Model for the maximum impact force and penetration depth of ellipsoidal rockfall impacting sand cushion: experimental, numerical and theoretical insights

The maximum impact force (F_max) and maximum penetration depth (δ_max) are critical parameters in the design of rockfall protection structures. Current methods for calculating rockfall impact force often simplify the rockfall as a sphere, thereby neglecting the significant effects of rockfall shape and impact angle, which can lead to estimation inaccuracies. Through experimental and numerical analyses, we demonstrate that the F_max occurs at a 90° impact angle, with the F_max of an ellipsoidal rockfall (sphericity = 0.6) being 1.42 times greater than that of a spherical rockfall. Conversely, at a 0° impact angle, the δ_max of ellipsoidal rockfall is 1.72 times greater than its spherical rockfall. This phenomenon indicates that calculating impact force by assuming a spherical rockfall shape may underestimate the actual impact force, resulting in inadequate safety of the protective structure. Based on these findings, we propose two innovative calculation methods for evaluating the F_max and δ_max exerted by an ellipsoidal rockfall impacting a sand cushion. The first method proposes the shape magnification coefficient, and establishes the quantitative conversion relationship for F_max and δ_max between ellipsoidal and spherical rockfalls through experiments and simulations. The second method, based on Hertz contact theory, the analytical solution including shape parameters is derived. The suggested values of F_max and δ_max for ellipsoidal rockfalls are given. Verification indicates that new models enable more accurate prediction of F_max and δ_max for ellipsoidal rockfalls. The results can be applied to the design of protective structures such as shed tunnels, effectively enhancing the accuracy and economy of rockfall disaster prevention and control.