## 3 Verification of the Algorithm
### Values of the Effective Ratio for the Transverse Bending Rigidity Based on the Full-Scale Test  
As shown in Table 3, under the condition that \(L^{\prime}=L\), including cases for reducing the bending rigidity by the reduction of E and reducing \(I\) by the first method of thickness reduction, the effective ratios of the transverse bending rigidity for a modified uniform rigidity ring can be calculated using Equation (12) or (14). Using the first method of thickness reduction, the thickness reduction coefficients can be calculated with Equation (18).  
Under the condition that \(L^{\prime}>L\), which represents the second method of thickness reduction, the thickness of a modified uniform rigidity ring can be obtained by solving Equation (21). As listed in Table 3, the thickness reduction coefficients and the corresponding effective ratios of the transverse bending rigidity for the modified uniform rigidity rings under different loading conditions can be obtained according to Equation (17). It is noted that the effective ratios and thickness reduction coefficients under the condition of \(L^{\prime}>L\) are slightly larger than those under the condition of \(L^{\prime}=L\).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Loading Conditions** & \(\mathbf{k_{1}/10^{6}}\) **N\(\cdot\)m/rad** & \(\mathbf{k_{2}/10^{6}}\) **N\(\cdot\)m/rad** & \(\mathbf{k_{3}/10^{6}}\) **N\(\cdot\)m/rad** \\ \hline
1 st & 34 & 28 & 50 \\
2 nd & 13 & 10 & 21 \\
3 rd & 6 & 4 & 9 \\
4 th & 4.2 & 2.5 & 7 \\ \hline \end{tabular}
\end{table}
Table 2: Joints’ bending stiffness obtained through back analysis.  
Figure 3: Beam-spring model.  
However, [PERSON] [20] and [PERSON] [24] proposed other methods to calculate the effective ratio of the transverse bending rigidity for the modified uniform rigidity ring, which can be expressed as Equations (22) and (23), respectively:  
\[\eta_{1}=\frac{D_{1}+\Delta D_{ht}}{D_{1}+\Delta D_{hi}} \tag{22}\]  
\[\eta_{2}=\frac{\Delta D_{ht}}{\Delta D_{hi}} \tag{23}\]  
where \(\eta_{1}\) is the algorithm of [PERSON] about the effective ratio of the transverse bending rigidity for the modified uniform rigidity ring, \(\Delta D_{ht}\) is the horizontal convergence deformation of the uniform rigidity ring, \(\Delta D_{hi}\) is the horizontal convergence deformation of the installed segment ring, and \(\eta_{2}\) is the algorithm of [PERSON] about the effective ratio of the transverse bending rigidity for the modified uniform rigidity ring.  
In order to use the method proposed by [PERSON] and [PERSON] to calculate the effective ratio, the convergence-confinement method is used to obtain the convergence deformation of the uniform stiffness ring and installed segment ring under different load conditions [31], which are represented by the uniform stiffness ring model and beam-spring model, respectively. These deformations are listed in Table 4. In engineering practice, the relative deformation at the positions of \(90^{\circ}\) and \(270^{\circ}\) is regarded as the horizontal convergence deformation while the deformation at \(0^{\circ}\) and \(180^{\circ}\) is regarded as the vertical convergence deformation. Therefore, effective ratios calculated with Equations (22) and (23) are obtained, which are listed in Table 5. It is noted that the effective ratios in Table 3 under the condition of \(L^{\prime}=L\) and those in Table 5, which are calculated with Equation (22), are extremely similar. However, the effective ratios in Table 5, which are calculated with Equation (23), are larger than those calculated with Equation (22) and those listed in Table 3.