Theoretical study of rockfall impacts based on logistic curves

• A new theoretical rockfall impact model is proposed using logistic curves.
• The coefficient of restitution and the impact force can be calculated accurately.
• The penetration depth from any incident angle can thus be estimated correctly.
• New models improved the predictions of rockfall trajectories.
• Better management of rockfall incidents can be achieved with the proposed model.

The normal velocity curve of a falling boulder with time is fitted by a logistic curve. Without considering the rotational velocity, a logistic equation is used to fit the normal velocity curve and to derive a formula based on the momentum theorem for the normal impact force and penetration depth. Then the tangential force and the tangential coefficient of restitution (COR) are calculated based on the friction theory. The results of the proposed method indicate that the normal COR increases and the tangential COR decreases with the increasing of the stiffness of the ground material. When the total impact velocity is a constant, the incident angle has a great influence on the tangential COR, whereas the normal COR is mainly controlled by the properties of the ground material. The results also indicate that the maximum impact force suggested in this paper is comparable with that by the ASTRA in Switzerland and Japan Road Association (JRA). The penetration depth calculated using the proposed method is close to the result of the test and BIMPAM, while it is far less than that predicted by ASTRA or Pichler et al. (2005). The proposed method has the advantage of allowing systematic and theoretical calculation of the coefficient of restitution, the impact force and the penetration depth from any incident angle. The result will be more accurate if the rotational velocity is considered.