# Mechanical Analysis of Bolt-Lining Combined Support in Deeply Buried Circular Tunnel
## 3 Comparison with Numerical Solution  
Twelve bolts were installed every 30 degrees. The relevant parameters are listed in the following.  
The numerical solution was calculated with ANSYS. The size of the model was 60 m \(\times\) 60 m, the lining and rock mass were simulated using the Plane42 element (2D entity unit) with different material properties, as shown in Table 1, the point-anchored rock bolts were simulated using the Combin14 element (spring unit), and the contact relationship between the lining and the surrounding was simulated using the target169 and contact171 elements, which is able to set different contact relationships through options such as complete contact. The boundary condition was to limit the displacement of the symmetry axis of the model in the \(x\) and y directions, respectively. The process of tunnel excavation was simulated through the \"stress release method\" proposed by [PERSON] and [PERSON] [37]. In addition, [PERSON]'s law was adopted as the constitutive model in this paper. Figure 3 shows the numerical model. Figures 4-6 show a comparison of the analytical and numerical solutions, including the axial forces of bolts, tangential stresses, and radial displacements of points at the inner boundary of the lining. \(\lambda=\sigma_{v}/\sigma_{h}\) is the coefficient of lateral pressure. The compressive stress and the displacement toward to hole were positive.  
\begin{table}
\begin{tabular}{c c c c} \hline Radius of circular tunnel (\(R_{0}\)) & 3.0 m & Vertical in situ stress (\(\sigma_{\text{v}}\)) & 2.0 MPa \\ \hline Radius of inner boundary of the lining (\(R_{1}\)) & 2.8 m & [PERSON]’s modulus of rock mass (\(E_{1}\)) & 1 GPa \\ \hline Bolt length (\(L\)) & 1.5 m & [PERSON]’s modulus of the lining (\(E_{2}\)) & 10 GPa \\ \hline Row spacing of bolts (\(S\)) & 1.0 m & [PERSON]’s modulus of bolt (\(E_{3}\)) & 210 GPa \\ \hline Bolt diameter (\(d\)) & 2.5 cm & [PERSON]’s ratio of rock mass and lining (\(m_{1}=m_{2}\)) & 0.25 \\ \hline Coefficient of displacement released (\(h\)) & 0.0 & Term of the series (\(Ne\)) & 120 \\ \hline \end{tabular}
\end{table}
Table 1: The values of relevant parameters.  
Figure 4: A comparison of axial forces of the bolts.  
Figure 5: A comparison of tangential stresses of points at the inner boundary of the lining.  
Figure 3: The ANSYS model of circular tunnel with support.  
It can be seen from the Figure 4 that the axial force distribution of the bolts obtained by the proposed method was in line with the actual laws, i.e., the maximum axial force was reached when the bolt was installed along the direction of the maximum in situ stress because of maximum deformation and the axial forces of the bolts were the same at different angles when \(\lambda=1\). As compared with the numerical solution, there were only marginal differences along the direction of minimum in situ stress when \(\lambda=0.5\) and \(\lambda=1.5\), and the numerical solution was larger. In addition, they were almost equal when \(\lambda=1\), and only at certain angles was the numerical solution slightly larger. As for the tangential stress and the radial displacement of the point at the inner boundary of the lining, as shown in Figures 5 and 6, the analytical solution almost coincided with the numerical solution. Therefore, to a large extent, the effectiveness of this model is proved.