# 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
Based on the displacement compatibility condition, we have  
\[(1-\eta)\Delta u_{3}^{\rho}-\Delta u_{0}^{\rho}-\Delta u_{2}^{\rho}=\Delta u( \delta);\ (\delta=1,2,\cdot\cdot\cdot,n) \tag{28}\]  
where \(\Delta u(\delta)\) is the elongation of the \(\delta\)-th rock bolt, which can be calculated as follows:  
\[\Delta u(\delta)=\frac{4N_{\delta}L}{E_{3}\pi d^{2}};\ (\delta=1,2,\cdot\cdot \cdot,n) \tag{29}\]  
where \(E_{3}\), \(d\) represent the Young's modulus and the diameter of the rock bolt, respectively. By combining Equations (2), (4), (12)-(15), (28), and (29), the displacement compatibility condition between the \(\delta\)-_th_ rock bolt and the rock mass can be obtained as follows:  
\[ \begin{array}{l}\frac{1}{4\pi(1+\kappa_{1})G_{1}}\sum\limits_{j=1}^{n}\left\{ N_{\beta}\sum\limits_{l=1}^{10}\left[h_{l}(\beta_{j},\alpha_{j},R_{0},\alpha_{ \delta})-h_{l}(\beta_{j},\alpha_{j},R_{0}+L,\alpha_{\delta})\right]\right\}\\ +\frac{\cos{\alpha_{\delta}}-\sin{\alpha_{\delta}}}{2C_{1}}\left\{\sum\limits _{k=1}^{Ne}a_{k}\left[(1+L/R_{0})^{-k}(\cos{k\alpha_{\delta}}-i\sin{k\alpha_{ \delta}})\right.\right.\\ \left.\left.+k(1+L/R_{0})^{-k}(\cos(k+2)\alpha_{\delta}+i\sin(k+2)\alpha_{ \delta})\right]\\ -\sum\limits_{k=1}^{Ne}b_{k}[(1+L/R_{0})^{-k}(\cos{k\alpha_{\delta}}-i\sin{k \alpha_{\delta}})]\right\}\\ -\frac{\cos{\alpha_{\delta}}-\sin{\alpha_{\delta}}}{2C_{1}}\left\{\sum\limits _{k=1}^{Ne}a_{k}\left[\cos{k\alpha_{\delta}}-i\sin{k\alpha_{\delta}}+k(\cos(k+ 2)\alpha_{\delta}\right.\\ \left.+i\sin(k+2)\alpha_{\delta})\right]-\sum\limits_{k=1}^{Ne}b_{k}(\cos{k \alpha_{\delta}}-i\sin{k\alpha_{\delta}})\right\}-\frac{4N_{\delta}L}{E_{3} \pi d^{2}}\\ =\frac{(1-\eta)}{2C_{1}}\left[-\frac{1+\kappa_{1}}{2}\frac{R_{0}(\sigma_{v}- \sigma_{h})}{(1+L/R_{0})}\cos{2\alpha_{\delta}}+\frac{R_{0}(\sigma_{v}-\sigma _{h})}{2(1+L/R_{0})}\cos{2\alpha_{\delta}}+\frac{R_{0}(\sigma_{v}+\sigma_{h}) }{2(1+L/R_{0})}\right]\\ -\frac{(1-\eta)}{2C_{1}}\left[-\frac{1+\kappa_{1}}{2}R_{0}(\sigma_{v}-\sigma _{h})\cos{2\alpha_{\delta}}+\frac{R_{0}(\sigma_{v}-\sigma_{h})}{2}\cos{2 \alpha_{\delta}}+\frac{R_{0}(\sigma_{v}+\sigma_{h})}{2}\right]\end{array}  \tag{30}\]  
To solve \(\varphi_{0}(\xi)\), \(\psi_{0}(\xi)\), \(\varphi_{1}(\xi)\), \(\psi_{1}(\xi)\) and \(N_{\beta}(j=1,\ 2,\ \ldots,\ n)\), \(6 Ne+4+n\) unknown numbers need to be solved, \(a_{0}\), \(a_{k}\), \(b_{0}\), \(b_{k}\), \(d_{0}\), \(e_{k}\), \(f_{k}\), \(g_{0}\), \(p_{k}\), \(q_{k}\) (\(k=1,\ 2,\ \ldots,\ Ne\)) as does \(N_{\beta}(j=1,\ 2,\ \ldots,\ n)). Accordingly, \(6 Ne+4+n\) linear equations can be established, i.e., \(2 Ne+1\) linear equations describing the stress condition of the inner boundary of the lining in Equation (18), \(2 Ne+1\) linear equations describing the contact stress condition on the interface between the lining and the surrounding rock in Equation (20), linear equations based on the displacement compatibility condition of the interface between the lining and the surrounding rock in Equation (22) (\(\theta=2k\pi/(2 Ne+1)\), (\(k=1,\ 2,\ldots,\ 2 Ne+1\)) respectively, (avoid being too close to the anchor head), one linear equation describing the influence of the lining on the surrounding rock at infinity in Equation (27), and \(n\) linear equations describing the displacement compatibility condition between the bolts and the surrounding rock in Equation (30) (let \(\delta\) be equal to \(1,\ 2,\ \ldots,\ n\), respectively).