Stability of production in block environments

When solving the issue of the stability of workings, they often face a situation where destruction occurs only due to the movement of solid blocks, and not their destruction, due to the strength of the rock and rock pressure. In this case, the question arises as to whether the shape of the block (and the worked-out space next to it) is capable of moving under the influence of gravity or rock pressure into the workings. It is important to take into account the role of friction forces and determine the relative number of dangerous blocks that can fall into the mine. Similar problems arise when studying faults, when protruding blocks can impede movement along the fault. To solve problems related to the kinematics of the block, taking into account these forces, the solutions presented in the works of Goodman and Shi-Gen-Hua were developed. This article provides a brief overview of the Goodman method with modified proofs of the main theorems, as well as the tasks associated with determining the average number of dangerous blocks. It is assumed that cracks are grouped into a finite number of systems of mutually parallel cracks, which are modeled by planes. Two models are considered - Poisson and equidistant, differing in the distribution of distances between cracks.