Analytical Solution for Deep Circular Tunnels in Rock with Consideration of Disturbed zone, 3D Strength and Large Strain

Abstract

This paper presents a new analytical solution for deep circular tunnels in rock with consideration of disturbed zone, 3D strength and large strain. The rock is assumed to be elastic–brittle–plastic and governed by a 3D Hoek–Brown yield criterion. To take the large displacement around a tunnel into account, the large-strain theory is adopted to determine the displacement of rock in the plastic zone. Based on the equilibrium equation, constitutive law and large-strain theory, the governing equations for the stresses and radial displacement around the tunnel were derived and solved by using MATLAB. The proposed solution was validated by using it to analyze a tunnel and comparing the results with those from numerical analysis using a finite difference code. Finally, extensive parametric studies were performed on tunnels in both poor-quality and good-quality rock masses with respect to stresses and radial displacement. The results indicate that the disturbed zone and the flow rule both have significant effects on the stress and displacement distributions around the tunnel.

Appendix

1.1 Derivatives of Yield Function and Potential Function

For the consistency condition, Eq. (21), the derivatives in the three directions can be given as

$$\frac{\partial f}{\partial {\stackrel{\sim }{\sigma }}_{i}}=\frac{\partial f}{\partial {\stackrel{\sim }{I}}_{1}^{*}}+\frac{\partial f}{\partial {\stackrel{\sim }{I}}_{2}^{*}}\left({\stackrel{\sim }{I}}_{1}^{*}-{\stackrel{\sim }{\sigma }}_{i}\right)+\frac{\partial f}{\partial {\stackrel{\sim }{I}}_{3}^{*}}\frac{{\stackrel{\sim }{I}}_{3}^{*}}{{\stackrel{\sim }{\sigma }}_{i}};\quad i=r,\theta ,x$$

(30)

where

$$\frac{{\partial f}}{{\partial \tilde{I}_{1}^{*} }} = \frac{{m_{b} \tilde{I}_{1}^{*} }}{{2\sqrt {\tilde{I}_{1}^{{*2}} - 3v_{2}^{*} } }} - \frac{{m_{b} \left( {\tilde{I}_{1}^{{*3}} \tilde{I}_{2}^{{*2}} - 14\tilde{I}_{1}^{{*2}} \tilde{I}_{2}^{*} \tilde{I}_{3}^{*} + 9\tilde{I}_{1}^{*} \tilde{I}_{3}^{{*2}} + 12\tilde{I}_{2}^{{*2}} \tilde{I}_{3}^{*} } \right)}}{{4\left( {\tilde{I}_{1}^{*} \tilde{I}_{2}^{*} - 9\tilde{I}_{3}^{*} } \right)^{2} \tilde{\sigma }_{m}^{*} }} + \frac{{\tilde{I}_{1}^{*} }}{{a\left( {\tilde{I}_{1}^{{*2}} - 3\tilde{I}_{2}^{*} } \right)^{{1 - \frac{1}{{ {2a}}}}} }}$$

(31)

$$\frac{\partial f}{\partial {\stackrel{\sim }{I}}_{2}^{*}}=-\frac{3{m}_{b}}{4\sqrt{{\stackrel{\sim }{I}}_{1}^{*2}-3{\stackrel{\sim }{I}}_{2}^{*}}}+\frac{{m}_{b}\left(8{\stackrel{\sim }{I}}_{1}^{*3}{\stackrel{\sim }{I}}_{3}^{*}+3{\stackrel{\sim }{I}}_{1}^{*2}{\stackrel{\sim }{I}}_{2}^{*2}-54{\stackrel{\sim }{I}}_{1}^{*}{\stackrel{\sim }{I}}_{2}^{*}{\stackrel{\sim }{I}}_{3}^{*}+27{\stackrel{\sim }{I}}_{3}^{*2}\right)}{8{\left({\stackrel{\sim }{I}}_{1}^{*}{\stackrel{\sim }{I}}_{2}^{*}-9{\stackrel{\sim }{I}}_{3}^{*}\right)}^{2}{\stackrel{\sim }{\sigma }}_{m}^{*}} +\frac{3}{2a{\left({\stackrel{\sim }{I}}_{1}^{*2}-3{\stackrel{\sim }{I}}_{2}^{*}\right)}^{1-\frac{1}{2a}}}$$

