A New Procedure for Calculation of Ground Response Curve of a Circular Tunnel Considering the Influence of Young’s Modulus Variation and the Plastic Weight Loading

Abstract

In this paper, an analytical–numerical solution is presented for a circular tunnel in the hydrostatic condition using a generalized nonlinear Hoek–Brown failure criterion. To a deep insight into the problem, the influence of the plastic zone weight, and the Young's modulus on the ground response curve of a deep tunnel in a strain-softening media is highlighted. Experimental results show that the magnitude of the confined stress can have a direct effect on the critical softening parameter. Thus, a variable critical softening parameter is also assumed. That is, as the radial stress magnitude increases, the critical softening parameter is enhanced. The equations are derived for different directions (e.g. crown, floor, and sidewall) around the circular tunnel. The accuracy of the proposed method is validated by some examples. The results show that assuming a constant Young's modulus in the plastic zone imposes a significant error in the predicting of the tunnel convergence.

Appendix: Ground Response Curve Calculation

Input data:

$\sigma_{ci}$ = uniaxial compression strength of the rock mass; GSI ^peak = geological strength index of the rock mass at the peak state; $m_{i}$ = constant parameter of intact rock; $r_{o}$ = external tunnel radius; $\sigma_{0}$ = initial hydrostatic stress;$\gamma$ = unit weight of rock mass; $\theta$ = angle measured counter-clockwise from the spring line of the tunnel; $D$ = degree of disturbance of the rock mass; $v_{e}$ = Poisson's ratio of rock mass in the elastic zone; $\alpha$ = curvature of the behavior model; $n$ = number of rings in the plastic zone; $P_{i}$ = internal support pressure; $R_{P}$ = plastic radius.

Preliminary calculation:

• 1. $F_{r} = \gamma sin\theta$

• 2. $GSI^{res} = 17.25e^{{0.0107 \times GSI^{peak} }} for 25 < GSI^{peak} < 75$

• 3. $s_{{\left( {peak} \right)}} = s_{{\left( {i = 1} \right)}} = e^{{\frac{{GSI^{peak} - 100}}{9 - 3D}}} ,s_{{\left( {res} \right)}} = e^{{\frac{{GSI^{res} - 100}}{9 - 3D}}}$

• 4. $m_{{b\left( {peak} \right)}} = m_{{b\left( {i = 1} \right)}} = m_{i} e^{{\frac{{GSI^{peak} - 100}}{28 - 14D}}} , m_{{b\left( {res} \right)}} = m_{i} e^{{\frac{{GSI^{res} - 100}}{28 - 14D}}}$

• 5. $a_{{\left( {peak} \right)}} = a_{{\left( {i = 1} \right)}} = 0.5 + \frac{1}{6}\left( {e^{{ - \frac{{GSI^{peak} }}{15}}} - e^{{ - \frac{20}{3}}} } \right), a_{{\left( {res} \right)}} = 0.5 + \frac{1}{6}\left( {e^{{ - \frac{{GSI^{res} }}{15}}} - e^{{ - \frac{20}{3}}} } \right)$

• 6. $E_{{e\left( {peak} \right)}} \left( {MPa} \right) = E_{{e\left( {i = 1} \right)}} = 100000\left( {\frac{{1 - \frac{D}{2}}}{{1 + e^{{\frac{{75 + 25D - GSI^{peak} }}{11}}} }}} \right)$

• 7. $E_{{e\left( {res} \right)}} \left( {MPa} \right) = 100000\left( {\frac{{1 - \frac{D}{2}}}{{1 + e^{{\frac{{75 + 25D - GSI^{res} }}{11}}} }}} \right)$

