## 3 Verification of the Algorithm
### Verification Analysis
For the installed segment ring, the central angle of the bottom segment is 84\({}^{\circ}\), and the bending stiffness of the longitudinal joints adjacent to the bottom segment (\(k_{3}\)) is the largest under any loading condition; therefore, the bending rigidity in the bottom zone of the beam-spring model is the largest. Thus, under the same loading condition, the bending moments of the beam-spring model in the bottom zone are larger than those of the corresponding modified uniform rigidity ring model. In contrast, the central angle of the top segment is 16\({}^{\circ}\), and the bending stiffness of the longitudinal joints adjacent to the top segment (\(k_{1}\)) is less than \(k_{3}\) under any loading condition. Accordingly, the bending rigidity in the top zone of the beam-spring model is less than that of the modified uniform rigidity ring model. Thus, under the same loading condition, the bending moments in the top zone of the beam-spring model are less than those of the corresponding modified uniform rigidity ring model. Furthermore, in the haunch zone of the installed segment ring, the minimum bending stiffness of any loading conditions for the longitudinal joints occurs at positions of \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Loading**} & \multicolumn{4}{c}{**Axial Forces of Beam-Spring Model/kN**} & \multicolumn{4}{c}{**Axial Forces of Modified**} \\ \cline{2-9}  & \(\mathbf{0}^{\circ}\) & \(\mathbf{90}^{\circ}\) & \(\mathbf{180}^{\circ}\) & \(\mathbf{270}^{\circ}\) & \(\mathbf{0}^{\circ}\) & \(\mathbf{90}^{\circ}\) & \(\mathbf{180}^{\circ}\) & \(\mathbf{270}^{\circ}\) \\ \hline
1 st & \(-227.4\) & \(-287.4\) & \(-209.8\) & \(-287.4\) & \(-218.6\) & \(-287.3\) & \(-218.6\) & \(-287.3\) \\
2 nd & \(-392.3\) & \(-490.5\) & \(-347.2\) & \(-490.5\) & \(-369.8\) & \(-490.3\) & \(-369.8\) & \(-490.3\) \\
3 rd & \(-515.1\) & \(-640.2\) & \(-448.9\) & \(-640.2\) & \(-482\) & \(-639.9\) & \(-482\) & \(-639.9\) \\
4 th & \(-611.7\) & \(-749.9\) & \(-526.2\) & \(-749.9\) & \(-568.9\) & \(-749.6\) & \(-568.9\) & \(-749.6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Simulated axial forces under different loading conditions (unit: kN). 73\({}^{\circ}\) and 287\({}^{\circ}\) (\(k_{2}\)). Therefore, the bending stiffness in the haunch zone of the beam-spring model is lower than that of the corresponding modified uniform rigidity ring model. Thus, the absolute values of the bending moments in the haunch zone of the beam-spring model are less than those of the corresponding uniform rigidity model. As analyzed above, it is noted that the joint bending stiffness, joint number, and joint distribution of an installed segment ring greatly influence the distribution of the bending moments of the beam-spring model. All these factors should be taken into consideration in the shield tunnel design. Because the distribution of the bending rigidity for a modified uniform rigidity ring model is completely different than that of the beam-spring model, the modified uniform rigidity ring model is not applied to the design of the shield tunnel. The axial forces of the beam-spring model and the modified uniform rigidity ring model are compared in Figure 6. As shown in Figure 6 and Table 9, the axial forces of both segment ring models are approximately equivalent, and both models have high accuracy, which indicates that the distribution of the bending rigidity has little effect on the axial force.