(32)

$$\frac{\partial f}{\partial {\stackrel{\sim }{I}}_{3}^{*}}=\frac{2{m}_{b}{\stackrel{\sim }{I}}_{1}^{*}{\stackrel{\sim }{I}}_{2}^{*}\left(-{\stackrel{\sim }{I}}_{1}^{*2}+3{\stackrel{\sim }{I}}_{2}^{*}\right)}{2{\left({\stackrel{\sim }{I}}_{1}^{*}{\stackrel{\sim }{I}}_{2}^{*}-9{\stackrel{\sim }{I}}_{3}^{*}\right)}^{2}{\stackrel{\sim }{\sigma }}_{m}^{*}}$$

(33)

$${\stackrel{\sim }{\sigma }}_{m}^{*}=\sqrt{\frac{\left({\stackrel{\sim }{I}}_{1}^{*}{\stackrel{\sim }{I}}_{2}^{*}-{\stackrel{\sim }{I}}_{3}^{*}\right)\left({\stackrel{\sim }{I}}_{1}^{*2}-3{\stackrel{\sim }{I}}_{2}^{*}\right)}{{\stackrel{\sim }{I}}_{1}^{*}{\stackrel{\sim }{I}}_{2}^{*}-{9\stackrel{\sim }{I}}_{3}^{*}}}$$

(34)

According to the flow rule, the plastic strain rate in the three directions can be given as

$${\dot{\varepsilon }}_{r}^{p}=\lambda \frac{\partial g}{\partial {\sigma }_{r}}=\lambda \frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{r}}\frac{\partial {\stackrel{\sim }{\sigma }}_{r}}{\partial {\sigma }_{r}}=\lambda \frac{1}{{\sigma }_{c}}\frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{r}}$$

(35)

$${\dot{\varepsilon }}_{\theta }^{p}=\lambda \frac{\partial g}{\partial {\sigma }_{\theta }}=\lambda \frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{\theta }}\frac{\partial {\stackrel{\sim }{\sigma }}_{\theta }}{\partial {\sigma }_{\theta }}=\lambda \frac{1}{{\sigma }_{c}}\frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{\theta }}$$

(36)

$${\dot{\varepsilon }}_{x}^{p}=\lambda \frac{\partial g}{\partial {\sigma }_{x}}=\lambda \frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{x}}\frac{\partial {\stackrel{\sim }{\sigma }}_{x}}{\partial {\sigma }_{x}}=\lambda \frac{1}{{\sigma }_{c}}\frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{x}}$$

(37)

For associated flow rule

$$\frac{\partial g}{\partial {\stackrel{\sim }{\sigma }}_{i}}=\frac{\partial f}{\partial {\stackrel{\sim }{\sigma }}_{i}};\quad i=r,\theta ,x$$

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while for unassociated flow rule with a dilation angle, one has

$$\frac{\partial g/\partial {\stackrel{\sim }{\sigma }}_{r}}{\partial g/\partial {\stackrel{\sim }{\sigma }}_{\theta }}=-\frac{1-\mathrm{sin}\varPsi }{1+\mathrm{sin}\varPsi }$$

(39)

where $\varPsi$ is the dilation angle.

1.2 Equations for Iterative Algorithms

For the cylindrical cavity contraction problem, the yield function in terms of ${\sigma }_{rp}$ can be simplified as

$$f\left({\sigma }_{rp}^{\prime}\right)=\frac{1}{{\sigma }_{c}^{\left(1/a-1\right)}}{\left[\sqrt{3}\left({\sigma }_{0}^{\prime}-{\sigma }_{rp}^{\prime}\right)\right]}^{1/a}+\frac{\sqrt{3}{m}_{b}}{2}\left({\sigma }_{0}^{\prime}-{\sigma }_{rp}^{\prime}\right)-\frac{{m}_{b}}{2}\sqrt{4{\sigma }_{0}^{2}-{\left({\sigma }_{rp}^{\prime}-{\sigma }_{0}^{\prime}\right)}^{2}}$$

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The corresponding derivation of yield function $f$ with respect to ${\sigma }_{rp}^{\prime}$ is

$${f}^{\prime}\left({\sigma }_{rp}^{\prime}\right)=-\frac{{3}^{\frac{1}{\left(2a\right)}}}{a{\sigma }_{c}^{\left(1/a-1\right)}}{\left({\sigma }_{0}^{\prime}-{\sigma }_{rp}^{\prime}\right)}^{1/a-1}-\frac{\sqrt{3}{m}_{b}}{2}+\frac{{m}_{b}}{2}\frac{{\sigma }_{rp}^{\prime}-{\sigma }_{0}^{\prime}}{\sqrt{4{\sigma }_{0}^{\prime2}-{\left({\sigma }_{rp}^{\prime}-{\sigma }_{0}^{\prime}\right)}^{2}}}$$

(41)