• 8. $\sigma_{cm}^{^{\prime}} = \sigma_{ci} .\frac{{\left( {m_{{b\left( {peak} \right)}} + 4s_{{\left( {peak} \right)}} - a_{{\left( {peak} \right)}} \left( {m_{{b\left( {peak} \right)}} - 8s_{{\left( {peak} \right)}} } \right)} \right)\left( {\frac{{m_{{b\left( {peak} \right)}} }}{4} + s_{{\left( {peak} \right)}} } \right)^{{a_{{\left( {peak} \right)}} - 1}} }}{{2\left( {1 + a_{{\left( {peak} \right)}} } \right)\left( {2 + a_{{\left( {peak} \right)}} } \right)}}$

• 9. $\sigma_{3n}^{^{\prime}} = \frac{{0.47\sigma_{cm}^{^{\prime}} \left( {\frac{{\sigma_{cm}^{^{\prime}} }}{{\sigma_{0} }}} \right)^{ - 0.94} }}{{\sigma_{ci} }}$

• 10. $\varphi_{{\left( {peak} \right)}} = sin^{ - 1} \left[ {\frac{{6a_{{\left( {peak} \right)}} m_{{b\left( {peak} \right)}} (s_{{\left( {peak} \right)}} + m_{{b\left( {peak} \right)}} \sigma_{3n}^{^{\prime}} )^{{a_{{\left( {peak} \right)}} - 1}} }}{{2\left( {1 + a_{{\left( {peak} \right)}} } \right)\left( {2 + a_{{\left( {peak} \right)}} } \right) + 6a_{{\left( {peak} \right)}} m_{{b\left( {peak} \right)}} (s_{{\left( {peak} \right)}} + m_{{b\left( {peak} \right)}} \sigma_{3n}^{^{\prime}} )^{{a_{{\left( {peak} \right)}} - 1}} }}} \right]$

• 11. $\psi_{{\left( {peak} \right)}} = \frac{{5 GSI^{peak} - 125}}{1000}\varphi_{{\left( {peak} \right)}} , for 25 < GSI^{peak} < 75$

• 12. $K_{{\Psi \left( {peak} \right)}} = K_{{\Psi \left( {i = 1} \right)}} = \frac{{1 + sin\Psi_{{\left( {peak} \right)}} }}{{1 - sin\Psi_{{\left( {peak} \right)}} }}$

• 13. $\sigma_{{r\left( {i = 1} \right)}} = \sigma_{{r\left( {R_{P} } \right)}}$

• 14. $\sigma_{{\theta \left( {i = 1} \right)}} = 2\sigma_{0} - \sigma_{{r\left( {R_{P} } \right)}}$

• 15. $\varepsilon_{{\theta \left( {i = 1} \right)}}^{ } = \varepsilon_{\theta \left( r \right)}^{e} = \frac{{1 + v_{e} }}{{E_{e} }}\left( {\sigma_{0} - \sigma_{{r\left( {i = 1} \right)}} } \right) , \varepsilon_{{\theta \left( {i = 1} \right)}}^{P} = 0$

• 16. $\varepsilon_{{r\left( {i = 1} \right)}}^{ } = \varepsilon_{{r\left( {i = 1} \right)}}^{e} = - \frac{{1 + v_{e} }}{{E_{e} }}\left( {\sigma_{0} - \sigma_{{r\left( {i = 1} \right)}} } \right) , \varepsilon_{{r\left( {i = 1} \right)}}^{P} = 0$

• 17. $\eta_{{\left( {i = 1} \right)}}^{P} = 0$

• 18. $\rho_{{\left( {i = 1} \right)}} = 1$

• 19. $M_{{\left( {i = 1} \right)}} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - E_{e} \left[ {0.0046e^{{0.0768 \times GSI^{peak} }} } \right]\left( {\frac{{\sigma_{{r\left( {i = 1} \right)}}^{ } }}{{\sigma_{ci} \sqrt {s^{peak} } }}} \right)^{ - 1} } & {if \frac{{\sigma_{{r\left( {i = 1} \right)}}^{ } }}{{\sigma_{ci} \sqrt {s^{peak} } }} \ge 0.1} \\ { - E_{e} \left[ {0.0046e^{{0.0768 \times GSI^{peak} }} } \right]\left( {\frac{{\sigma_{{r\left( {i = 1} \right)}}^{ } }}{{2\sigma_{ci} \sqrt {s^{peak} } }} + 0.05} \right)^{ - 1} } & {if \frac{{\sigma_{{r\left( {i = 1} \right)}}^{ } }}{{\sigma_{ci} \sqrt {s^{peak} } }} \le 0.1} \\ \end{array} } \\ \\ \end{array} } \right.$

