## 3 Verification of the Algorithm
### Verification Analysis  
To verify the algorithm proposed in this article, the modified uniform rigidity ring models are established according to the effective ratios of the transverse bending rigidity obtained by the methods proposed in this article (i.e., \(E^{\prime}<E\) and \(I^{\prime}=I\)), by [PERSON] and [PERSON], and simulated in a way similar to that of the beam-spring model established in the back analysis. Under the loads (at the 1 st, 2 nd, 3 rd, and 4 th loading conditions) applied in the full-scale test, the convergence deformations of the modified uniform rigidity rings are listed in Table 6. It is noted that under different loading conditions, the convergence \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loading** & \multicolumn{2}{c}{**Uniform Rigidity Ring**} & \multicolumn{2}{c}{**Installed Segment Ring**} \\ \cline{2-5}
**Conditions** & \(\Delta D_{ht}\) & \(\Delta D_{ht}\) & \(\Delta D_{ht}\) & \(\Delta D_{vi}\) \\ \hline
1 st & 3.61 & \(-\)3.86 & 9.08 & \(-\)9.76 \\
2 nd & 6.34 & \(-\)6.76 & 31.52 & \(-\)33.17 \\
3 rd & 8.31 & \(-\)8.85 & 84.94 & \(-\)87.71 \\
4 th & 9.5 & \(-\)10.14 & 139.18 & \(-\)143.01 \\ \hline \hline \end{tabular} \(\Delta D_{ht}\) is the vertical convergence deformation of the uniform rigidity ring, \(\Delta D_{vi}\) is the vertical convergence deformation of the installed segment ring.  
\end{table} 
Table 4: Convergence deformations for the uniform rigidity ring and installed segment ring under different loading conditions (unit: mm). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Loading Conditions** & \(\eta_{1}\)=\(\frac{D_{1}+\Delta D_{ht}}{D_{1}+\Delta D_{hi}}\) & \(\eta_{2}\)=\(\frac{\Delta D_{ht}}{\Delta D_{hi}}\) \\ \hline
1 st & 0.9991 & 0.3976 \\
2 nd & 0.996 & 0.2013 \\
3 rd & 0.9878 & 0.0979 \\
4 th & 0.9795 & 0.0683 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Effective ratios of the transverse bending rigidity calculated using different methods. deformations of the uniform rigidity rings obtained using the methods proposed in this article are extremely similar to those of the uniform rigidity rings obtained using [PERSON]'s method [24] and are also similar to those of the installed segment rings shown in Table 4. On the other hand, the convergence deformations of the modified uniform rigidity rings established using the method proposed by [PERSON] [20] are visibly different from other modified uniform rigidity rings. Under the same loading condition, the interactions between the tunnel rings embedded in the ground and the surrounding earth can be similar only if the convergence deformations of a modified uniform rigidity ring are similar to those of an installed segment ring. Thus, the extent of the closeness between the convergence deformations of a modified uniform rigidity ring and those of the corresponding installed segment ring can be used to decide whether the effective ratio of the transverse bending rigidity is appropriate. Based on that decision, the methods for calculating the effective ratios proposed by [PERSON] and developed in this article are feasible. Compared to the existing methods for obtaining the effective ratio of the transverse bending rigidity, the algorithm proposed in this article is simpler and more conceptually elegant. Moreover, the algorithm proposed in this article has established the correlation between the longitudinal joint bending stiffness (i.e., the key parameter of the installed segment ring) and the effective ratio (i.e., the decisive parameter of the modified uniform rigidity ring). Additionally, the algorithm can express the correlation between the effective ratio and other influencing factors, such as the diameter of the tunnel ring [32], thickness of the segment [33], and number of joints [34], and this algorithm is easier to use than the existing methods.