# Numerical Simulation on the Leakage-Induced Collapse of Segmental Tunnels
## 4 Tunnel Collapse Process Analysis
### Structural Response
#### 4.2.2 Tunnel Collapse Process  
In the simulation of tunnel collapse, the external loads were applied one by one according to the six key loads extracted from the stratigraphic analysis, which corresponded to the moments from 1 to 6. The internal forces at the six moments are shown in Figure 14, where the bending moments are positive in tension at the inner side of the tunnel and the axial forces are positive in compression. Taking the loading moment as the vertical axis coordinate and the relative displacement between the top and bottom of the tunnel as the horizontal axis coordinate, the damage development process of the tunnel structure is plotted in Figure 15. To facilitate the analysis and comparison, the results of the numerical simulation were transformed into the data of the prototype according to the similarity relationship. The structural damage development process was analyzed as follows:  
1. Moment 1: at this time, the tunnel was subjected to normally distributed external soil and water pressure, which resulted in a 1.14 mm top-bottom convergence, and no structural damage had occurred yet.
2. Moment 2: with the formation of the initial soil arch, the load at the top and bottom of the tunnel increased significantly. Under the dual effects of waist load decrease  
Figure 13: External load distribution at different moments (kPa): (**a**) Moment 1 to 3; (**b**) Moment 4 to 6.  
and top and bottom loads increase, the bending moment generated at moment 2 increased significantly, thus leading to large top and bottom convergence deformation of 20.53 mm.
3. Moment 3: after the initial soil arch destabilization, the pressure at the top of the tunnel decreased. Therefore, the bending moment decreased, resulting in the relative displacement between the top and bottom of the tunnel decreasing to 3.92 mm.
4. Moment 4: after the first impact load was applied to the top of the tunnel, the load on the top of the tunnel increased suddenly, and the bending moments of the top and bottom of the tunnel and the waist of the tunnel increased significantly, resulting in the crushing of the gypsum at the 12\({}^{\circ}\) and 168\({}^{\circ}\) joints.
5. Moment 5: after the second impact load was applied to the top of the tunnel, the gypsum of the 80\({}^{\circ}\) and 304\({}^{\circ}\) joints at the waist of the tunnel was crushed. Up to now, four plastic hinges appeared in the tunnel and the tunnel was destabilized. At this time, the convergent deformation at the top and bottom of the tunnel had reached 242.82 mm.  
A comparison of the collapse phenomena of the tunnel calculated by numerical simulation and in the model tests is shown in Figure 16. As can be seen from the comparison, the damage phenomena of the tunnel obtained from numerical simulation and the test results have a high degree of compatibility. Both are manifested in the overall destabilization of the upper part of the tunnel. Plastic hinges were formed at the joints, and the top two segments were destabilized and rotated, while the bottom deformation of the tunnels was relatively small. This proves the reliability of the numerical simulation results.  
Figure 14: Internal force response of the tunnel: (**a**) Bending moment (moment 1 to 3) (kN\(\cdot\)m); (**b**) Bending moment (moment 4 and 5) (kN\(\cdot\)m); (c) Axial force (kN).