# Research on the Impact Force of Rockfall Impacting Sand Cushions with Different Shapes
## 4 Numerical Model and Discussion
### Large-Scale Model  
The accuracy of the numerical model is corrected by comparing the numerical simulation results at the laboratory scale with the experimental results. Considering the actual engineering scale, the rockfall impact energy level is much greater than the impact energy level corresponding to laboratory experiments. Therefore, based on this model, considering a larger and closer rockfall impact model to the actual engineering scale, the rockfall impact process with an impact energy level of 180 kJ is simulated. It is worth noting that an energy level of 180 kJ has a variety of conditions. This paper only selected three conditions. With rock block and cushion material parameters unchanged, the falling rock shape is still set to five kinds. The falling stone masses are 1000 kg, 1500 kg, and 2000 kg. The falling height corresponds to 18 m, 12 m, and 9 m. The cushion size is 5 m \(\times\) 5 m, and the cushion thickness is set to1.5 m, which is commonly used in engineering. The specific size of falling stone is shown in Table 6.  
When the energy level is fixed to 180 kJ, the impact force results of falling rocks with five shapes of different masses at corresponding heights are shown in Figure 10. The results are compared with those calculated by using the impact force results of spherical rockfall in numerical simulation as the benchmark and combining with the modified formula proposed in this paper. Figure 9: Comparison of the experimental results and numerical results of five shapes of falling rocks at five falling heights. (**a**–**e**) correspond to five different shapes of rockfall, respectively. It can be seen from the numerical simulation results of Case 1, Case 2, and Case 3 in Figure 10 that the shape of rockfall does affect the impact force of rockfall. Rockfall impact force decreases with the increase in shape coefficient \(S\). When the impact energy level increases, the effect of rockfall shape on rockfall impact force is also obvious. When the impact energy is the same, the impact force corresponding to different mass and height combinations is approximately equal. However, the impact force corresponding to the condition of large mass and small height is slightly larger than that of small mass and large height. That is, the impact force of the three conditions is: Case 1 \(<\) Case 5 \(<\) Case 3. As shown in Figure 10, the numerical results are compared with the results of the formula in this paper. Consider the three cases of Case1, Case2, and Case3. For the tip ellipsoid with a shape factor greater than 1, the difference between the two results is slightly larger, and the maximum difference is about 18.7%. When the shape coefficient \(S\) is less than 1, the two results are close. Especially when \(S\) = 0.5 in Case 2, the difference is only 0.2%. Therefore, it can be considered that the impact force fitting formula considering rockfall shape proposed in this paper is not only limited to the small-scale model but also applicable to the case of scale increase, which can provide guidance for engineering practices. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Falling** & **Shape** & **Vertical** & **Horizontal** \\  & **Height(m)** & **Coefficient** & & **Radius (m)** & **Radius (m)** \\ \hline \multirow{4}{*}{Case 1} & & 4 & 992.582 & 1.12 & 0.28 \\  & & 2 & 1011.456 & 0.71 & 0.355 \\  & 18 & 1 & 1002.852 & 0.446 & 0.446 \\  & & 0.5 & 992.582 & 0.28 & 0.56 \\  & & 0.25 & 1011.456 & 0.1775 & 0.71 \\  & & 4 & 1481.638 & 1.28 & 0.32 \\  & & 2 & 1501.852 & 0.81 & 0.405 \\ Case 2 & 12 & 1 & 1499.487 & 0.51 & 0.51 \\  & & 0.5 & 1481.638 & 0.32 & 0.64 \\  & & 0.25 & 1501.852 & 0.2025 & 0.81 \\  & & 4 & 2022.913 & 1.42 & 0.355 \\  & & 2 & 2060.154 & 0.9 & 0.45 \\ Case 3 & 9 & 1 & 1985.163 & 0.56 & 0.56 \\  & & 0.5 & 2022.913 & 0.355 & 0.71 \\  & & 0.25 & 2060.154 & 0.225 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Large size simulation of falling rock size.