## 3 Verification of the Algorithm
### Verification Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{
\begin{tabular}{c} **Loading** \\ **Conditions** \\ \end{tabular} } & \multicolumn{4}{c}{**Bending Moments of Beam-Spring Model/kN\(\cdot\)m**} & \multicolumn{4}{c}{**Bending Moments 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 & \(86.3\) & \(-89.4\) & \(137.9\) & \(-89.4\) & \(108.3\) & \(-93.2\) & \(108.3\) & \(-93.2\) \\
2 nd & \(141.9\) & \(-145.7\) & \(273.8\) & \(-145.7\) & \(189.4\) & \(-164.1\) & \(189.4\) & \(-164.1\) \\
3 rd & \(190.2\) & \(-176.2\) & \(383.8\) & \(-176.2\) & \(248.1\) & \(-215.1\) & \(248.1\) & \(-215.1\) \\
4 th & \(222.6\) & \(-182.4\) & \(472.8\) & \(-182.4\) & \(284.7\) & \(-245.4\) & \(284.7\) & \(-245.4\) \\ \hline \end{tabular}
\end{table} 
Table 8: Simulated bending moments under different loading conditions (unit: kN\(\cdot\)m). The sign conventions are adopted such that the deformations are positive when the deformations of the segment are far away from the center of the model, the bending moments are positive when the internal surface of the segment is subjected to tension, and the compressive axial forces are negative. The deformations of the beam-spring model and the modified uniform rigidity ring model, which are determined using the effective ratio obtained by the algorithm developed in this article, are shown in Figure 4. Note that to clearly distinguish the deformation shapes of the segment ring models, as shown in Figure 4,d, the deformations of all the nodes are magnified 15 times. For the vertical and horizontal restraints applied in the numerical models, the reference points of the vertical deformations are at the positions of 90\({}^{\circ}\) and 270\({}^{\circ}\), and those of the horizontal deformations are at the positions of 0\({}^{\circ}\) and 180\({}^{\circ}\). The simulated results under different loading conditions are shown in Table 7. Figure 4 shows that even though the convergence deformations are similar to each other, the deformation shapes of both models are different from each other. The largest horizontal deformations for the beam-spring model are located at the positions of 73\({}^{\circ}\) and 287\({}^{\circ}\) compared with the positions of 90\({}^{\circ}\) and 270\({}^{\circ}\) for the modified uniform rigidity ring model. As indicated in Figure 5 and Table 8, under the same loading condition, the distribution forms of the bending moment for both models are visibly different. For the beam-spring model, the bending moment at the top position (At the position of 0\({}^{\circ}\)) is less than that at the bottom position (At the position of 180\({}^{\circ}\)). However, for the modified uniform rigidity ring model, the bending moments at the top and bottom positions are the same but are larger than that at the top position and less than that at the bottom position of the beam-spring model. As shown in Table 8, the absolute values of the bending moments for the modified uniform rigidity ring model at the positions of 90\({}^{\circ}\) and 270\({}^{\circ}\) are larger than those of the beam-spring model. However, the results show that the summation of the absolute values for the bending moments at the positions of 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\) for the beam-spring model and the corresponding modified uniform rigidity ring model are the same. Both the beam-spring model and the modified uniform rigidity ring model are hyperstatic structures; therefore, the distribution of the bending moments is affected by the distribution of the bending rigidity around the structure [35]. Generally, under the same loading condition, a greater bending rigidity corresponds to a larger bending moment.