## 4 Model Validation  
To validate the proposed stiffness reduction method and the fitted formulas, a full-scale physical PUTCGE model test under circumferential static loading was used [36]. It should be mentioned that the physical test was performed for only a single-cell PUTCGE model due to constraints available for these testing resources, and there have been no physical tests on double-cell PUTCGEs that can be found in the literature. However, the calculation method and procedure as outlined above are essentially the same, irrespective of a single-cell or double-cell model, in terms of analysis of the PUTCGE. Therefore, a single-cell model could be used for this validation purpose. The specimen was composed of two upper and lower groove-shaped prefabricated elements assembled using prestressed tendons, and the loading was carried out in a circumferential direction. Figure 10 shows the experimental setup. The calculation steps were as follows:  
1. Calculate the aspect ratio and the joint stiffness ratio. The width of the model specimen was 3.3 m, and the height was 3.8 m, so the aspect ratio \(\beta\) was 3.3/3.8 = 0.87. The wall thickness was 0.3 m, the longitudinal segment length was 1 m, and the elastic modulus was 35.6 \(\times\) 10\({}^{6}\) kPa, so the sidewalls' linear stiffness \(I\) was \(EI/h\) = 35.6 \(\times\) 10\({}^{6}\)\(\times\) 1 \(\times\) 0.3\({}^{3}\)/(12 \(\times\) 3.8) = 21,079 kN\(\cdot\)m. The joint rotational stiffness was 24,500 kN\(\cdot\)m/rad, so the joint stiffness ratio \(\gamma\) was \(k/i\) = 24,500/21,079 = 1.16, according to Equation (7). All the original data can be found in [PERSON] and [PERSON] [36]. 2. Calculate the stiffness reduction coefficient and the bending moment adjustment coefficients. By substituting the values of \(\beta\) and \(\gamma\) (obtained in Step 1) into Equation (8), the stiffness reduction coefficient \(\eta\) was obtained and was equal to 0.68, which lay between 0.6 and 0.8, a typical range for shield tunnels [33]. Similarly, the upper bending moment \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitting Coefficient** & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{R^{2}}\) \\ \hline \(\eta\) & 0.125 & \(-\)0.103 & 0.680 & 0.95 \\ \(\xi_{t}\) & \(-\)0.106 & 0 & 0.219 & 0.96 \\ \(\xi_{b}\) & \(-\)0.090 & \(-\)0.026 & 0.216 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting parameters for calculation method of stiffness reduction. Figure 10: Experimental test setup: (**a**) actual setup and (**b**) schematic setup. adjustment coefficient \(\xi_{t}\) was obtained and was equal to 0.15, and the lower bending moment adjustment coefficient \(\xi_{b}\) was obtained and was equal to 0.13. Considering that the test was carried out with two concentrated loads with the same magnitudes applied to the sidewalls, the bending moments at the upper and lower ends of the sidewalls were nearly the same, so the adjustment coefficient was taken as the average value (0.14) of the upper and lower ends. Obtain initial bending moments \(M_{t1}\), \(M_{b1}\), and \(M_{m1}\) according to FE simulation results. The stiffness reduction model of the PUTCGE was established using SAP2000. The concentrated force \(P\), with a value of 73.0 kN [36], the loading locations, and configuration followed Figure 10b. The simulation results showed that \(M_{t1}=M_{b1}=53.0\) kN\(\cdot\)m, \(M_{m1}=31.0\) kN\(\cdot\)m.  
Calculate final adjusted bending moments \(M_{t2}\), \(M_{b2}\), and \(M_{m2}\). By substituting the relevant values from Steps 3 and 2, i.e., \(M_{t1}=M_{b1}=53.0\) kN\(\cdot\)m, \(M_{m1}=31.0\) kN\(\cdot\)m, and \(\xi_{t}=\xi_{b}=0.14\), into Equations (4)-(6), the final adjusted bending moments could be easily obtained, which gave \(M_{t2}=M_{2}=60.5\) kN\(\cdot\)m and \(M_{m2}=23.5\) kN\(\cdot\)m.  
Compare with the experimental values. Table 2 shows a comparison of the calculated values and experimental values [36] in terms of bending moments at the ends of the sidewalls and at the middle-height of the sidewalls.