## 3 Results and Analysis
### Characteristics of Stress Distribution around a Single Roadway
#### 3.2.2 Solution of the Complex Stress Function  
According to the theory of [PERSON] [3], in the Cartesian coordinate system, the stress components (\(\sigma_{x}\), \(\sigma_{y}\), \(\tau_{xy}\)) on the Z-plane are expressed as follows:  
\[ \begin{cases}\sigma_{x}+\sigma_{y}=4\text{Re}\big{[}\varphi_{1}^{\prime}(z) \big{]}\\ \sigma_{y}-\sigma_{x}+2i\tau_{xy}=2\big{[}\overline{z}\varphi_{1}^{\prime \prime}(z)+\psi_{1}^{\prime}(z)\big{]}\end{cases}  \tag{5}\]  
where \(\psi(z)=\theta_{1}^{\prime}(z)\). In polar coordinates, the stress components (\(\sigma_{\rho}\), \(\sigma_{\theta}\), \(\tau_{r\theta}\)) can be transformed according to the following formula:  
\[ \begin{cases}\sigma_{\rho}+\sigma_{\theta}=\sigma_{x}+\sigma_{y}\\ \sigma_{\theta}-\sigma_{\rho}+2i\tau_{r\theta}=\big{(}\sigma_{y}-\sigma_{x}+2i \tau_{xy}\big{)}e^{2i\theta}\end{cases}  \tag{6}\]  
Figure 5: The relationship curve between the accuracy of the mapping function and \(m\). Let the mapping function from the external area of the opening on the \(Z\)-plane to the external area of the unit circle on the \(\zeta\)-plane be \(w\), by substituting Equation (6) into Equation (5), the following expression is obtained:  
\[\left\{ \begin{aligned} \sigma_{\rho}+\sigma_{\theta}=4\text{Re}[ \Phi(\zeta)]\\ \sigma_{\theta}-\sigma_{\rho}+2i\tau_{\rho\theta}=\frac{2\zeta^{2} }{\rho^{2}\omega^{\prime}(\zeta)}\big{[}\overline{\omega}(\zeta)\Phi^{\prime} (\zeta)+\omega^{\prime}(\zeta)\Psi(\zeta)\big{]}\end{aligned} \right. \tag{7}\]  
In the equation, \(\sigma_{\rho}\), \(\sigma_{\theta}\)\(\tau_{r\theta}\) represent the radial stress, hoop stress shear stress, respectively, at a certain point outside the opening on the \(Z\)-plane. Measures (\(\rho\), \(\theta\)) denote the coordinates of the mapped point on the \(\zeta\)-plane. \(\Phi\) and \(\Psi\) are the stress-related complex potential functions with the expressions as follows:  
\[\left\{ \begin{aligned} \Phi(\zeta)&=\frac{\phi^{\prime}( \zeta)}{\omega^{\prime}(\zeta)}\\ \Psi(\zeta)&=\frac{\phi^{\prime}(\zeta)}{\omega^{ \prime}(\zeta)}\end{aligned} \right. \tag{8}\]  
where:  
\[\left\{ \begin{aligned} \varphi(\zeta)&=B\omega( \zeta)+\varphi_{0}(\zeta)\\ \psi(\zeta)&=(B^{\prime}+iC^{\prime})\omega(\zeta)+ \psi_{0}(\zeta)\end{aligned} \right. \tag{9}\]  
The constants \(B\), \(B^{\prime}\)\(C^{\prime}\) reflect the situation of the far-field surrounding rock stress can be determined by the following formula:  
\[\left\{ \begin{aligned} B&=\frac{\sigma_{x}^{\infty}+ \sigma_{y}^{\infty}}{2}\\ B^{\prime}&=\frac{\sigma_{y}^{\infty}-\sigma_{y}^{ \infty}}{2}\\ C^{\prime}&=\tau_{xy}^{\infty}\end{aligned} \right. \tag{10}\]  
Figure 6: Comparison of mapped shapes and original shapes with different \(m\) values. (black dashed lines indicate the original shapes and colored solid lines indicate the mapped shapes). Let \(\sigma_{x}^{\infty}=p\), lateral pressure coefficient \(\lambda=\sigma_{x}^{\infty}/\sigma_{y}^{\infty}\). For the case in this study, \(\tau_{x}^{\infty}=0\); therefore:  
\[\left\{ \begin{aligned} B=\frac{1+\lambda}{4}p\\ B^{\prime}=\frac{1-\lambda}{2}p\\ C^{\prime}=0\end{aligned} \right. \tag{11}\]  
The unknown analytic functions \(\varphi_{0}(\zeta)\) and \(\psi_{0}(\zeta)\) can be represented by Laurent series:  
\[\left\{ \begin{aligned} \varphi_{0}(\zeta)=\sum\limits_{n=1}^{ \infty}a_{n}\zeta^{-n}\\ \psi_{0}(\zeta)=\sum\limits_{n=1}^{\infty}b_{n}\zeta^{-n}\end{aligned} \right.