Analytical formulas for the geometric and inertia quantities of the largest removable blocks around tunnels

摘要

This paper presents algorithms for determining the vertices of the maximum removable block (MB) created by a joint pyramid (JP) around a tunnel when discontinuities are fully persistent. It is shown that an MB cannot be formed by more than 4 discontinuities and this drastically limits the proliferation of rock blocks that need to be analysed. The non-convex block obtained after the MB is tunnelled through (real maximum block, RMB) is partitioned into a set of tetrahedra, and procedures are given for determining the vertices of these tetrahedra. Geometric and inertia quantities needed for stability analysis and support/ reinforcement design are determined as functions of the calculated vertices' co-ordinates. These quantities are: RMB's volume, face areas, perimeter of the excavated surface, centroid and inertia tensor. The algorithms for their calculation are at least two times faster than other algorithms previously proposed in other applications. It is shown that the formulations presented by Goodman and Shi for translational analysis and by Tonon for rotatability analysis can be used to analyse the RMBs using the geometric quantities presented. A numerical example is presented among those used to verify these analytical procedures and their implementation.