• 20. $\eta_{{\left( {i = 1} \right)}}^{P*} = \left( {1 + \frac{{K_{{\Psi \left( {\eta_{{\left( {i = 1} \right)}}^{P} } \right)}} }}{2}} \right)\left[ {\sigma_{1}^{peak} \left( {\sigma_{{r\left( {i = 1} \right)}}^{ } } \right) - \sigma_{1}^{res} \left( {\sigma_{{r\left( {i = 1} \right)}}^{ } } \right)} \right]\left[ {\frac{1}{{E_{e} }} - \frac{1}{{M_{{\left( {i = 1} \right)}} }}} \right]$

• 21. $\Delta \sigma_{{r\left( {i = 1} \right)}} = \Delta \sigma_{{\theta \left( {i = 1} \right)}} = \Delta \varepsilon_{{r\left( {i = 1} \right)}}^{e} = \Delta \varepsilon_{{\theta \left( {i = 1} \right)}}^{e} = \Delta \rho_{{\left( {i = 1} \right)}} = \overline{\rho }_{{\left( {i = 1} \right)}} = \Delta \varepsilon_{{\theta \left( {i = 1} \right)}}^{P} = \Delta \varepsilon_{{r\left( {i = 1} \right)}}^{P} = \Delta \varepsilon_{{\theta \left( {i = 1} \right)}} = \Delta \varepsilon_{{r\left( {i = 1} \right)}} = 0$

Sequence of calculation for each ring ($i \ge 2$):

• 1. $\Delta \sigma_{r\left( i \right)} = \frac{{P_{i} - \sigma_{{r\left( {R_{P} } \right)}} }}{n}$

• 2. $\sigma_{r\left( i \right)} = \sigma_{{r\left( {i - 1} \right)}} + \Delta \sigma_{r\left( i \right)}$

• 3. $\sigma_{\theta \left( i \right)} = \sigma_{r\left( i \right)} + \sigma_{ci} \left( {\frac{{m_{{b\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} \sigma_{r\left( i \right)} }}{{\sigma_{ci} }} + s_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} } \right)^{{a_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}$

• 4. $\Delta \sigma_{\theta \left( i \right)} = \sigma_{\theta \left( i \right)} - \sigma_{{\theta \left( {i - 1} \right)}}$

• 5. $\Delta \varepsilon_{r\left( i \right)}^{e} = \frac{{1 + v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}{{E_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}\left[ {\Delta \sigma_{r\left( i \right)} \left( {1 - v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} } \right) - \Delta \sigma_{\theta \left( i \right)} v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} } \right]$

• 6. $\Delta \varepsilon_{\theta \left( i \right)}^{e} = \frac{{1 + v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}{{E_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}\left[ {\Delta \sigma_{\theta \left( i \right)} \left( {1 - v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} } \right) - \Delta \sigma_{r\left( i \right)} v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} } \right]$

• 7. $H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right) = \sigma_{ci} \left( {\frac{{m_{{b_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }} \left[ {\frac{{\sigma_{r\left( i \right)} + \sigma_{{r\left( {i - 1} \right)}} }}{2}} \right]}}{{\sigma_{ci} }} + s_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} } \right)^{{a_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}$

• 8. $C_{4} = \sqrt {\left[ {\Delta \sigma_{r\left( i \right)} - 2F_{r} R_{P} \rho_{i - 1} + 2H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right)} \right]^{2} - 8\Delta \sigma_{r\left( i \right)} H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right)}$

