## 2 Deterministic Analysis Method
### Soil-Existing Tunnel Interaction
 \tag{15}\]  
Then, the bending moment and shear force can be expressed as follows:  
\[M=-\frac{(EI)_{\text{eq}}D_{\text{e}}}{(\kappa GA)_{\text{eq}}}\Bigg{(}\frac {(\kappa GA)_{\text{eq}}}{D_{\text{e}}}\frac{\text{d}^{2}w}{\text{d}x^{2}}+k (x)U-k(x)w\Bigg{)} \tag{16}\]  
\[Q=-\frac{(EI)_{\text{eq}}D_{\text{e}}}{(\kappa GA)_{\text{eq}}}\Bigg{(}\frac {(\kappa GA)_{\text{eq}}}{D_{\text{e}}}\frac{\text{d}^{3}w}{\text{d}x^{3}}+ \frac{\text{d}k(x)}{\text{d}x}U+k(x)\frac{\text{d}U}{\text{d}x}-\frac{\text{d }k(x)}{\text{d}x}w-k(x)\frac{\text{d}w}{\text{d}x}\Bigg{)} \tag{17}\]  
Equation (15) is solved by adopting the finite difference method (FDM), and the finite discretization of the tunnel is illustrated in Figure 1. The tunnel is separated into \(n\) real segments and four virtual segments along the longitudinal direction. Therefore, the total number of discretization nodes is \(n+5\). Based on the finite difference theory, the central differential form for the governing differential equation can be given as follows:  
\[ \begin{array}{l}(EI)_{\text{eq}}\frac{6w_{j}-4(w_{j+1}+w_{j-1})+(w_{j+2}+w_ {j-2})}{\text{d}x}\\ -\frac{(EI)_{\text{eq}}D_{\text{e}}}{(\kappa GA)_{\text{eq}}}\Big{(}\frac{(k_{i+ 1}-2k_{i}+k_{i-1})w_{i}}{\text{d}^{2}}+\frac{(k_{i+1}-k_{i-1})(w_{j+1}-w_{j-1}) }{2^{2}}+\frac{k_{i}(w_{i+1}-2w_{j}+w_{j-1})}{\text{d}^{2}}\Big{)}+k_{i}w_{i}D _{\text{e}}\\ =k_{i}U_{i}D_{\text{e}}-\frac{(EI)_{\text{eq}}D_{\text{e}}}{(\kappa GA)_{\text {eq}}}\Big{(}\frac{(k_{i+1}-2k_{i}+k_{i-1})U_{i}}{\text{d}^{2}}+\frac{(k_{i+1} -k_{i-1})(U_{i+1}-U_{i-1})}{2^{2}}+\frac{k_{i}(U_{i+1}-2U_{i}+U_{i-1})}{\text{ d}^{2}}\Big{)}\end{array}  \tag{18}\]  
where \(w_{i}\) is the settlement of the tunnel at node \(i\); \(k_{i}\) is the subgrade reaction coefficient at node \(i\); \(U_{i}\) is the greenfield soil settlement at node \(i\); \(l=S/n\) with \(S\) as the tunnel length, \(i=0,\,1,\,2,\,3,\,\ldots,\,n\). The two ends of the tunnel are assumed to be unconstrained, and the boundary conditions are given as  
\[M_{0}=-\frac{(EI)_{\text{eq}}}{(\kappa GA)_{\text{eq}}}\Bigg{(}(\kappa GA)_{ \text{eq}}\frac{w_{1}-2w_{0}+w_{-1}}{\text{$l^{2}$}}+k_{0}U_{0}D_{\text{e}}-k_ {0}D_{\text{e}}w_{0}\Bigg{)}=0 \tag{19}\]  
\[M_{n}=-\frac{(EI)_{\text{eq}}}{(\kappa GA)_{\text{eq}}}\Bigg{(}(\kappa GA)_{ \text{eq}}\frac{w_{n+1}-2w_{n}+w_{n-1}}{\text{$l^{2}$}}+k_{n}U_{n}D_{\text{e}}-k _{n}D_{\text{e}}w_{n}\Bigg{)}=0 \tag{20}\]  
\[Q_{0}=-\frac{(EI)_{\text{eq}}D_{\text{e}}}{(\kappa GA)_{\text{eq}}}\Bigg{(} \frac{(\kappa GA)_{\text{eq}}\frac{w_{2}-2w_{1}+2w_{-1}-w_{-2}}{\text{$l^{2}$}} \frac{k_{i-1}-k_{-1}}{\text{$l^{2}$}}w_{0}-k_{0}\frac{w_{1}-w_{-1}}{\text{$l^{2}$}} \Bigg{)}}{+k_{0}\frac{U_{1}-U_{1}}{\text{$l^{2}$}}-\frac{k_{1}-k_{-1}}{\text{$l^{ 2}$}}w_{0}-k_{0}\frac{w_{1}-w_{-1}}{\text{$l^{2}$}}}\Bigg{)}=0 \tag{21}\]\[Q_{n}=-\frac{(EI)_{\text{eq}}D_{\text{e}}}{\left(\kappa GA\right)_{\text{eq}}} \left( \begin{array}{c}\frac{(\kappa GA)}{D_{\text{e}}}\frac{w_{n+2-2w_{n+1}+2 \eta_{n-1}-w_{n-2}}}{2^{2}}+\frac{k_{n+1}-k_{n-1}}{2l}U_{n}\\ +k_{n}\frac{U_{n+1}-U_{n-1}}{2l}-\frac{k_{n+1}-k_{n-1}}{2l}w_{n}-k_{n}\frac{w_ {n+1}-w_{n-1}}{2l}\end{array} \right)=0 \tag{22}\]  
The settlement of the virtual nodes located at both ends of the tunnel can be determined through the boundary conditions and expressed in the following way:  
\[w_{-1}=\left(\frac{k_{0}l^{2}}{\left(\kappa GA\right)_{\text{eq}}}+2\right)w_ {0}-w_{1}-\frac{k_{0}U_{0}D_{\text{e}}l^{2}}{\left(\kappa GA\right)_{\text{eq}}} \tag{23}\]  
\[w_{n+1}=\left(\frac{k_{n}l^{2}}{\left(\kappa GA\right)_{\text{eq}}}+2\right) w_{n}-w_{n-1}-\frac{k_{n}U_{n}D_{\text{e}}l^{2}}{\left(\kappa GA\right)_{\text{eq}}} \tag{24}\]  
\[