# Mechanical Properties of the Functionally Graded Lining for a Deep Buried Subway Tunnel
## 3 Variance analysis of influence factors
### Verification of the Numerical Simulation  
In the numerical calculation, the lining of subway tunnels has a certain thickness, but in the theoretical calculation, the lining is simplified as a curved rod structure. Therefore, under the condition of the different thickness diameter ratio, the displacement calculation results of the inner wall, outer wall and axis of the lining in the numerical calculation are \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Position**} & \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Dof**} & \multirow{2}{*}{**Square of Deviance**} & \multirow{2}{*}{**F**} & \multicolumn{3}{c}{**Confidence**} & \multirow{2}{*}{**Significance**} \\ \cline{5-6}  & & & & & **99\%** & & & **90\%** \\ \hline \multirow{5}{*}{0} & \(a\) & 2 & 0.188 & 12.698 & 99 & 19 & 9 & * \\  & \(b\) & 2 & 0.190 & 12.843 & 99 & 19 & 9 & * \\  & \(\lambda\) & 2 & 37.505 & 2533.057 & 99 & 19 & 9 & *** \\  & Error & 2 & 0.015 & & & & & \\ \hline \multirow{5}{*}{15} & \(a\) & 2 & 0.188 & 12.448 & 99 & 19 & 9 & * \\  & \(b\) & 2 & 0.190 & 12.590 & 99 & 19 & 9 & * \\  & \(\lambda\) & 2 & 28.128 & 1862.404 & 99 & 19 & 9 & *** \\  & Error & 2 & 0.015 & & & & & \\ \hline \multirow{5}{*}{30} & \(a\) & 2 & 0.188 & 12.698 & 99 & 19 & 9 & * \\  & \(b\) & 2 & 0.190 & 12.843 & 99 & 19 & 9 & * \\  & \(\lambda\) & 2 & 9.378 & 633.349 & 99 & 19 & 9 & *** \\  & Error & 2 & 0.015 & & & & & \\ \hline \multirow{5}{*}{45} & \(a\) & 2 & 0.188 & 12.704 & 99 & 19 & 9 & * \\  & \(b\) & 2 & 0.190 & 12.838 & 99 & 19 & 9 & * \\  & \(\lambda\) & 2 & 0.000 & 0.000 & 99 & 19 & 9 & - \\  & Error & 2 & 0.015 & & & & & \\ \hline \multirow{5}{*}{60} & \(a\) & 2 & 0.188 & 12.698 & 99 & 19 & 9 & * \\  & \(b\) & 2 & 0.190 & 12.843 & 99 & 19 & 9 & * \\  & \(\lambda\) & 2 & 9.373 & 633.011 & 99 & 19 & 9 & *** \\  & Error & 2 & 0.015 & & & & & \\ \hline \multirow{5}{*}{75} & \(a\) & 2 & 0.188 & 12.471 & 99 & 19 & 9 & * \\  & \(b\) & 2 & 0.190 & 12.565 & 99 & 19 & 9 & * \\ \cline{1-1}  & \(\lambda\) & 2 & 28.119 & 1861.611 & 99 & 19 & 9 & *** \\ \cline{1-1}  & Error & 2 & 0.015 & & & & & \\ \hline \multirow{5}{*}{90} & \(a\) & 2 & 0.188 & 12.698 & 99 & 19 & 9 & * \\ \cline{1-1}  & \(b\) & 2 & 0.190 & 12.843 & 99 & 19 & 9 & ** \\ \cline{1-1}  & \(\lambda\) & 2 & 37.495 & 2532.381 & 99 & 19 & 9 & *** \\ \cline{1-1}  & Error & 2 & 0.015 & & & & & \\ \hline \end{tabular}
\end{table} 
Table 7: Variance analysis of influence factors of section moment. compared with the theoretical calculation results, so as to determine the corresponding position of the radius \(R_{\rm H}\) in the theoretical calculation. The calculation results are shown in Figure 12.  
From Figure 12, it can be observed that the radial displacement at the inner edge of the cylinder is the largest, followed by the axis, and the outer edge is the smallest, indicating that the inner edge of the cylinder structure should be used as the basis for failure under the action of external force. When \(t/R\leq 0.2\), the theoretical calculation results are close to the radial displacement of the inner edge of the cylinder in the numerical simulation results. With the increase in the thickness diameter ratio, the numerical simulation results of the radial displacement of the inner edge at \(0^{\circ}\) and \(90^{\circ}\) gradually increase. When \(t/R=0.2\), the numerical simulation results of the radial displacement of the inner edge at \(0^{\circ}\) and \(\alpha\)= \(90^{\circ}\) are closest to the theoretical calculation results. However, when \(t/R>0.2\), the numerical simulation results of the radial displacement of the inner edge at \(0^{\circ}\) and \(90^{\circ}\) exceed the theoretical calculation results, and the gap between them gradually becomes larger. It shows that when \(t/R\leq 0.2\), the theoretical calculation results can be used to calculate the radial displacement of the inner edge, and the calculation results are conservative.