# Granular Flow Impact on Shed Tunnels and the Buffering Effect of Cushion Layers
## 4 Discussion
### Energy Characteristics  
In this study, the energy transfer mechanism throughout the entire process of granular flow motion is evaluated using the built-in energy tracking program in PFC3D. According to the principle of energy conservation, gravitational potential energy is converted into frictional energy, collision energy, kinetic energy, and other forms of energy consumption during the motion of granular flows. The analysis focused on dynamic impact and buffering effects from an energy perspective. During a dynamic analysis, the kinetic energy at every time step is calculated as follows:  
\[E_{k}=\frac{1}{2}m_{k}{v_{k}}^{2} \tag{1}\]  
where \(E_{k}\) is the kinetic energy, \(m_{k}\) is the elemental mass of particles, and \(v_{k}\) is the velocity at the midpoint of the element. The total energy dissipated by collisions between particles is represented by the dashpot energy, calculated as follows:  
\[E_{p}=\frac{1}{2}m_{k}^{p}\Big{(}\eta_{n}v_{k}^{pn}\Big{)}^{2}+\frac{1}{2}m_{k} ^{p}\Big{(}\eta_{t}v_{k}^{pt}\Big{)}^{2} \tag{2}\]  
where \(E_{p}\) is the dashpot energy; \(m_{k}\) is the elemental mass of particles; \(\eta_{n}\) and \(\eta_{t}\), respectively, represent the normal and tangential damping ratios; and \(v_{k}^{pn}\) and \(v_{k}^{pt}\) are the velocity components of particle \(k\) along the normal and tangential directions, respectively. Figure 12 shows the time history curves of the kinetic energy and dashpot energy of the granular flow at different slope angles without a cushion layer. From the graph, it can be observed that as the slope angle increases, the kinetic energy of the granular flow rapidly increases. The peak kinetic energy at a slope angle of \(60^{\circ}\) is 3.74 times higher than that at \(40^{\circ}\). Additionally, with increasing the slope angle, the time to reach the peak kinetic energy gradually advances. In the time history curve of the dashpot energy, there is a distinct differentiating point. Before this point, there was not much difference in the dashpot energy of the granular flow at different angles. However, after this point, a phenomenon occurs where the dashpot energy increases with increasing the slope angle. The dashpot energy reflects the intensity of collisions between particles and between particles and the walls in the linear parallel bond model. Therefore, it can be inferred that as the slope angle increases and the granular flow impacts the shed tunnel, the collisions between particles and between particles and the shed tunnel become more intense, resulting in higher dashpot energy. This conclusion aligns well with the trend of the impact force on the shed tunnel varying with the slope angle. Figure 13a depicts the time history curves of the kinetic energy for different cushion layer thicknesses. Before reaching the peak, the upward trend of the kinetic energy is mostly consistent across the five scenarios. After the peak, the downward trend shows  
Figure 12: Kinetic energy and dashpot energy time history at different slope angles. (**a**) Kinetic energy; (**b**) dashpot energy. little variation among the different cases. The peak value of the kinetic energy decreases with the increasing cushion layer thickness, indicating that the cushion layer can reduce the overall kinetic energy of the debris flow to some extent through particle collisions and friction. Ultimately, this helps to decrease the impact force. From Figure 13b, it can be observed that the dashpot energy shows little difference among the four cases with cushion layer thicknesses of 0 mm, 50 mm, 100 mm, and 150 mm. However, when the cushion layer thickness reaches 200 mm and covers the top of the shed tunnel, the latter half of the dashpot energy curve differs from the other four cases. This is because when the cushion layer does not cover the top of the shed tunnel, although the impact on the frontal portion of the shed tunnel decreases, the top of the shed tunnel still experiences direct impact. As a result, the dashpot energy shows little variation.