Theoretical notes

The model used for the computation of the trajectories of rock falls considers the blocks like point block that have an impact on an elevation attributed plane. This plane is formed by a grid of tridimensional nodes that form a triangular mesh which represents the entire area between the launch and the stopping of the blocks.

The elements of the grid must be defined so that, in the inside of the perimeter, the inclination and the direction of the slope together with the physical parameters of the model ( the normal and tangential restitution parameters), defined as ratio of the post and pre - impact energy, can be considered constants.

The triangles must be dimensioned right because they can be considered "big" regarding the volume of the blocks and "little" regarding the area involved.

The definition of the launch area assumes a previously analysis of the launch niche so that for every block it can be defined a launch velocity as a function of the initial route along the wall.

This model also carries out the analysis for the positioning and sizing of protective works. The initial computation verifies that the block knocks against the barrier and does not climb over it, so it verifies that the impact kinetic energy of the block can be completely absorbed by the protective work.

This model for the computation of trajectories of rock falls is mostly used for the design of embankments as protective works because, thanks to their high expansion, the embankments can be able to intercept a great number of blocks before requiring maintenance interventions or reconstruction, and assure a remarkable absorption energy, typically generated by long fall routes.

For the design of more precise works, like rock fall nets, the studies on the fall route of blocks are more accurate in the two-dimensional models that are more sophisticated in the description of the physical phenomenon.

The trajectory of the block can be determined by using the motion equations of a block. Referred to a cartesian orthogonal axis system this equations are:

s = v × t + s0

z = - 1/2 × g × t2 + v × t + z0

where:

v = block's velocity;

t = time;

g = gravitational constant;

s = crossed space;

In this way the trajectory of the motion results to be formed by a series of parabolas drawn from the launch point to the point where the block knocks first the slope, in the initial phase of the motion, and between two consecutive impact points on the slope, or at the foot, until the stopping point.

By indicating with vn and vt the components (normal and tangential) of the velocity before the impact, v'n and v't after the impact can be calculated using the following relations:

v'n= vn × λn

v't= vt × λt

where:

λn and λt are the restitution coefficients, varying in the interval 0-1