Relative probabilities of occurrence of points of intersection and lines of intersection of joints in a rock mass are derived and shown to depend only on the mean frequencies and relative orientations of the joint sets. Relative probabilities of occurrence of blocks in a tunnel are also derived and are shown to depend further on the tunnel direction. An application to rock engineering is suggested, in which the derived probability expression is combined with the methods of block theory to find the expected value of maximum support force in a tunnel as a function of tunnel azimuth.
The geometric elements of an idealized discontinuous rock mass include joint planes, lines of intersection of joints and points of intersection of joints. The lines of intersection of joints play a significant role in the hydrologic behavior of jointed rock. The points of intersection of joints have a direct correspondence with the blocks comprising the rock mass, and therefore relate to the mechanical behavior of the rock mass in the vicinity of a free surface. In this paper we derive expressions for the relative probabilities of occurrence of intersection lines and points for each joint pair and triple, respectively. We proceed to find the relative probability that any given combination of three joints creates blocks in a tunnel. Finally, we show how these results can be used as the basis for a probabilistic approach to block theory.