## 3 Elastic Solutions to Stress Surrounding Rock of Rectangular Roadways with Super-Large Sections
### Basic Theory  
According to the theory of complex variable function, the problem of non-circular roadway needs to be calculated by mapping the outside of a non-circular roadway into a unit circle through conformal transformation. Accordingly, rectangular roadways with super-large sections were mapped into a unit circle via the mapping function \(z=\omega(\zeta)\) for surrounding rock stress calculation. The analytical solution to the stress upon the surrounding rock of rectangular roadways with super-large sections was calculated by using the theory on elastic mechanics [22]. First of all, two analytic functions \(\varphi(\xi)\) and \(\psi(\xi)\) were determined, and their fundamental equations went as follows:  
\[\sigma_{\theta}+\sigma_{\rho}=2[\Phi(\zeta)+\overline{\Phi(\zeta)}]=4\text{Re }\Phi(\zeta) \tag{1}\]  
\[\sigma_{\theta}-\sigma_{\rho}+2i\tau_{\theta\theta}=\frac{2\zeta^{2}}{\rho^{2 }\overline{\omega^{\prime}(\xi)}}[\overline{\omega(\xi)}\Phi^{\prime}(\zeta)+ \omega^{\prime}(\xi)\Psi(\zeta)] \tag{2}\]  
where \(\varphi(\xi)\), \(\psi(\xi)\), \(\Phi(\xi)\) and \(\Psi(\xi)\) are complex analytic functions of the complex variable \(\xi\); \(\sigma_{\theta}\) is the tangential stress of rectangular roadways, MPa; \(\sigma_{\rho}\) is the radial stress of that roadway, MPa; and \(\tau_{\rho\theta}\) is the shear stress of that roadway, MPa.  
\[\begin{array}{l}\varphi(\zeta)=\frac{1+\mu}{8\pi}(X+iY)\ln\zeta+B\omega(\zeta )+\varphi_{0}(\zeta)\\ \psi(\zeta)=-\frac{3-\mu}{8\pi}(X-iY)\ln\zeta+(B^{\prime}+iC^{\prime})\omega( \zeta)+\psi_{0}(\zeta)\end{array} \tag{3}\]  
\[\left\{\begin{array}{l}\Phi(\zeta)=\frac{\varphi^{\prime}(\zeta)}{\omega^{ \prime}(\zeta)}\\ \Psi(\zeta)=\frac{\varphi^{\prime}(\zeta)}{\omega^{\prime}(\zeta)}\end{array}\right. \tag{4}\]  
where \(X\) and \(Y\) represent resultant surface forces, MPa, along the \(x\) and \(y\) axes on the boundary, respectively; and \(B\), \(B^{\prime}\) and \(C^{\prime}\) are constants associated with \(\sigma_{\theta}\) and \(\sigma_{\rho}\).  
Figure 3: Computation model of rectangular roadways’ surrounding rock.  
\[\varphi_{0}(\zeta)=\sum\limits_{n=1}^{\infty}\alpha_{n}\zeta^{n},\psi_{0}(\zeta)= \sum\limits_{n=1}^{\infty}\beta_{n}\zeta^{n}\]  
are analytic functions about \(\zeta\) within the unit circle at the external tangent center of rectangular roadways with super-large sections and continuous within the circle and on the circumference, and they can be expressed as follows:  
\[\varphi_{0}(\zeta)=\frac{1}{2\pi i}\int_{\sigma}\frac{f_{0}}{\sigma-\zeta} \mathrm{d}\sigma-\frac{1}{2\pi i}\int_{\sigma}\frac{\omega(\sigma)}{\omega^{ \prime}(\sigma)}\frac{\overline{\varphi_{0}^{\prime}(\zeta)}}{\sigma-\zeta} \mathrm{d}\sigma \tag{5}\]  
\[\psi_{0}(\zeta)=\frac{1}{2\pi i}\int_{\sigma}\frac{\overline{f_{0}}}{\sigma- \zeta}\mathrm{d}\sigma-\frac{1}{2\pi i}\int_{\sigma}\frac{\overline{\omega( \sigma)}}{\omega^{\prime}(\sigma)}\frac{\varphi_{0}^{\prime}(\zeta)}{\sigma- \zeta}\mathrm{d}\sigma \tag{6}\]  
where \(f_{0}\) is a notation introduced to facilitate calculation:  
\[f_{0}=i\int\left(\overline{X}+i\overline{Y}\right)ds-\frac{X+iY}{2\pi}\ln \sigma-\frac{1+\mu}{8\pi}(X-iY)\frac{\omega(\sigma)}{\omega^{\prime}(\sigma)} \sigma-2B\omega(\sigma)-(B^{\prime}-iC^{\prime})\overline{\omega(\sigma)} \tag{7}\]