## 2 Simplified Analytical Solution
### Closed-Form Solution of Tunnel through Soft-Hard Stratum Junction with Flexible Joints  
In this section, the analytical solution to the problem in Figure 2 is developed. Four lining segments in Figure 2 are considered. For the sake of solution, the deformation of each part is separately studied, and local coordinate systems \(O_{1}\)-\(x_{1}y_{1}\), \(O_{2}\)-\(x_{2}y_{2}\), \(O_{3}\)-\(x_{3}y_{3}\) and \(O_{4}\)-\(x_{4}y_{4}\) are built, as shown in Figure 2. Hence, the expression of the Green's functions \(G(x,x_{0})\) of the model appears as a four-step piecewise function. The free boundaries are applied in the mechanical model, as presented by [PERSON] et al. [25], so the Green's functions of the beams are:  
\[\begin{array}{l}G_{1}(x_{1},x_{10})=H(x_{1}-x_{10})\phi_{11}(x_{1}-x_{10})+A _{1}\phi_{12}(x_{1})+B_{1}\phi_{13}(x_{1})+C_{1}\phi_{14}(x_{1})+D_{1}\phi_{1 5}(x_{1})\\ G_{2}(x_{2},x_{20})=H(x_{2}-x_{20})\phi_{21}(x_{2}-x_{20})+A_{2}\phi_{22}(x_{  
Due to the existence of joints, the discontinuous displacement and rotation angle at joint can be written as:  
\[\begin{array}{l}W_{k}|_{x_{k}=t_{k^{-}}}=W_{k+1}|_{x_{k+1}=0^{+}}-\frac{1}{C_{k ^{0}k}}\kappa GA_{k+1}(W_{k+1}^{\prime}|_{x_{k+1}=0^{+}}-\[PERSON]_{x_{k+1}}|_{x_{k+1 }=0^{+}}),\\ \[PERSON]_{k}|_{x_{k}=t_{k^{-}}}=\[PERSON]_{k+1}|_{x_{k+1}=0^{+}}-\frac{1}{C_{k^{0}k}} \mathbb{E}_{k+1}\[PERSON]_{k+1}^{\prime}|_{x_{k+1}=0^{+}},-E_{k}\[PERSON]_{k}^{\prime}|_{ x_{k}=t_{k^{-}}}=-E_{k+1}\[PERSON]_{k+1}^{\prime}|_{x_{k+1}=0^{+}},\\ \kappa GA_{k}(W_{k}^{\prime}|_{x_{k}=t_{k^{-}}}-\[PERSON]_{k}|_{x_{k}=t_{k^{-}}})= \kappa GA_{k+1}(W_{k+1}^{\prime\prime}|_{x_{k+1}=0^{+}}-\[PERSON]_{k+1}|_{x_{k+1}=0^ {+}})\end{array} \tag{17}\]  
Substituting Equation (16) into Equation (17), the following relation can be obtained:  
\[U_{k+1}=(T_{k})^{-1}S_{k}U_{k}-(T_{k})^{-1}M_{k}\ (k=1,2) \tag{18}\]  
where \(U_{k}=(A_{k},\ B_{k},\ C_{k},\ D_{k})^{T}\), \(U_{k+1}=(A_{k+1},\ B_{k+1},\ C_{k+1},\ D_{k+1})^{T}\), and other parameters in Equation (18) are given in Appendix A.  
The continuity conditions at the joint \(k=3\) are different from those of other joints due to the difference between its local coordinate and others, and it can be expressed as:  
\[\begin{array}{l}W_{3}|_{x_{3}=t_{3^{-}}}=W_{4}|_{x_{k}=t_{k^{-}}}+F_{x3} \kappa GA_{1}(W_{k+1}^{\prime}|_{x_{k}=t_{k^{-}}}-\[PERSON]_{x_{k}}|_{x_{k}=t_{k^{- }}}),\\ \[PERSON]_{3}|_{x_{3}=t_{3^{-}}}=-\[PERSON]_{4}|_{x_{k}=t_{k^{-}}}-F_{9}\mathbb{E}_{1} \[PERSON]_{4}^{\prime}|_{x_{k}=t_{k^{-}}}-E_{1}\[PERSON]_{3}^{\prime}|_{x_{3}=t_{k^{-}} }=-E_{1}\[PERSON]_{4}^{\prime}|_{x_{k}=t_{k^{-}}},\\ \kappa GA_{3}(W_{3}^{\prime}|_{x_{3}=t_{3^{-}}}-\[PERSON]_{3}|_{x_{3}=t_{3^{-}}})= -\kappa GA_{1}(W_{4}^{\prime}|_{x_{k}=t_{k^{-}}}-\[PERSON]_{4}|_{x_{k}=t_{k^{-}}}) \end{array} \tag{19}\]  
Substituting Equation (16) into Equation (19), the following equation is obtained:  
\[U_{4}=(T_{3})^{-1}S_{3}U_{3}-(T_{3})^{-1}M_{3} \tag{20}\]  
where \(U_{3}=(A_{3},\ B_{3},\ C_{3},\ D_{3})^{T}\), \(U_{4}=(A_{4},\ B_{4},\ C_{4},\ D_{4})^{T}\), and other parameters in Equation (20) are given in Appendix A.  
The unascertained coefficients can be calculated via Equations (18) and (20). Then, inserting these coefficients into Equation (16), the Green's functions in the local coordinate systems can be obtained. Subsequently, the Green's functions in the global coordinate systems can be expressed as:  
\[G(x;x_{0})=\left\{\begin{array}{l}G_{1}(x,x_{0})\,\ \ \ \ x\in[0,L_{1}^{-})\\ G_{2}(x-L_{1},x_{0}-L_{1}),\ x\in(L_{1}^{+},(L_{1}+L_{2})^{-})\\ G_{3}(x-L_{1}-L_{2},x_{0}-L_{1}-L_{2}),\ x\in((L_{1}+L_{2})^{+},(L_{1}+L_{2}+L _{3})^{-}]\\ G_{3}(L-x,L-x_{0}),\ x\in((L_{1}+L_{2}+L_{3})^{+},L]\end{array}\right. \tag{21}\]  
Thus, the responses of the tunnel through soft-hard stratum junction equipped with flexible joints can be obtained by Equation (7) and Equations (13)-(15).