# Mechanical Analysis of Bolt-Lining Combined Support in Deeply Buried Circular Tunnel
## 2 Fundamental Principles and Equations  
### Two Analytic Functions Considering Lining Effect  
Firstly, the conformal transformation is adapted to map the lining and the outer area of the tunnel into a concentric circle and the outer area of a unit circle, respectively. See Figure 2. Moreover, the mapping function is as follows:  
\[z=\omega(\zeta)=R_{0}\zeta \tag{1}\]  
where \(R_{0}\) is the radius of the circular tunnel; \(\zeta=\rho e^{i\Theta}\) and \(\zeta=\sigma=e^{i\Theta}\) (the tunnel boundary); \(i=\sqrt{-1}\); \(\rho\) is the radius in the \(\zeta\) plane, and \(\rho=r/R_{0}\), where \(r\) is the polar radius in the \(z\) plane; \(\theta\) is the polar angle in the \(\zeta\) plane, as is the polar angle in the \(z\) plane. The outer radius of the ring is set to 1 and the inner radius \(R\) is equal to \(R_{1}/R_{0}\); \(R_{1}\) is the radius of the inner boundary of the lining. The inner and outer boundaries of the lining are indicated by \(L^{-}\) and \(L^{+}\) in the \(z\) plane, and \(\gamma^{-}\) and \(\gamma^{+}\) in the \(\zeta\) plane, respectively.  
Figure 1: The equivalent action of point-anchored rock bolts.  
After the application of the lining, the surrounding rock interacts with the lining. The two analytic functions representing the effect of the lining on the surrounding rock can be written as follows:  
\[\varphi_{0}(\zeta)=a_{0}+\sum_{k=1}^{\infty}a_{k}\zeta^{-k},\psi_{0}(\zeta)=b_{0} +\sum_{k=1}^{\infty}b_{k}\zeta^{-k} \tag{2}\]  
Correspondingly, the two analytic functions representing the effect of the surrounding rock on the lining can be written as follows:  
\[\varphi_{1}(\zeta)=d_{0}+\sum_{k=1}^{\infty}e_{k}\zeta^{-k}+\sum_{k=1}^{\infty }f_{k}\zeta^{k},\psi_{1}(\zeta)=g_{0}+\sum_{k=1}^{\infty}p_{k}\zeta^{-k}+\sum_{ k=1}^{\infty}q_{k}\zeta^{k} \tag{3}\]  
The above \(a_{0}\), \(b_{0}\), \(d_{0}\), \(g_{0}\), \(a_{k}\), \(b_{k}\), \(e_{k}\), \(f_{k}\), \(p_{k}\), and \(q_{k}\) are all real numbers to be determined. The series is taken as a finite term, i.e., \(Ne\) in this paper, and when \(Ne\) is large enough, the result is sufficiently accurate.  
Using \(\varphi_{0}(\zeta)\), \(\psi_{0}(\zeta)\), \(\varphi_{1}(\zeta)\) and \(\psi_{1}(\zeta)\), the displacement of any point within the surrounding rock restricted by the lining and the displacements of any point within the lining due to the action of the surrounding rock can be solved as follows:  
\[2G_{1}(u_{0}^{\rho}+iu_{0}^{\theta})=\frac{\overline{\zeta}}{\rho}\frac{ \overline{\omega^{\prime}(\zeta)}}{\overline{\omega^{\prime}(\zeta)}}\Bigg{[} x_{1}\varphi_{0}(\zeta)-\frac{\omega(\zeta)}{\overline{\omega^{\prime}(\zeta)}} \overline{\overline{\varphi^{\prime}_{0}(\zeta)}}-\overline{\psi_{0}(\zeta)} \Bigg{]} \tag{4}\]  
\[2G_{2}(u_{1}^{\rho}+iu_{1}^{\theta})=\frac{\overline{\varphi}}{\rho}\frac{ \overline{\omega^{\prime}(\zeta)}}{\overline{\omega^{\prime}(\zeta)}}\Bigg{[} x_{2}\varphi_{1}(\zeta)-\frac{\omega(\zeta)}{\overline{\omega^{\prime}(\zeta)}} \overline{\overline{\varphi^{\prime}_{1}(\zeta)}}-\overline{\psi_{1}(\zeta)} \Bigg{]} \tag{5}\]  
Figure 2: Lined circular tunnel and mapped result.  
where \(u_{0}^{\rho}\) and \(u_{0}^{\theta}\) are the radial and tangential displacement components in the \(z\) plane, respectively, \(K_{1}=3\)-\(4\)\(\mu_{1}\), \(K_{2}=3\)-\(4\)\(\mu_{2}\); (\(\mu_{1}\), \(G_{1}\), \(E_{1}\)), (\(\mu_{2}\), \(G_{2}\), \(E_{2}\)) are [PERSON]'s ratio, shear modulus, and Young's modulus of the rock mass and lining, respectively; and \(G_{1}=E_{1}/[2(1+m_{1})]\), \(G_{2}=E_{2}/[2(1+m_{2})]\).