## 3 Theoretical Derivations
### Bolts
#### 3.1.3 Displacement of the Tunnel Wall Derivation  
The wedge element of the surrounding rock containing only one bolt was analyzed, and the supporting force of the bolt on the surrounding rock was simplified to the radial volume force \(f(r)\). The resultant force \(dQ\) on the microend \(dr\) interface of the bolt is  
\[dQ=-\pi d_{b}\tau(r)dr \tag{22}\]  
The volume \(dV\) of this microend is  
\[dV=Dr\theta dr \tag{23}\]  
\[f(r)=\frac{dQ}{dV}=-\frac{\pi d_{b}}{D_{b}\theta}\times\frac{\tau(r)}{r} \tag{24}\]  
where \(D_{b}\) and \(\theta\) are the longitudinal spacing and circumferential included angle of the bolts along the tunnel, respectively. According to [PERSON]'s solution, assuming that the cement paste and the rock are elastic materials with the same properties, the rock is regarded as half space, the bolt is infinite, and the displacement of the rock mass at the orifice is equal to the total elongation of the bolt, the bond stress distribution equation of the fully bonded bolt is derived \(\tau(r)\):  
\[\tau(r)=\frac{p_{b}tr}{2\pi a_{b}}e^{-\frac{1}{2}tr^{2}} \tag{25}\]  
\[t=\frac{1}{(1+\mu)(3-2\mu)a_{b}^{2}}(\frac{E_{0}}{E_{b}}) \tag{26}\]  
where \(p_{b}\) is the ultimate pulling force of the bolt, \(a_{b}\) is the radius of the bolt, and \(E_{0}\) and \(\mu\) are the elastic modulus and Poisson's ratio of the soil, respectively. \(p_{b}\) can be calculated by the following formula:  
\[p_{b}=2\pi a_{b}L\tau_{ult} \tag{27}\]  
where \(\tau_{ult}\) is the ultimate shear strength of the rock mass. The surrounding rock in the anchorage zone will meet the equilibrium differential equation:  
\[\frac{d\sigma_{\tau}}{dr}+\frac{\sigma_{\tau}-\sigma_{\theta}}{r}+f(r)=0 \tag{28}\]  
where \(\sigma_{\tau}\) is the radial stress in the tunnel anchorage zone and \(\sigma_{\theta}\) is the circumferential stress in the anchorage zone. Replace Equation (28). In conjunction with the Mohr-Coulomb criterion, the following is obtained:  
\[\frac{\sigma_{r}+C_{b}\cot\varphi_{b}}{\sigma_{b}+C_{b}\cot\varphi_{b}}=\frac{1- \sin\varphi_{b}}{1+\sin\varphi_{b}} \tag{29}\]  
where \(\varphi_{b}\) and \(C_{b}\) are the internal friction angle and cohesion of the surrounding rock, respectively, through the following constant variation method:  
\[ \begin{array}{l}\sigma_{r}=r^{n-1}[A_{1}-\int f(r)r^{1-n}dr]-C_{b}\cot \varphi_{b}\\ n=\frac{1-\sin\varphi_{b}}{1+\sin\varphi_{b}}\end{array}  \tag{30}\]  
where \(A_{1}\) is the integral constant, which is determined by the boundary conditions, when \(r\) = \(r_{0}\), \(\sigma_{r}\) = \(p_{i}\), where \(p_{i}\) is the support force of the shotcrete layer. If the shotcrete layer is not applied, \(p_{i}\) = 0. Based on this, the following can be obtained:  
\[A_{1}=\frac{p_{i}+C_{b}\cot\varphi_{b}}{r_{0}^{n-1}}+\int^{r_{0}}f(r)r^{1-n}dr \tag{31}\]  
Replace Equation (31) and substitute Equation (30) and obtain  
\[\sigma_{r}=r^{n-1}[\frac{p_{i}+C_{b}\cot\varphi_{b}}{r_{0}^{n-1}}+\int^{r_{0} }f(r)r^{1-n}dr-\int f(r)r^{1-n}dr]-C_{b}\cot\varphi_{b} \tag{32}\]  
The anchorage zone can be considered as the support resistance \(p_{i}\)\({}^{\prime}\) to the deep surrounding rock, and the tunnel radius can be considered as \(r\) = \(r_{0}\) + \(L\).