Statistical design of rockfall protective structures using a stochastic trajectory analysis model

Classically used trajectory analysis models do not really account for the stochastic nature of the trajectory of falling rocks related to the variability of the impact process. The presented work focuses on the bouncing phase modelling in the case of the interaction of a boulder with a coarse granular soil. Impact simulations are first held using a Discrete Element Method. A statistical analysis of the numerical results is performed in order to build a stochastic impact model relating boulder velocities before impact and after impact. The stochastic impact model is impended within a trajectory analysis software and a validation procedure using real-scale experiments is carried out. Finally, the stochastic trajectory analysis model developed allows characterizing probability distributions functions that quantify hazard levels on endangered slopes and allows defining a probabilistic framework for the optimization of protective structures design. characterization of rockfall hazard and to the design of protection structures in terms of functional and structural efficiency. 2 STOCHASTIC IMPACT MODEL Numerical modelling of the impact Assuming that rocks composing the talus slope can be considered as rigid locally deformable two-dimensional bodies, the software Particle Flow Code 2D (Itasca, 1999) based on a Discrete Element Method (Cundall and Strack, 1979) is used. In the Discrete Element Method, contact forces are applied to neighboring particles in contact. In this paper, the normal and tangent contact forces acting between two particles are calculated using the Hertz-Mindlin model (Mindlin and Deresiewicz, 1953). The contact law only models frictional energy dissipation during the interaction of two rocks. The mean diameter of soil particles is Rm = 0.3 m which corresponds to a relevant value for most of observed rockfall events. In addition, in the case of the impact of a boulder on a coarse granular soil, boulder and soil particles sizes are nearly the same. Boulder radius Rb therefore varies from Rm to 5Rm. Sample properties are defined following the procedure used in Bourrier et al (2008). Once the sample is generated, impact simulations are held for varying impact points and incident kinematic parameters. Incident kinematic conditions are fully defined by the magnitude of the incident velocity V, the incident angle α and the incident rotational velocity ω (Figure 1). Finally, reflected velocities are collected when the normal component of the boulder velocity reaches its maximum. It is important to note that the relevance of the DEM model has been proved compared to results from the literature (Bourrier et al., 2007) and from half-scale experiments of impacts on a coarse soil (Bourrier et al., 2008)]. Fig. 1. Incident kinematics conditions.