# Research on the Impact Force of Rockfall Impacting Sand Cushions with Different Shapes
## 3 Experimental Results
The impact force curve of various shapes of rockfall impacting the sand cushion was fitted in the form of a power function and the impact force \(P=\alpha^{*}H\) (falling height) \(\hat{}\) \  
is proportional to the 3/5 power of the falling height, indicating that the experimental error is within an acceptable range and the experimental results are reliable. Due to the independence of the falling height, it can be considered that although only five kinds of falling heights were set in this paper to explore the impact of falling stone shapes on the impact force, it was sufficient to explain the situation of multiple falling heights. Based on the current reliable experimental results, the impact force calculation of different shapes of rockfall impacting sand cushions is considered. Since there are many reasonable formulas for calculating the impact force of spherical rockfall, this paper takes the impact force results of spherical D3 as the benchmark and intends to use the shape coefficient \(S\) to modify the formula for calculating the impact force. The formula for calculating the impact force \(P\) of rockfall is expressed as follows:  
\[P=S^{k}P_{\text{sp}} \tag{1}\]  
where \(P\) is the rockfall impact force (kN), \(S\) is a coefficient describing the shape of rockfall, \(k\) is a constant coefficient obtained by fitting, and \(P_{\text{sp}}\) is the impact force (kN) of spherical rockfall obtained under the same mass and same working conditions. Considering the five falling heights shown in Figure 4, the average value of the impact force ratio of each rockfall shape to spherical rockfall was taken and fitted. The fitting results are shown in Figure 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & \(\alpha\) & \(\beta\) \\ \hline D1 & 0.41 & 0.60 \\ D2 & 0.68 & 0.64 \\ D3 & 1.12 & 0.56 \\ D4 & 1.65 & 0.61 \\ D5 & 2.62 & 0.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: First fitting parameters. Figure 4: Impact force results of drop hammers with different shapes impacting the sand cushion. By fitting \(k\) to approximately \(-0.6055\), the \(k\) value was substituted into Equation (1), and the following formula was obtained:  
\[P=S^{-0.6055}P_{\text{sp}} \tag{2}\]  
Equation (2) was used to calculate the impact force of impact sand cushions with different shapes of falling stones at different falling heights, and the results were compared with the experimental results. The impact force corresponding to D3 in Figure 4 is taken from \(P_{\text{sp}}\), and the results are shown in Figure 6.  
From the fitting results in Figure 6, except for individual results, the difference between the two results was less than 20%. In particular, for the D5 rockfall, that is, the maximum impact force and the most unsafe situation, the fitting effect was better, and the difference was only approximately 2%. It can be considered that the impact force calculation formula considering the rockfall shape obtained by Equation (2) has a certain rationality and correctness. To further verify the rationality of the above fitting formula, the fitting results of the formula were compared with those from other large-scale experiments or numerical simulations. The results are shown in Table 4.  
Figure 5: Second fitting results. Figure 6: Comparison between fitting results and test results. [PERSON] et al. [17] used a numerical method to study the difference in the impact effect between four kinds of tip ellipsoids and spheres. The weight of rockfall was 1358 kg, and the impact velocities were 15.66 m/s, 26.22 m/s, and 36.94 m/s. The impact object was a reinforced concrete slab. Considering the three landing heights, the ratios of the impact force results of the four tip ellipsoids to the sphere were averaged, which were approximately 0.93, 0.88, 0.76, and 0.51. Combined with the concept of the shape coefficient mentioned in this paper, the shape coefficients corresponding to the four tip ellipsoids were 1.15, 1.41, 1.71, and 2.16, respectively.