{\prime}\left[ {\left( {h - f} \right)t^{h + 1} + 1} \right] + \left\{ {\frac{{u_{{r_{1} }} }}{f + 1} + M^{\prime}\left[ {\left( {2f - h + 1} \right)t^{h + 1} + 1} \right] - \left( {f + 1} \right)r_{1} - r_{2} } \right\}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f} $$

(74)

where $M^{\prime} = \frac{{2\left( {p_{{r_{b} }} - p_{{r_{1} }} } \right)r_{b}^{2} r_{2} }}{{E_{s} \beta \left( {h + 1} \right)\left( {f + 1} \right)}}$.

The radial displacement $u_{{r_{1} }}$ at $r = r_{1}$ can be obtained by

$$ u_{{r_{1} }} = \frac{1 + \mu }{{\beta E_{s} }}r_{1} \left[ {\left( {1 - 2\mu } \right)\left( {p_{{r_{b} }} r_{b}^{2} - p_{{r_{1} }} r_{1}^{2} } \right) - \left( {p_{{r_{1} }} - p_{{r_{b} }} } \right)r_{b}^{2} } \right] $$

(75)

The variables $r_{1}$, $r_{2}$, $p_{{r_{b} }}$ and $p_{{r_{1} }}$ required in Eqs. (74) and (75) and the radial stresses $p_{{r_{2} }}$ at $r = r_{2}$ and $p_{{r_{c} }}$ at $r = r_{c}$ can be obtained from the following equation set:

$$ \left\{ \begin{gathered} p_{{r_{b} }} = \frac{{2\left( {1 - \mu } \right)\left[ {\alpha E_{s} p_{{r_{b} }} r_{c}^{2} + \eta E_{d} p_{i} r_{0}^{2} } \right]}}{{\alpha E_{s} \left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{c}^{2} } \right] + \eta E_{d} \left[ {\left( {1 - 2\mu } \right)r_{b}^{2} + r_{0}^{2} } \right]}} \hfill \\ p_{{r_{c} }} = \frac{{2\left[ {E\left( {1 - \mu } \right)p_{{r_{b} }} r_{b}^{2} + \eta E_{d} \left( {1 - \mu } \right)p_{0} } \right]}}{{\eta E_{d} + E\left[ {\left( {1 - 2\mu } \right)r_{c}^{2} + r_{b}^{2} } \right]}} \hfill \\ p_{{r_{2} }} = \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} - \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }} + \left[ {p_{i} - \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ t^{h + 1} = \frac{1}{{A_{2} }}\left[ {p_{{r_{2} }} \left( {\xi_{r} - \xi_{0} } \right) + B^{\prime}_{2} - B_{2} + A^{\prime}_{2} } \right] \hfill \\ p_{{r_{1} }} = \frac{{B_{2} }}{{1 - \xi_{0} }}\left( {1 - t^{{\xi_{0} - 1}} } \right) + \frac{{A_{2} }}{{h + \xi_{0} }}\left( {t^{h + 1} - 1} \right) + p_{{r_{2} }} t^{{\xi_{0} - 1}} \hfill \\ r_{1} = \left\{ {\frac{{p_{{r_{b} }} r_{b}^{2} }}{{p_{{r_{1} }} }} - \frac{1}{2}\left[ {\left( {1 + \xi_{0} } \right) + \frac{{A_{2} + B_{2} }}{{p_{{r_{1} }} }}} \right]\beta } \right\}^{0.5} \hfill \\ \end{gathered} \right. $$

(76)

B.8 The solution for Form B8

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{2} + dr}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - dr}} } \right. = u_{r} \left| {_{{r = r_{2} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(77)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{2Dr_{0} }}{f + 1}t^{h + 1} \left[ {\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} + \frac{{D\left( {f - h} \right)}}{{\left( {f + 1} \right)\left( {h + 1} \right)}}} \right] + \frac{{Dr_{0} \left( {h - 1} \right)}}{h + 1} $$

(78)

