# Numerical Simulation on the Leakage-Induced Collapse of Segmental Tunnels
## 3 Numerical Model
### Numerical Tunnel Model  
The tunnel model was analyzed using a single-ring tunnel. Since the bearing capacity of the tunnel is greatly reduced due to the opening on the waist, it is unable to reflect the process in which leakage occurs and eventually induces the collapse of the whole structure. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Particle density (kg/m\({}^{3}\)) & 2700 \\ Damping & 0.7 \\ Friction coefficient of particles & 0.8 \\ Tensile strength of particle contacts (Pa) & 36 \\ Shear strength of particle contacts (Pa) & 36 \\ Normal stiffness of particle contacts (N/m) & \(1.0\times 10^{8}\) \\ Shear stiffness of particle contacts (N/m) & \(5.0\times 10^{7}\) \\ Porosity of soils & 0.4 \\ Normal stiffness of contacts between wall and particles (N/m) & \(1.0\times 10^{8}\) \\ Shear stiffness of contacts between wall and particles (N/m) & \(1.0\times 10^{8}\) \\ Friction coefficient between wall and particles & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters in the discrete element model. Figure 6: Monitoring points arrangement. In the coupled fluid-solid strata model, although the loads were extracted from the tunnel with an opening, the loads can also be assumed to be identical to the loads of the adjacent intact tunnel. The reason is that the dimensions of the destabilized falling soil (maximum of about 600 mm in the model tests) were much larger than the width of the model tunnel (24 mm), so the impact loads applied to the opened tunnel ring and the adjacent ring are close to each other, which has been proven in the model tests [25]. In summary, the tunnel was analyzed using a complete single-ring model without opening. The earth pressure was applied to the tunnel, thus reflecting the mechanism of structural collapse without weakening the lining bearing capacity. The geometric nonlinearity of the tunnel structure needs to be considered in the analysis because the tunnel is subjected to large deformations and eventually collapses. In addition, the material nonlinearity problem also needs to be considered in the analysis because the material of some sections and joints enters the elastic-plastic state when the structure collapses. Therefore, a nonlinear analysis model of the shield tunnel was established. The tunnel model was constructed using LS-DYNA 971 [34], a versatile finite element software package known for its proficiency in simulating dynamic phenomena. Hughes-Liu beam elements [34] with cross-section integration were employed to represent segment behavior. The segment section was discretized into eight layers, comprising six layers of concrete and two layers of reinforcing bars. The thickness and positioning of the steel bar layers were determined based on the actual segment's steel bar positions and areas. Longitudinal joints were modeled using a combination of nonlinear springs and rigid beams. The joint surfaces were treated as rigid beams, with lengths matching the segment thickness. Multiple compressive springs and one tensile spring were utilized between the two rigid beams to simulate compression between the segment concrete and bolt tensile behavior. [PERSON] et al. [35] verified the validity of the model by full-scale model tests, so the verification of the tunnel model will not be repeated here. The dimensions and joint angles are consistent with the model tests, as shown in Figure 7.  
In the strata analysis model, the tunnel was simulated with a rigid wall at a fixed position. Therefore, the external loads of the tunnel captured in the simulation were from the active loads generated by the soil, and there were no passive soil resistance forces due to the deformation or settlement of the tunnel. In the simulation of the tunnel structure, to consider the passive resistance of the strata, a series of soil springs, which are only subjected to compression, were established around the tunnel.