# Mechanical Analysis of Bolt-Lining Combined Support in Deeply Buried Circular Tunnel
## 2 Fundamental Principles and Equations
### Two Analytic Functions Considering the Excavation of the Tunnel
#### 2.4.3 Displacement Compatibility Condition between Lining and Surrounding Rock
\begin{array}{l}t_{j}=\left[\begin{array}{c}-\frac{\cos a_{j}-i\sin a_{j} }{2\pi(1+\kappa_{1})}\frac{R_{0}\sigma}{R_{0}+c_{j}!\sigma}+\frac{\kappa_{1 }(\cos a_{j}-i\sin a_{j})}{2\pi(1+\kappa_{1})}\frac{R_{0}\sigma^{2}}{R_{0} \sigma+c_{j}}\\ +\frac{\cos a_{j}+i\sin a_{j}}{2\pi(1+\kappa_{1})}\frac{c_{j}!-R_{0}^{2}}{c_{j }}\frac{R_{0}\sigma^{2}}{(R_{0}\sigma+c_{j})^{2}}\end{array}\right]\\ -\left[\begin{array}{c}-\frac{\cos a_{j}-i\sin a_{j}}{2\pi(1+\kappa_{1})} \frac{R_{0}\sigma}{R_{0}+c_{j}!\sigma}+\frac{\kappa_{1}(\cos a_{j}-i\sin a_{j} )}{2\pi(1+\kappa_{1})}\frac{R_{0}\sigma^{2}}{R_{0}\sigma^{2}+c_{j+n}}\\ +\frac{\cos a_{j}+i\sin a_{j}}{2\pi(1+\kappa_{1})}\frac{c_{j}!+\sigma^{2}\!-R_ {0}^{2}}{c_{j}+n}\frac{R_{0}\sigma^{2}}{(R_{0}\sigma^{2}+c_{j+n})^{2}}\end{array} \right]\\ w_{j}=\left[\begin{array}{c}\frac{\kappa_{1}(\cos a_{j}+i\sin a_{j})}{2 \pi(1+\kappa_{1})}\ln\Bigl{(}1+\frac{R_{0}}{c_{j}!\sigma}\Bigr{)}+\frac{\cos a _{j}+i\sin a_{j}}{2\pi(1+\kappa_{1})}\ln\Bigl{(}1+\frac{R_{0}\sigma}{c_{j}} \Bigr{)}\\ -\frac{\kappa_{1}(\cos a_{j}-i\sin a_{j})}{2\pi(1+\kappa_{1})}\frac{c_{j}!}{R_ {0}\sigma^{2}+c_{j}}\frac{\cos a_{j}-i\sin a_{j}}{2\pi(1+\kappa_{1})}\frac{R_{0 }\sigma}{c_{j}!}\\ -\frac{\kappa_{1}(\cos a_{j}-i\sin a_{j})}{2\pi(1+\kappa_{1})}\frac{R_{0}\sigma ^{3}}{R_{0}\sigma^{2}+c_{j}}-\frac{\cos a_{j}+i\sin a_{j}}{2\pi(1+\kappa_{1} )}\frac{c_{j}!\sigma^{2}\!-R_{0}^{2}}{c_{j}}\frac{R_{0}\sigma^{3}}{(R_{0} \sigma+c_{j})^{2}}\end{array}\right]\\ -\left[\begin{array}{c}\frac{\kappa_{1}(\cos a_{j}+i\sin a_{j})}{2 \pi(1+\kappa_{1})}\ln\Bigl{(}1+\frac{R_{0}}{c_{j}!\sigma}\Bigr{)}+\frac{\cos a _{j}+i\sin a_{j}}{2\pi(1+\kappa_{1})}\ln\Bigl{(}1+\frac{R_{0}\sigma}{c_{j+n} }\Bigr{)}\\ -\frac{\cos a_{j}-i\sin a_{j}}{2\pi(1+\kappa_{1})}\frac{c_{j}!+\sigma}{\bar{c} _{j}\sigma^{2}+n}+\frac{\cos a_{j}-i\sin a_{j}}{2\pi(1+\kappa_{1})}\frac{R_{0 }\sigma}{c_{j+n}}\\ -\frac{\kappa_{1}(\cos a_{j}-i\sin a_{j})}{2\pi(1+\kappa_{1})}\frac{R_{0} \sigma^{3}}{R_{0}\sigma^{2}+c_{j+n}}-\frac{\cos a_{j}+i\sin a_{j}}{2\pi(1+ \kappa_{1})}\frac{c_{j}!\sigma^{2}\!-R_{0}^{2}}{c_{j}}\frac{R_{0}\sigma^{3}}{(R_{0} \sigma+c_{j+n})^{2}}\end{array}\right]\end{array}  \tag{25}\]  
Moreover, the influence caused by the lining on the surrounding rock drops when \(\zeta\) increases, namely, when \(\zeta\rightarrow\infty,\;u_{0}^{\prime}\to 0\) and \(u_{0}^{\prime\theta}\to 0\). Therefore, the following relationship can be obtained:  
\[\lim_{\zeta\rightarrow\infty}[2G_{1}(u_{0}^{\rho}+iu_{0}^{\beta})]=\lim_{\zeta \rightarrow\infty}\left\{\frac{\zeta}{\rho}\frac{\overline{\omega^{\prime}( \zeta)}}{\overline{\omega^{\prime}(\zeta)}}\biggl{[}\kappa_{1}\varphi_{0}(\zeta)- \frac{\omega(\zeta)}{\overline{\omega^{\prime}(\zeta)}}\overline{\varphi^{ \prime}_{0}(\zeta)}\overline{\varphi^{\prime}_{0}(\zeta)}-\overline{\psi_{0}( \zeta)}\biggr{]}\right\}=0 \tag{26}\]  
Since \(\lim_{\zeta\rightarrow\infty}[\varphi_{0}(\zeta)]=a_{0}\), \(\lim_{\zeta\rightarrow\infty}[\frac{\omega(\zeta)}{\overline{\omega^{\prime}( \zeta)}}\overline{\varphi^{\prime}_{0}(\zeta)}]=0\), and \(\lim_{\zeta\rightarrow\infty}[\psi_{0}(\zeta)]=d_{0}\), according to Equation (26), we have  
\[a_{0}=d_{0}/\kappa_{1} \tag{27}\]2.4.4. Displacement Compatibility Condition between Bolts and Surrounding Rock  
Due to the excavation of the tunnel, point-anchored rock bolts interact with the surrounding rock. Before the failure of the bolts, the points of connection (the bolt head and midpoint of the anchorage section) between the surrounding rock and the adjacent bolts will always be bonded and deform together.