The softening zone radius $r_{1}$ and the residual zone radius $r_{2}$ required in Eq. (78) can be obtained from the following equation set:

$$ \left\{ \begin{gathered} \frac{1}{{1 - \xi_{0} }}\left[ {B_{1} - \sigma_{p} - \frac{{2DE^{\prime}}}{h + 1}} \right] + \left( {p_{{r_{2} }} - \frac{{B_{1} }}{1 - \xi } + \frac{{A_{1} }}{{h + \xi_{0} }}t^{h + 1} } \right)\left( {\frac{{r_{c} }}{{r_{2} }}} \right)^{{\xi_{0} - 1}} \hfill \\ = \frac{2M}{{1 - \xi_{0} }}\left( {\frac{{r_{c} }}{{r_{1} }}} \right)^{{\xi_{0} - 1}} + \frac{1}{{h + \xi_{0} }}\left( {\frac{{2DE^{\prime}}}{h + 1} + A_{1} } \right)\left( {\frac{{r_{1} }}{{r_{c} }}} \right)^{h + 1} \hfill \\ p_{{r_{2} }} = \frac{{B^{\prime}_{1} }}{{1 - \xi_{r} }} - \frac{{A^{\prime}_{1} }}{{f + \xi_{r} }} + \left[ {p_{{r_{b} }} - \frac{{B^{\prime}_{1} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{1} }}{{f + \xi_{r} }}\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ p_{{r_{b} }} = \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} - \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{b} }}} \right)^{f + 1} + \left[ {p_{i} - \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ r_{1} = r_{2} \left[ {\left( {\sigma_{p} - \sigma_{pr} } \right)\frac{h + 1}{{2DE^{\prime}}} + 1} \right]^{{\frac{1}{h + 1}}} \hfill \\ \end{gathered} \right. $$

(79)

B.9 The solution for Form B9

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{2} + dr}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - dr}} } \right. = u_{r} \left| {_{{r = r_{2} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(80)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{u_{{r_{1} }} f}}{f + 1} - M^{\prime\prime}\left[ {\left( {h - f} \right)t^{h + 1} + 1} \right] + \left\{ {\frac{{u_{{r_{1} }} }}{f + 1} + M^{\prime\prime}\left[ {\left( {2f - h + 1} \right)t^{h + 1} + 1} \right] - \left( {f + 1} \right)r_{1} - r_{2} } \right\}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f} $$

(81)

where $M^{\prime\prime} = \frac{{2\left( {p_{{r_{c} }} - p_{{r_{1} }} } \right)r_{c}^{2} r_{2} }}{{E_{d} \left( {\beta + \eta } \right)\left( {h + 1} \right)\left( {f + 1} \right)}}$.

The radial displacement $u_{{r_{1} }}$ at $r = r_{1}$ can be obtained by

$$ u_{{r_{1} }} = \frac{1 + \mu }{{\left( {\beta + \eta } \right)E_{d} }}r_{1} \left[ {\left( {1 - 2\mu } \right)\left( {p_{{r_{c} }} r_{c}^{2} - p_{{r_{1} }} r_{1}^{2} } \right) - \left( {p_{{r_{1} }} - p_{{r_{c} }} } \right)r_{c}^{2} } \right] $$

(82)

The variables $r_{1}$, $r_{2}$, $p_{{r_{1} }}$ and $p_{{r_{c} }}$ required in Eq. (82) can be obtained from the equation set:

$$ \left\{ \begin{gathered} p_{{r_{2} }} = \frac{{B^{\prime}_{1} }}{{1 - \xi_{r} }} - \frac{{A^{\prime}_{1} }}{{f + \xi_{r} }} + \left[ {p_{{r_{b} }} - \frac{{B^{\prime}_{1} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{1} }}{{f + \xi_{r} }}\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ p_{{r_{c} }} = \frac{{2\left( {1 - \mu } \right)\left[ {Ep_{{r_{1} }} r_{1}^{2} + E_{d} \left( {\beta + \eta } \right)\left( {1 - \mu } \right)p_{0} } \right]}}{{\left( {\beta + \eta } \right)E_{d} + E\left[ {\left( {1 - 2\mu } \right)r_{c}^{2} + r_{1}^{2} } \right]}} \hfill \\ p_{{r_{c} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{c} }}} \right)^{h + 1} = \left[ {p_{{r_{2} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}t^{h + 1} } \right]\left( {\frac{{r_{c} }}{{r_{2} }}} \right)^{{\xi_{0} - 1}} \hfill \\ p_{{r_{b} }} = \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} - \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{b} }}} \right)^{f + 1} + \left[ {p_{i} - \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{b} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ p_{{r_{1} }} = \frac{{B_{1} }}{{1 - \xi_{0} }}\left( {1 + t^{{\xi_{0} - 1}} } \right) + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {t^{h + 1} - 1} \right) + p_{{r_{2} }} t^{{\xi_{0} - 1}} \hfill \\ p_{{r_{1} }} \left( {1 + \xi_{0} } \right) + A_{1} + B_{1} = \frac{{2\left( {p_{{r_{c} }} r_{c}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)}}{{r_{c}^{2} - r_{1}^{2} }} \hfill \\ \end{gathered} \right. $$

