# Mechanical Analysis of Bolt-Lining Combined Support in Deeply Buried Circular Tunnel
## 2 Fundamental Principles and Equations
### Two Analytic Functions Considering Bolts Effect  
After the rock bolts are installed into the rock mass, they will restrict deformation. Meanwhile, the bolts are stretched. As mentioned above, the effect of point-anchored rock bolts on the surrounding rock can be regarded as pairs of concentrated forces of equal magnitude acting in opposite directions along the length of the bolts. The analytic functions representing the action of the \(j\)-th concentrated force on the surrounding rock can be expressed as follows [22]:  
\[ \begin{array}{ll}\varphi_{(j)}(\zeta)=&-\frac{X_{j}+Y_{j}}{2\pi(1+\kappa_{1 })}\ln\Bigl{(}1+\frac{R_{0}}{c_{j}}\zeta\Bigr{)}-\frac{\kappa_{1}(X_{j}+Y_{j} )}{2\pi(1+\kappa_{1})}\ln\Bigl{(}1+\frac{R_{0}}{c_{j}^{2}}\frac{1}{\zeta} \Bigr{)}\\ &+\frac{X_{j}-Y_{j}}{2\pi(1+\kappa_{1})}\biggl{(}\frac{c_{j}c_{j}^{2}-R_{0}^{ 2}}{c_{j}^{2}}\biggr{)}\frac{\zeta}{R_{0}+c_{j}^{2}\zeta}\end{array}  \tag{6}\]  
\[ \begin{array}{ll}\psi_{(j)}(\zeta)=&\frac{\kappa(X_{j}-Y_{j})}{2\pi(1+\kappa )}\ln\Bigl{(}1+\frac{R_{0}}{c_{j}}\zeta\Bigr{)}+\frac{X_{j}-Y_{j}}{2\pi(1+ \kappa)}\ln\Bigl{(}1+\frac{R_{0}}{c_{j}}\frac{1}{\zeta}\Bigr{)}\\ &-\frac{X_{j}+Y_{j}}{2\pi(1+\kappa)}\frac{c_{j}^{2}}{c_{j}+R_{0}\zeta}+\frac{ X_{j}+Y_{j}}{2\pi(1+\kappa)}\frac{R_{0}}{c_{j}}\frac{1}{\zeta}-\frac{\kappa(X_{j}+Y_{j} )}{2\pi(1+\kappa)}\frac{R_{0}}{c^{2}\bigl{(}R_{0}+c_{j}\zeta\bigr{)}}\\ &-\frac{X_{j}-Y_{j}}{2\pi(1+\kappa)}\frac{c_{j}c_{j}^{2}-R_{0}^{2}}{c_{j}^{2}} \frac{R_{0}}{c\bigl{(}R_{0}+c_{j}^{2}\zeta\bigr{)}^{2}}\end{array}  \tag{7}\]  
where \(X_{j}\), \(Y_{j}\) respectively represent the component forces of the \(j\)-th concentrated force along the \(x\) and \(y\) axis; and \(c_{j}\) is radius vector from the position of the concentrated force to the center of circle tunnel. Based on the simplified bolts model, one of the concentrated forces acts outwards at the boundary of the tunnel, and the other acts inwards at the midpoint of the bond length (see Figure 1). Let \(N_{j}\) denote the axial force of the \(j\)-th rock bolt. Then, we have the following equations:  
\[\left\{ \begin{array}{ll}X_{j}=N_{j}\cos\alpha_{j}/S;\ Y_{j}=N_{j}\sin\alpha_ {j}/S\\ X_{j+n}=-N_{j}\cos\alpha_{j}/S;\ Y_{j+n}=-N_{j}\sin\alpha_{j}/S\ \ (j=1,2,\ldots,n) \end{array} \right. \tag{8}\]  
\[\left\{ \begin{array}{ll}c_{j}=(-R_{0}\cos\alpha_{j})+i(-R_{0}\sin\alpha_{j} )\\ c_{j+n}=\bigl{[}-(R_{0}+L)\cos\alpha_{j}\bigr{]}+i\bigl{[}-(R_{0}+L)\sin\alpha_ {j}\bigr{]}\end{array} \right.\ (j=1,2,\ddots,n) \tag{9}\]  
where \(\alpha_{j}\) is the \(j\)-th (\(j=1,2,\ldots,n\)) rock bolt installment angle; \(n\) is the installation number of the point-anchored rock bolts; \(L\) is the length between the effective ends of the bolts; and \(S\) is the spacing of the rock bolts. According to the superposition principle of elasticity, the analytic functions representing the effect of the point-anchored rock bolts on the surrounding rock is as follows:  
\[\varphi_{2}(\zeta)=\sum_{j=1}^{2n}\varphi_{(j)}(\zeta),\psi_{2}(\zeta)=\sum_{j =1}^{2n}\psi_{(j)}(\zeta) \tag{10}\]  
Therefore, the displacements of any point within the surrounding rock caused by bolts is  
\[2G_{1}(u_{2}^{\rho}+iu_{2}^{\theta})=\frac{\overline{\zeta}}{\rho}\frac{ \overline{\omega^{\prime}(\zeta)}}{\overline{\omega^{\prime}(\zeta)}}\Biggl{[} \kappa_{1}\varphi_{2}(\zeta)-\frac{\omega(\zeta)}{\overline{\omega^{\prime}( \zeta)}}\overline{\overline{\varphi^{\prime}}_{2}(\zeta)}-\overline{\psi_{2}( \zeta)}\Biggr{]} \tag{11}\]  
where \(u_{2}^{\rho}\) and \(u_{2}^{\theta}\) are the radial and tangential displacement components, respectively.