## 3 Verification of the Algorithm
### Verification Analysis
To discuss the difference between the beam-spring model and the modified uniform rigidity ring model developed in this article, the deformations, bending moments, and internal forces of both models under different loading conditions are compared in Figures 4-6. The simulated results under different loading conditions are shown in Tables 7-9. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Loading**} & \multicolumn{3}{c}{\(\frac{\text{(D_{1}+\text{D})_{2}}\pi}{\text{(D_{1}+\text{D})_{2}}\pi+2E\sum_{1 }^{\pi}\frac{1}{\pi}}\)} & \multicolumn{3}{c}{\(\frac{\text{AD}_{\text{us}}}{\text{AD}_{\text{us}}}\)} & \multicolumn{3}{c}{\(\frac{\text{D}_{1}+\text{AD}_{\text{us}}}{\text{D}_{1}+\text{AD}_{\text{us}}}\)} \\ \cline{2-7}  & \(\overline{\text{(D_{1}+\text{D})_{2}}\pi+2E\sum_{1}^{\pi}\frac{1}{\pi}}\) & & \(\frac{\text{AD}_{\text{us}}}{\text{AD}_{\text{us}}}\) & & \(\frac{\text{D}_{1}+\text{AD}_{\text{us}}}{\text{D}_{1}+\text{AD}_{\text{us}}}\) \\ \cline{2-7}
**Conditions** & **Horizontal** & **Vertical** & **Horizontal** & **Vertical** & **Horizontal** & **Vertical** \\ \hline
1 st & \(8.8\) & \(-9.18\) & \(9.08\) & \(-9.7\) & \(3.61\) & \(-3.86\) \\
2 nd & \(30.44\) & \(-31.34\) & \(31.52\) & \(-33.57\) & \(6.37\) & \(-6.78\) \\
3 rd & \(82.6\) & \(-84.48\) & \(84.94\) & \(-90.44\) & \(8.42\) & \(-8.96\) \\
4 th & \(135.28\) & \(-138.14\) & \(139.18\) & \(-148.5\) & \(9.7\) & \(-10.35\) \\ \hline \end{tabular}
\end{table} 
Table 6: Convergence deformations for the modified uniform rigidity rings with effective ratios of the transverse bending rigidity calculated using different methods (unit: mm). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Loading**} & \multicolumn{3}{c}{**Deformations of Beam-Spring Model/mm**} & \multicolumn{3}{c}{**Deformations of Modified**} \\  & \multicolumn{3}{c}{**Uniform Rigidity Ring Model/mm**} \\ \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 & \(-6.37\) & \(4.54\) & \(-3.39\) & \(4.54\) & \(-4.59\) & \(4.4\) & \(-4.59\) & \(4.4\) \\
2 nd & \(-23.5\) & \(15.76\) & \(-9.67\) & \(15.76\) & \(-15.67\) & \(15.22\) & \(-15.67\) & \(15.22\) \\
3 rd & \(-65.13\) & \(42.47\) & \(-22.58\) & \(42.47\) & \(-42.24\) & \(41.3\) & \(-42.24\) & \(41.3\) \\
4 th & \(-107.57\) & \(69.59\) & \(-35.44\) & \(69.59\) & \(-69.07\) & \(67.64\) & \(-69.07\) & \(67.64\) \\ \hline \end{tabular}
\end{table} 
Table 7: Simulated deformations under different loading conditions (unit: mm). Figure 4: Simulated deformations under different loading conditions (unit: mm): (**a**) Beam-spring model; (**b**) modified uniform rigidity ring model; (**c**) deformation’s comparison of the beam-spring model; (**d**) deformation’s comparison of the modified uniform rigidity ring model (in order to better reflect the deformation of (**a**,**b**), the different segment ring model’s deformation are amplified 15 times in (**c**,**d**)). Figure 5: Simulated bending moments under different loading conditions (unit: kN\(\cdot\)m): (**a**) Beam-spring model; (**b**) modified uniform rigidity ring model. Figure 6: Simulated axial forces under different loading conditions (unit: kN): (**a**) Beam-spring model; (**b**) modified uniform rigidity ring model.