## 2 Three-Dimensional Finite Element Modelling and Validation
### Validation  
[PERSON] et al. [12] conducted full-scale model tests on shield tunnel lining structures, as illustrated in Figure 5a, investigating the deformation characteristics and failure mechanisms of these structures under various load conditions. The dimensions of the lining structure, the number of lining rings, and the assembly were selected based on full-scale model tests in this study. This study aimed to enhance the verification of the numerical model's accuracy; thus, the distribution and magnitude of loads in the numerical sim  
Figure 4: Schematic diagram of interactions. ulation were directly based on the experimental setup, as illustrated in Figure 5b. The loading method of the experimental loading device is a concentrated load application, utilizing 24 concentrated loads to simulate the earth pressure on the tunnel, as illustrated in Figure 5c. \(P_{1}\) is primarily employed to replicate the earth pressure at the top of the tunnel, \(P_{2}\) simulates the lateral earth pressure on the tunnel, and \(P_{3}\) represents the earth pressure on the tunnel's shoulder transition section. Table 2 lists the load magnitudes in the segment full-scale model test. Two sets of jacks and corresponding loading beams were used to apply each load on the segments, while half-ring segments were loaded with a single set of jacks and a loading beam. The segments experienced a longitudinal load of 1500 kN, distributed evenly using 16 jacks. The numerical model replicates a three-ring staggered shield tunnel segment, similar to the full-scale test in terms of the circumferential load application. It also simulates an equivalent longitudinal load distribution of 280 kPa acting on the segment's longitudinal surface. Figure 6 shows the comparison of tunnel convergence between the numerical simulation and full-scale tests. The comparison reveals that the simulation closely aligns with the test results for loading conditions L1 to L3. Discrepancies between the numerical calculations and full-scale experimental results for conditions L3 to L4 are acknowledged. Conditions L1 through L4 represent the loading stages, and, as illustrated in Figure 6, the lining structure's elastic deformation stages from L1 to L3 exhibit a high degree of agreement between the numerical simulations and experiments. This concurrence suggests that the boundary conditions and the elastic mechanical parameters of the materials used in the numerical simulations are consistent with those in the experiments. However, during the transition from L3 to L4, which marks the plastic deformation stage of the lining structure, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Conditions** & \(P_{1}\) **(kN)** & \(P_{2}\) **(kN)** & \(P_{3}\) **(kN)** \\ \hline L0 & 0 & 0 & 0 \\ L1 & 356 & 255 & 270 \\ L2 & 610 & 427 & 445 \\ L3 & 750 & 530 & 558 \\ L4 & 746 & 414 & 436 \\ L5 & 450 & 310 & 320 \\ L6 & 434 & 450 & 436 \\ L7 & 293 & 130 & 167 \\ L8 & 413 & 278 & 283 \\ L9 & 413 & 148 & 220 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The loading procedure in the segment full-scale model test. Figure 5: Schematic diagram of segment full-scale model’s test loading. (a) Segment full-scale model test; (b) distribution of loading beam; (c) distribution of test loading. the discrepancy between the numerical simulations and experimental results begins to widen. The primary factors in this discrepancy are identified as the plastic mechanical parameters of the concrete and the constitutive model's simulation of concrete's plastic mechanical development. This study employs the Concrete Damaged Plasticity (CDP) model, which, to a certain extent, accurately simulates the plastic characteristics of concrete. Nonetheless, it fails to replicate issues observed in experiments, such as concrete spalling and detachment, which likely contribute to the main source of error. The error is more significant for conditions L4 to L6, with deviations of 26% and 47% for L5 and L6, respectively.