## 3 Theoretical Derivations
### Analysis of Primary Support Bearing Capacity
#### 4.3.2 The Influence of Space between the Bolts  
The length of the bolts was determined to be 6 m, and the results of analyzing the tunnel wall displacement and bearing capacity under different bolt spacings using the method proposed in this article are shown in Table 8.  
From the analytical results, it can be seen that as the spacing between the bolts increases, the displacement \(u_{2}\) of the tunnel wall after applying the bolts gradually increases. It can be seen that the smaller the spacing between the bolts, the more they can constrain the loosening of the surrounding rock. The parameters of the shotcrete and steel arch frame  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Comparison**} & \multicolumn{2}{c}{**Length of The**} & \multirow{2}{*}{\(u_{0}\) (mm)} & \multirow{2}{*}{\(u_{1}\) (mm)} & \multirow{2}{*}{\(u_{2}\) (mm)} & \multirow{2}{*}{\(u_{3}\) (mm)} & \multirow{2}{*}{\(u_{4}\) (mm)} \\  & & Bolts (m) & & & & & \\ \hline Analytical & & 45.74 & 11.7 & 34.05 & 17.2 & 16.85 \\ Numerical simulation & 6 & 48.75 & 13.59 & 35.16 & 20.56 & 14.6 \\ Error & & 3.01 & 1.89 & 1.11 & 3.36 & 2.25 \\ Analytical & & 45.74 & 13.63 & 32.11 & 17.83 & 14.28 \\ Numerical simulation & 8 & 48.75 & 17.17 & 31.58 & 19.26 & 12.32 \\ Error & & 3.01 & 3.54 & 0.53 & 1.43 & 1.96 \\ Analytical & & 45.74 & 14.21 & 31.53 & 19.59 & 11.94 \\ Numerical simulation & 10 & 48.75 & 19.2 & 29.55 & 19.37 & 10.18 \\ Error & & 3.01 & 4.99 & 198 & 0.22 & 1.76 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison between the analytical results of the tunnel wall displacement and the numerical simulation results.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
\begin{tabular}{c} **Bolts\({}^{\prime}\)** \\ **Length** \\ \end{tabular} & \multirow{2}{*}{\(u_{0}\) (mm)} & \multirow{2}{*}{\(u_{1}\) (mm)} & \multirow{2}{*}{\(u_{2}\) (mm)} & \multirow{2}{*}{\(u_{3}\) (mm)} & \multirow{2}{*}{\(u_{4}\) (mm)} & \multirow{2}{*}{\(Q_{s}\) (MPa)} & \multirow{2}{*}{\(T\) (MPa)} & \multirow{2}{*}{\(q\) (MPa)} \\ \hline
6 & 45.74 & 11.7 & 34.05 & 17.2 & 16.85 & 0.074 & 0.57 & 0.411 \\
8 & 45.74 & 13.63 & 32.11 & 17.83 & 14.28 & 0.096 & 0.59 & 0.473 \\
10 & 45.74 & 14.21 & 31.53 & 19.59 & 11.94 & 0.121 & 0.64 & 0.574 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Analytical results of the wall displacement and bearing capacity of different bolt lengths.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
\begin{tabular}{c} **Bolts\({}^{\prime}\)** \\ **(m)** \\ \end{tabular} & \multirow{2}{*}{\(u_{0}\) (mm)} & \multirow{2}{*}{\(u_{1}\) (mm)} & \multirow{2}{*}{\(u_{2}\) (mm)} & \multirow{2}{*}{\(u_{3}\) (mm)} & \multirow{2}{*}{\(u_{4}\) (mm)} & \multirow{2}{*}{\(Q_{s}\) (MPa)} & \multirow{2}{*}{\(T\) (MPa)} & \multirow{2}{*}{\(q\) (MPa)} \\ \hline
1.4 & 45.74 & 13.5 & 32.24 & 18.49 & 13.75 & 0.092 & 0.62 & 0.524 \\
1.6 & 45.74 & 11.7 & 34.05 & 17.2 & 16.85 & 0.074 & 0.57 & 0.411 \\
1.8 & 45.74 & 9.53 & 36.21 & 17.09 & 19.12 & 0.052 & 0.55 & 0.374 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Analysis results of the wall displacement and bearing capacity of different bolt spacings.  
have not changed, but the surrounding rock \(u_{4}\) of the tunnel wall after support also shows a gradually increasing trend, indicating that the change in the bolt spacing will also affect the bearing capacity of the subsequent support measures.