• 9. $\rho_{i} = \left\{ {\begin{array}{*{20}c} {\frac{{ - \Delta \sigma_{r\left( i \right)} + 2H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right) - C_{4} }}{{2F_{r} R_{P} }}} & {if F_{r} \ne 0} \\ {\rho_{i - 1} \left[ {\frac{{\Delta \sigma_{r\left( i \right)} + 2H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right)}}{{ - \Delta \sigma_{r\left( i \right)} + 2H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right)}}} \right]} & {if F_{r} = 0} \\ \end{array} } \right.$

• 10. $\Delta \rho_{\left( i \right)} = \rho_{\left( i \right)} - \rho_{{\left( {i - 1} \right)}}$

• 11. $\overline{\rho }_{\left( i \right)} = \frac{{\rho_{\left( i \right)} + \rho_{{\left( {i - 1} \right)}} }}{2}$

• 12. $\Delta \varepsilon_{\theta \left( i \right)}^{P} = \frac{{ - \frac{{\Delta \varepsilon_{\theta \left( i \right)}^{e} }}{{\Delta \rho_{\left( i \right)} }} - \frac{{1 + v_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}{{E_{{\left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }} \frac{{H\left( {\overline{\sigma }_{r\left( i \right)} ,\eta_{{\left( {i - 1} \right)}}^{P} } \right)}}{{\overline{\rho }_{\left( i \right)} }} - \frac{{\varepsilon_{{\theta \left( {i - 1} \right)}}^{P} - \varepsilon_{{r\left( {i - 1} \right)}}^{P} }}{{\overline{\rho }_{\left( i \right)} }}}}{{\frac{1}{{\Delta \rho_{\left( i \right)} }} + \frac{{1 + K_{{\Psi (\eta_{{\left( {i - 1} \right)}}^{P} )}} }}{{\overline{\rho }_{\left( i \right)} }}}}$

• 13. $\Delta \varepsilon_{r\left( i \right)}^{P} = - K_{{\Psi (\eta_{{\left( {i - 1} \right)}}^{P} )}} \Delta \varepsilon_{\theta \left( i \right)}^{P}$

• 14. $\varepsilon_{\theta \left( i \right)}^{P} = \Delta \varepsilon_{\theta \left( i \right)}^{P} + \varepsilon_{{\theta \left( {i - 1} \right)}}^{P}$

• 15. $\varepsilon_{r\left( i \right)}^{P} = \Delta \varepsilon_{r\left( i \right)}^{P} + \varepsilon_{{r\left( {i - 1} \right)}}^{P}$

• 16. $\eta_{\left( i \right)}^{P} = \eta_{{\left( {i - 1} \right)}}^{P} + \Delta \varepsilon_{\theta \left( i \right)}^{P} - \Delta \varepsilon_{r\left( i \right)}^{P}$

• 17. $\Delta \varepsilon_{\theta \left( i \right)} = \Delta \varepsilon_{\theta \left( i \right)}^{e} + \Delta \varepsilon_{\theta \left( i \right)}^{P}$

• 18. $\Delta \varepsilon_{r\left( i \right)} = \Delta \varepsilon_{r\left( i \right)}^{e} + \Delta \varepsilon_{r\left( i \right)}^{P}$

• 19. $\varepsilon_{r\left( i \right)} = \varepsilon_{{r\left( {i - 1} \right)}} + \Delta \varepsilon_{r\left( i \right)}$

• 20. $\varepsilon_{\theta \left( i \right)} = \varepsilon_{{\theta \left( {i - 1} \right)}} + \Delta \varepsilon_{\theta \left( i \right)}$