(83)

B.10 The solution for Form B10

The boundary conditions are

$$ \left\{ \begin{gathered} \sigma_{r} \left| {_{{r = r_{0} }} } \right. = p_{i} \hfill \\ \sigma_{r} \left| {_{r = \infty } } \right. = p_{0} \hfill \\ \sigma_{r} \left| {_{{r = r_{b} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{b} + dr}} = p_{{r_{b} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{c} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{c} + dr}} = p_{{r_{c} }} } \right. \hfill \\ \sigma_{r} \left| {_{{r = r_{2} - dr}} } \right. = \sigma_{r} \left| {_{{r = r_{2} + dr}} } \right. = p_{{r_{2} }} \hfill \\ u_{r} \left| {_{{r = r_{1} - dr}} } \right. = u_{r} \left| {_{{r = r_{1} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{2} - dr}} } \right. = u_{r} \left| {_{{r = r_{2} + dr}} } \right. \hfill \\ u_{r} \left| {_{{r = r_{b} - dr}} } \right. = u_{r} \left| {_{{r = r_{b} + dr}} } \right. \hfill \\ \end{gathered} \right. $$

(84)

The tunnel wall displacement can be given by

$$ u_{{r_{0} }} = \frac{{u_{{r_{1} }} f}}{f + 1} - M^{\prime\prime}\left[ {\left( {h - f} \right)t^{h + 1} + 1} \right] + \left\{ {\frac{{u_{{r_{1} }} }}{f + 1} + M^{\prime\prime}\left[ {\left( {2f - h + 1} \right)t^{h + 1} + 1} \right] - \left( {f + 1} \right)r_{1} - r_{2} } \right\}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f} $$

(85)

where $u_{{r_{1} }}$ can be obtained by Eq. (82).

The variables $r_{1}$, $r_{2}$, $p_{{r_{1} }}$ and $p_{{r_{c} }}$ required in Eq. (85) can be obtained from the equation set:

$$ \left\{ \begin{gathered} p_{{r_{c} }} = \frac{{2\left( {1 - \mu } \right)\left[ {Ep_{{r_{1} }} r_{1}^{2} + E_{d} \left( {\beta + \eta } \right)\left( {1 - \mu } \right)p_{0} } \right]}}{{\left( {\beta + \eta } \right)E_{d} + E\left[ {\left( {1 - 2\mu } \right)r_{c}^{2} + r_{1}^{2} } \right]}} \hfill \\ p_{{r_{1} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }} = \left[ {p_{{r_{b} }} - \frac{{B_{1} }}{{1 - \xi_{0} }} + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} } \right]\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{{\xi_{0} - 1}} \hfill \\ p_{{r_{b} }} = \frac{{B_{2} }}{{1 - \xi_{0} }} - \frac{{A_{2} }}{{h + \xi_{0} }}\left( {\frac{{r_{1} }}{{r_{b} }}} \right)^{h + 1} + \left[ {p_{{r_{2} }} - \frac{{B_{2} }}{{1 - \xi_{0} }} + \frac{{A_{2} }}{{h + \xi_{0} }}t^{h + 1} } \right]\left( {\frac{{r_{b} }}{{r_{2} }}} \right)^{{\xi_{0} - 1}} \hfill \\ p_{{r_{1} }} = \frac{{B_{1} }}{{1 - \xi_{0} }}\left( {1 + t^{{\xi_{0} - 1}} } \right) + \frac{{A_{1} }}{{h + \xi_{0} }}\left( {t^{h + 1} - 1} \right) + p_{{r_{2} }} t^{{\xi_{0} - 1}} \hfill \\ p_{{r_{1} }} \left( {1 + \xi_{0} } \right) + A_{1} + B_{1} = \frac{{2\left( {p_{{r_{c} }} r_{c}^{2} - p_{{r_{1} }} r_{1}^{2} } \right)}}{{r_{c}^{2} - r_{1}^{2} }} \hfill \\ p_{{r_{2} }} = \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} - \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }} + \left[ {p_{i} - \frac{{B^{\prime}_{2} }}{{1 - \xi_{r} }} + \frac{{A^{\prime}_{2} }}{{f + \xi_{r} }}\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{f + 1} } \right]\left( {\frac{{r_{2} }}{{r_{0} }}} \right)^{{\xi_{r} - 1}} \hfill \\ \end{gathered} \right. $$

(86)