## Appendix A
The solution of the tunnel radial displacement at \(r=r_{0}\) is:  
\[u_{r_{0}}=h^{\prime}r_{0}\bigg{[}m^{\prime}(p_{i}+\beta^{\prime})+\frac{{\cal A}_{ 1}-{\cal A}_{2}}{N^{\prime}\psi+2}r_{0}+C^{\prime}r_{0}^{-N^{\varphi}_{\psi}-1}+{ \cal A}_{2}\bigg{]}, \tag{16}\]  
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