The joint intersection probability

In this paper a practical method to apply block theory is presented. Block theory provides the removable joint pyramids from a given free surface regardless of the number of joints in any joint intersection. While robust, the application of the theory in real practice is hampered by the large outcome space of possibly removable joint pyramids consisting of k mutually exclusive joints in a rock mass consisting of m joint sets. In this paper, we prove that the probability that k is greater than three in a three-dimensional space is zero. Consequently, only tetrahedral blocks need to be considered in the stability analysis for the analyzed free surface. The outcome space of theoretically removable joint pyramids can be further reduced by considering "safe" joint intersections, which consist of at least one line of intersection which is sub-parallel to the free surface. The block failure likelihood of the remaining joint intersections is proportional to two independent parameters: (1) the joint intersection probability and (2) the block instability parameter. We develop here a rigorous joint intersection probability expression based on simple frequency probability considerations which predicts that the probability for x in the rock mass to fall in joint intersection L_i,j,k is inversely proportional to the volume of the parallelepiped formed by joints i,j,k with mean spacing values x_i, x_j, x_k: P(x ∈ L_i,j,k) = 1/V_i,j,k/ ∑_l≠r≠s=1^m 1/V_l,r,s. Using the joint intersection probability and the instability parameter associated with each removable JP the critical key blocks of the excavation can be determined. In a brittle rock mass only the critical key blocks will require reinforcement. The paper concludes with a practical example which demonstrates the application of the concepts.