## 4 Key Parameters of Tunnelling Method for the FT
### Fundamental Theories and Equations for Solution  
The SAM aims to decompose the multi-connected domain problem into a series of single-connected domain problems and yields an approximate solution through several interactions. The solution procedure of the SAM for solving the double-connected domain problem is shown in Figure 10. First, the AT section has been excavated in the infinite plane. The stress on the FT-I section outline can be solved under the external load, which is a single-connected domain problem. Subsequently, the FT-I section would be excavated. To balance the forces on the surrounding rock of the FT-I section, an external balance force is added to make the external load approach zero. Similarly, the stress on the AT section outline can be obtained when the FT-I section has been excavated.  
Figure 9: Mathematic model of two sections.  
Figure 11 shows the vector relationship of the coordinate system of circle 1 and circle 2. The transformation relationship between the derivative of the complex stress function of the dual coordinate system is given by Equations (3) and (4).  
\[\varphi_{2}^{\prime}(z_{2})=\varphi_{1}^{\prime}(z_{2}+c) \tag{3}\]  
\[\psi_{2}^{\prime}(z_{2})=\psi_{1}^{\prime}(z_{2}+c)+\overline{c}\varphi_{1}^{ \prime\prime}(z_{2}+c) \tag{4}\]  
Integrating both sides of Equations (3) and (4) and deleting the integration constants, the transformation relationship of the complex stress functions is obtained as follows:  
\[\varphi_{2}(z_{2})=\varphi_{1}(z_{2}+c) \tag{5}\]  
\[\psi_{2}(z_{2})=\psi_{1}(z_{2}+c)+\overline{c}\varphi_{1}(z_{2}+c) \tag{6}\]  
\[\varphi_{1}(z_{1})=\varphi_{2}(z_{1}-c) \tag{7}\]  
\[\psi_{1}(z_{1})=\psi_{2}(z_{1}-c)+\overline{c}\varphi_{2}(z_{1}-c) \tag{8}\]  
The solving steps of the interactions are expressed in Appendix A. After the third iteration, the complex stress around section AT and FT-I is given as follows:  
\[\left\{\begin{array}{l}\varphi_{1}(z_{1})=\varphi_{1}^{(1)}(z_{1})+\varphi_ {1}^{(2)}(z_{1})+\varphi_{1}^{(3)}(z_{1})\\ \psi_{1}(z_{1})=\psi_{1}^{(1)}(z_{1})+\psi_{1}^{(2)}(z_{1})+\psi_{1}^{(3)}(z_{ 1})\\ \varphi_{2}(z_{2})=\varphi_{2}^{(1)}(z_{2})+\varphi_{2}^{(2)}(z_{2})+\varphi_ {2}^{(3)}(z_{2})\\ \psi_{2}(z_{2})=\varphi_{2}^{(1)}(z_{2})+\varphi_{2}^{(2)}(z_{2})+\psi_{2}^{(3 )}(z_{2})\end{array}\right. \tag{9}\]  
Figure 11: Illustration of the vector relationship between the coordinate system of circles 1 and 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number of** **Iterations**} & \multirow{2}{*}{**Step**} & \multirow{2}{*}{**Coordinate** **System**} & \multicolumn{4}{c}{**Content**} & \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Load**} \\ \cline{4-4} \cline{6-7}  & & & & & & **Content to** **Be Solved** & \\ \hline \multirow{4}{*}{3} & (e) & \(o_{1}x_{1}y_{1}\) & \(\surd\) & & Solve the complex stress around AT & \(\varphi_{1}^{(3)}(z_{1})\), & Balance external forces \\  & (f) & \(o_{2}x_{2}y_{2}\) & \(\surd\) & & Solve the complex stress around FT-I & \(\varphi_{2}^{(3)}(z_{2})\), & Balance external forces \\ \hline \multirow{4}{*}{4} & (g) & \(o_{2}x_{2}y_{2}\) & & \(\surd\) & Solve the complex stress around FT-I & \(\varphi_{2}^{(4)}(z_{2})\), & Balance external forces \\  & (h) & \(o_{1}x_{1}y_{1}\) & & \(\surd\) & Solve the complex stress around AT & \(\varphi_{1}^{(4)}(z_{1})\), & Balance external forces \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._In Equation (9), the explicit expressions of all the complex stress functions can be found in Equations (A1)-(A12). The stress component of any point in the surrounding rock is given by Equations (10) and (11) [32].  
\[\sigma_{\rho}+\sigma_{\theta}=4\text{Re}[\varphi_{1}^{\prime}(z_{1})] \tag{10}\]  
\[\sigma_{\theta}-\sigma_{\rho}+2i\tau_{\rho\theta}=2e^{2i\theta}[\pi\varphi_{1}^ {\prime\prime}(z_{1})+\psi_{1}^{\prime}(z_{1})] \tag{11}\]  
where \(\sigma_{\theta}\), \(\sigma_{\rho}\), and \(\tau_{\rho\theta}\) are the radial stress, hoop stress, and shear stress along the tunnel outline, respectively.