# Stress Analysis and Spalling Failure Simulation on Surrounding Rock of Deep Arch Tunnel
## 2 Stress Analysis of Surrounding Rock
### Stress Solution of the Surrounding Rock  
The determination of the analytical solution for the surrounding rock stress of a tunnel with an arch cross-section shape is more complex. It requires the use of complex functions and angle-preserving transformations in the elasticity theory. Replacing the corner points in the tunnel cross-section with circular angle approximations and the approximate elastic solutions for the stress distribution in the surrounding rock were obtained through mapping transformations. Given the stress at any point on the boundary of the unit circle on a complex plane [31], the stress around the tunnel on the physical plane is determined by the following:  
\[\left\{ \begin{array}{l}\sigma_{\rho}=\frac{1}{2}\bigg{\{}4\text{Re}(\Phi( \zeta))-\text{Re}\bigg{[}\frac{2\zeta^{2}}{\omega^{\prime}(\zeta)}[\overline{ \omega(\xi)}\Phi^{\prime}(\zeta)+\omega^{\prime}(\zeta)\Psi(\zeta)]\bigg{]} \bigg{\}}\\ \sigma_{\theta}=\frac{1}{2}\bigg{\{}4\text{Re}(\Phi(\xi))+\text{Re}\bigg{[} \frac{2\zeta^{2}}{\omega^{\prime}(\zeta)}[\overline{\omega(\xi)}\Phi^{\prime} (\zeta)+\omega^{\prime}(\zeta)\Psi(\zeta)]\bigg{]}\bigg{\}}\\ \tau_{\rho\theta}=\frac{1}{2i}\text{Im}\bigg{\{}\frac{2\zeta^{2}}{\omega^{ \prime}(\zeta)}[\overline{\omega(\xi)}\Phi^{\prime}(\zeta)+\omega^{\prime}( \zeta)\Psi(\zeta)]\bigg{\}}\end{array} \right. \tag{19}\]  
For a semi-circular arch with a rise-span ratio of \(f/B=1/2\) and aspect ratio of \(B/H=1\), the surrounding rock stresses of a three-centered arch with a rise-span ratio of \(f/B=1/3\) and \(f/B=1/4\), and aspect ratios of \(B/H=1.20\) and \(B/H=1.33\), the constant coefficients of  
Figure 2: The mapping diagram of outer domain of arch hole boundary on the z-plane to outer domain of unit circle boundary on the \(\zeta\)-plane. the mapping function for different tunnels were obtained using the BOX composite shape method from the literature [32] based on the geometrical parameters of the aforementioned tunnels, as shown in Table 1.  
Substituting the results in Table 1 into Equations (11) and (16) yields the following mapping function for each tunnel:  
\[\left\{ \begin{array}{l}\omega_{1}(\zeta)=27.12\zeta-2.95-0.18\zeta^{-1}+2.29 \zeta^{-2}-2.16\zeta^{-3}+0.90\zeta^{-4}+0.20\zeta^{-5}-0.23\zeta^{-6}\\ \omega_{2}(\zeta)=25.30\zeta-2.31-2.40\zeta^{-1}+1.89\zeta^{-2}-2.44\zeta^{-3}+ 0.72\zeta^{-4}+0.34\zeta^{-5}-0.28\zeta^{-6}\\ \omega_{3}(\zeta)=24.40\zeta-1.75-3.58\zeta^{-1}+1.32\zeta^{-2}-2.55\zeta^{-3}+ 0.65\zeta^{-4}+0.47\zeta^{-5}-0.21\zeta^{-6}\end{array} \right. \tag{20}\]  
According to Equation (20), the contours of the mapping function for the arched hole are shown in Figure 3 (the bottom boundary line shifts to the same level). The standard arch hole boundary and the hole profile boundary determined through the mapping function are extremely close and can be approximated to reflect the arch hole boundaries. Based on the mapping function derived from Equation (20), the following equation can be directly derived (for a semi-circular arch with a rise-span ratio \(f/B=1/2\) and an aspect ratio \(B/H=1\)):  
\[\left\{ \begin{array}{l}\omega_{1}(\sigma)=27.12\sigma-2.95-0.18\sigma^{-1}+2.29\sigma^{-2}-2.16\sigma^{-3}+0.90\sigma^{-4}+0.20\sigma^{-5}-0.23\sigma^{-6} \\ \dfrac{\omega^{\prime}{}_{1}(\sigma)}{\omega_{1}(\sigma)}=27.12+0.18\sigma^{-2 }-4.58\sigma^{-3}+6.48\sigma^{-4}-3.60\sigma^{-5}-10.00\sigma^{-6}+1.38\sigma^ {-7}\\ \dfrac{\omega^{\prime}{}_{1}(\sigma)}{\omega_{1}(\sigma)}=27.12\sigma^{-1}-2.9 5-0.18\sigma+2.29\sigma^{2}-2.16\sigma^{3}+0.90\sigma^{4}+0.20\sigma^{5}-0.2 3\sigma^{6}\\ \dfrac{\omega^{\prime}{}_{1}(\sigma)}{\omega_{1}^{\prime}(\sigma)}=27.12+0.18 \sigma^{2}-4.58\sigma^{3}+6.48\sigma^{4}-3.60\sigma^{5}-1.00\sigma^{6}+1.38 \sigma^{7}\end{array} \right. \tag{21}\]  
\[\dfrac{\omega(\sigma)}{\omega^{\prime}(\sigma)}=D_{6}\sigma^{-6}+D_{5}\sigma^{- 5}+D_{4}\sigma^{-4}+D_{3}\sigma^{-3}+D_{2}\sigma^{-2}+D_{1}\sigma^{-1}+D_{0}+ \sum_{k=1}^{\infty}d_{k}\sigma^{k} \tag{22}\]