## Appendix A
\begin{array}{l}\left\{\begin{array}{l}R_{1}=r_{0}\bigg{[}\frac{({\mathbb{C }}_{1}-1)p_{R_{1}}+{\cal C}_{2}+{\mathbb{I}}^{\prime}p_{a}}{({\mathbb{C}}_{1}- 1)p_{R_{1}}+{\cal C}_{2}+{\mathbb{I}}^{\prime}p_{a}}\bigg{]}^{\frac{1}{1}}\\ \,p_{R_{1}}=-\frac{\big{(}R_{1}^{2}-r_{2}^{2}\big{)}}{2R_{1}^{2}}\bigg{[}({\cal C }_{1}^{\prime}+1)(p_{i}+\beta^{\prime})\big{(}\frac{R_{1}}{r_{0}}\big{)}^{C_{ 1}-1}-\frac{2{\cal C}_{2}}{{\cal C}_{1}^{\prime}-1}-\frac{C_{1}+1}{{\cal C}_{1} -1}{\mathbb{I}}^{\prime}p_{a}+\frac{{\mathbb{I}}^{\prime}p_{a}}{1-\mu^{\prime }}{\rm In}\frac{r_{0}}{R_{1}}\bigg{]}+\frac{p_{a}r_{2}^{2}}{R_{1}^{2}}\\ \,h^{\prime}\bigg{[}m^{\prime}\big{(}p_{i}+\beta^{\prime}\big{)}\big{(}\frac{R _{1}}{r_{0}}\big{)}^{C_{1}-1}+\frac{{\cal A}_{1}^{\prime}-{\cal A}_{2}^{\prime }}{N^{\varphi}_{\psi}+2}r_{a}+{\cal C}_{n}^{\prime}r_{a}^{-N^{\varphi}_{\psi}- 1}+{\cal A}^{\prime}{}_{2}\bigg{]}=\\ \,np_{a}\bigg{[}\Big{(}\frac{R_{1}^{2}}{r_{2}^{2}}-2{\rm In}\frac{R_{0}}{r_{a}} -1\Big{)}+2\Big{(}\frac{R_{2}^{2}}{r_{2}^{2}}-1\Big{)}\bigg{]}-h(p_{0}+p_{a}- p_{r_{a}})\\ \,C^{\prime}=R_{1}^{N^{\varphi}_{\psi}+1}\bigg{[}\frac{(1-\mu^{\prime})\big{(}1 -{\mathbb{C}}_{1}^{2}\big{)}}{N^{\varphi}_{\psi}+{\mathbb{C}}_{1}}(p_{i}+ \beta^{\prime})\big{(}\frac{R_{1}}{r_{0}}\big{)}^{C_{1}-1}+\frac{{\cal A}_{1} ^{\prime}-{\cal A}_{2}^{\prime}}{N^{\varphi}_{\psi}+2}R_{1}\bigg{]}\end{array} \right.\end{array}  \tag{11}\]  
CFG III: In this configuration, the plastic zone only appears in the natural ground while the reinforced zone remains elastic. Besides Equation (10), the following boundary conditions are imposed:  
\[ \begin{array}{l}(\sigma_{r}+\sigma_{\theta})_{|r=R_{2}-dr}=(\sigma_{r}+ \sigma_{\theta})_{|r=R_{2}+dr}\\ \varepsilon_{\theta|r=R_{2}}^{p}=0\end{array} \,, \tag{12}\]  
where \(R_{1}\) is the plastic zone radius in the natural zone. The tunnel radial displacement at \(r=r_{0}\) is obtained as:  
\[u_{r_{0}}=\frac{h^{\prime}r_{0}}{({r_{a}}^{2}-{r_{0}}^{2})}\Big{[}\big{(}p_{i}- p_{r_{a}}){r_{a}}^{2}-(1-2\mu^{\prime})\Big{(}p_{r_{a}}{r_{a}}^{2}-p_{i}{r_{0}}^{2} \big{)}\Big{]}, \tag{13}\]  
where the plastic zone radius \(R_{2}\) and the radial stress \(p_{r_{a}}\) at \(r=r_{a}\) can be evaluated from the following set of relations:  
\[ \begin{array}{l}\left\{\begin{array}{l}p_{r_{a}}=\frac{1}{{\mathbb{C}}_{1} +1}\bigg{[}\frac{2{\mathbb{C}}_{2}({\mathbb{C}}_{1}-1)+p_{a}}{({\mathbb{C}}_{1} -1)^{2}}-\frac{p_{a}}{1-\mu}-2(p_{0}+p_{e})\bigg{]}\Big{(}\frac{r_{a}}{R_{2}} \Big{)}^{{\mathbb{C}}_{1}-1}-\beta\\ \,n^{\prime}l^{\prime}p_{a}\Big{(}\frac{r_{0}^{2}}{r_{a}^{2}}-2{\rm In}\frac{r_{0 }}{r_{a}}-1\Big{)}-h\bigg{[}m(p_{r_{a}}+\beta)-\frac{A_{1}-A_{2}}{N_{\psi}+2}r _{a}+C_{a}^{-N^{\varphi}_{0}-1}+A_{2}\bigg{]}\\ \,=\frac{h^{\prime}}{({r_{a}}^{2}-{r_{0}}^{2})}\bigg{(}1-2\mu^{\prime}\Big{)} \big{(}p_{r_{a}}{r_{a}}^{2}-{p_{i}}{r_{0}}^{2}\big{)}-(p_{i}-p_{r_{a}}){r_{0}}^{ 2}\bigg{]}\\ \,C=R_{2}^{N_{\psi}+1}\bigg{[}\frac{(1-\mu)\big{(}C_{1}^{2}-1)}{N_{\psi}+{ \mathbb{C}}_{1}}p_{r_{a}}+\beta\Big{(}\frac{R_{2}}{r_{a}}\Big{)}^{{\mathbb{C}} _{1}-1}-\frac{A_{1}-A_{2}}{N_{\psi}+2}R_{2}\bigg{]}\end{array}\right.\end{array} \,, \tag{14}\]  
where  
\[ \begin{array}{l}A_{1}=\frac{1}{{\mathbb{C}}_{1}-1}\bigg{[}\frac{2{\mu}-1}{1- \mu}{\mathbb{C}}_{2}-\frac{\mu({\mathbb{C}}_{1}+1)-1}{1-\mu}p_{a}\bigg{]}, \beta=\frac{{\mathbb{C}}_{2}+{\mathbb{I}}p_{a}}{{\mathbb{C}}_{1}-1},C_{1}= \frac{1+\sin\varphi}{1-\sin\varphi},N_{\psi}=\frac{1+\sin\psi}{1-\sin\psi}\\ A_{2}=\frac{1}{{\mathbb{C}}_{1}-1}\Big{[}\frac{2{\mu}-1}{1-\mu}{\mathbb{C}}_{2}+ \big{(}\frac{\mu}{1-\mu}-C_{1}\big{)}\big{]}p_{a}\Big{]},C_{2}=\frac{2c\cos \varphi}{1-\sin\varphi},m=\frac{(1-\mu)\big{(}1-{\mathbb{C}}_{1}^{2}\big{)}}{N_{ \psi}+{\mathbb{C}}_{1}}+C_{1}-\mu(C_{1}+1)\end{array}  \tag{15}\]  
CFG IV: In this configuration, the plastic zones appear in both the natural ground and the reinforced ground, and the reinforced ground has not been fully filled with the plastic zone. Hence, the corresponding boundary conditions include Equations (10), (11) and (12).