• 21. $M_{\left( i \right)} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - E_{e} \left[ {0.0046e^{{0.0768 \times GSI^{peak} }} } \right]\left( {\frac{{\sigma_{r\left( i \right)}^{ } }}{{\sigma_{ci} \sqrt {s^{peak} } }}} \right)^{ - 1} } & {if\; \frac{{\sigma_{r\left( i \right)}^{ } }}{{\sigma_{ci} \sqrt {s^{peak} } }} \ge 0.1} \\ { - E_{e} \left[ {0.0046e^{{0.0768 \times GSI^{peak} }} } \right]\left( {\frac{{\sigma_{r\left( i \right)}^{ } }}{{2\sigma_{ci} \sqrt {s^{peak} } }} + 0.05} \right)^{ - 1} } & {if\; \frac{{\sigma_{r\left( i \right)}^{ } }}{{\sigma_{ci} \sqrt {s^{peak} } }} \le 0.1} \\ \end{array} } \\ \\ \end{array} } \right.$

• 22. $\eta_{\left( i \right)}^{P*} = \left( {1 + \frac{{K_{{\Psi \left( {\eta_{{\left( {i - 1} \right)}}^{P} } \right)}} }}{2}} \right)\left[ {\sigma_{1}^{peak} \left( {\sigma_{r\left( i \right)}^{ } } \right) - \sigma_{1}^{res} \left( {\sigma_{r\left( i \right)}^{ } } \right)} \right]\left[ {\frac{1}{{E_{e} }} - \frac{1}{{M_{\left( i \right)} }}} \right]$

• 23. $K_{{\Psi (\eta_{\left( i \right)}^{P} )}} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {1 + \left( {K_{{\psi \left( {peak} \right)}} - 1} \right) \times e^{{\frac{{ - \eta_{\left( i \right)}^{P} }}{{\eta_{\left( i \right)}^{P*} }}}} } & {if\; 0 < \eta_{\left( i \right)}^{P} \le \eta_{\left( i \right)}^{P*} } \\ {1 + \left( {K_{{\psi \left( {peak} \right)}} - 1} \right) \times e^{ - 1} } & {if\; \eta_{\left( i \right)}^{P} > \eta_{\left( i \right)}^{P*} } \\ \end{array} } \\ \\ \end{array} } \right.$

• 24. $\omega_{{\left( {\eta_{\left( i \right)}^{P} } \right)}} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left( {\omega_{P} - \omega_{r} } \right)\left[ {\alpha \left( {\frac{{\eta_{\left( i \right)}^{P} }}{{\eta_{\left( i \right)}^{P*} }}} \right)^{2} - \left( {1 + \alpha } \right)\frac{{\eta_{\left( i \right)}^{P} }}{{\eta_{\left( i \right)}^{P*} }}} \right] + \omega_{P} } & {if 0 < \eta_{\left( i \right)}^{P} \le \eta_{\left( i \right)}^{P*} } \\ {\omega_{r} } & {if \eta_{\left( i \right)}^{P} > \eta_{\left( i \right)}^{P*} } \\ \end{array} } \\ \\ \end{array} } \right.$

• 25. If $\sigma_{r\left( i \right)} > P_{i}$, let i = i + 1 and repeat the above calculations for next ring

• 26. If $\sigma_{r\left( i \right)} = P_{i}$, consider the below note. Then, perform below steps

• 27. $R_{P} = \frac{{r_{o} }}{{\rho_{i} }}$

• 28. $u_{r\left( i \right)} = r_{o} \varepsilon_{\theta \left( i \right)}$

Note: It should be noted that when $F_{r}$ is not zero, the calculations depend on $R_{P}$ value. Due to the fact that this parameter is not initially determined, the calculation must be performed alternately to achieve an appropriate convergence. Thus, at the first step, an initial value for $R_{P}$ is assumed and calculations are performed. After the calculations, a new $R_{P}$ will be obtained and this value is used for calculations of subsequent stages. When the difference between the new value of this parameter and the last one becomes negligible, the computations will be stopped. An important point is that when the internal support pressure becomes smaller than some value, the magnitude of $R_{P}$ does not converge. This means that instability of the tunnel occurs.