## 3 Verification of the Algorithm  
### Verification Sample  
The sample of the installed segment ring used for the verification study is the tunnel lining of shield tunnels with non-staggered installation, which is widely used in Shanghai's single-line metro. The outer diameter of the segment ring is 6.2 m, and the thickness and width of the segment are 0.35 and 1.2 m, respectively. The segment is precasted using high-strength concrete (C55 grade) of which the Poisson's ratio and elastic modulus are 0.18 and 35.5 GPa, respectively. The segment ring is composed of six segments, which is shown in Figure 2. Straight bolts with a mechanical property grade of 5.8 are adopted for the circumferential and longitudinal connection of the segment rings.  
To obtain the longitudinal joint bending stiffness of the installed segment ring, a back analysis of a full-scale test simulating the interaction between the load and structure was conducted [30]. To simulate the surrounding earth pressure, there are 24 loading points spaced evenly around the segment ring. All the loads are divided into 3 groups: \(P_{1}\) (6 loading points), \(P_{2}\) (10 loading points), and \(P_{3}\) (8 loading points), which are also shown in Figure 2. Deformations measured in the full-scale test under four different loading conditions are used as a basis for the back analysis and are listed in Table 1. It is noted that the sign conventions adopted herein are positive for the outward deformations.  
The beam-spring model [14,15], well recognized for its ability to simulate an installed segment ring accurately, is established here in a cylindrical coordinate system using ANSYS. A typical configuration includes 366 node elements, 360 curved beam elements, and 6 spring elements. Because of the symmetrical conditions, the bending stiffness of the segment longitudinal joints are \(k_{1}\) at the positions of \(8^{\circ}\) and \(352^{\circ}\), \(k_{2}\) at the positions of \(73^{\circ}\) and \(287^{\circ}\), and \(k_{3}\) at the positions of \(138^{\circ}\) and \(222^{\circ}\). Additionally, the horizontal and vertical restraints are applied at the \(0^{\circ}\) and \(180^{\circ}\) positions and at the \(90^{\circ}\) and \(270^{\circ}\) positions, respectively, which are shown in Figure 3. Values for the longitudinal joint bending stiffness obtained from the back analysis under different loading conditions are listed in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Loading** & **Actual Load** & **Actual Load** & **Actual Load** & \multicolumn{3}{c}{**Deformation Results of Different Testing Point/mm**} \\ \cline{4-9}
**Conditions** & \(P_{1}\)**/kN** & \(P_{2}\)**/kN** & \(P_{3}\)**/kN** & \(0^{\circ}\) & \(74^{\circ}\) & \(105^{\circ}\) & \(180^{\circ}\) & \(255^{\circ}\) & \(286^{\circ}\) \\ \hline
1 st & 95.32 & 45.98 & 71.49 & \(-6.44\) & 6.45 & 3.93 & \(-2.48\) & 2.03 & 4.78 \\
2 nd & 163.15 & 76.83 & 122.36 & \(-22.18\) & 21.51 & 12.48 & \(-7.3\) & 8.66 & 18.54 \\
3 rd & 213.02 & 99.97 & 159.76 & \(-63.09\) & 55.66 & 32.18 & \(-18.79\) & 28.72 & 55.21 \\
4 th & 248.92 & 119.24 & 186.69 & \(-107.89\) & 94.36 & 55.21 & \(-31.54\) & 48.01 & 92.55 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Deformations measured in the full-scale test.  
Figure 2: Force distribution and segment ring division.