An analytical probability distribution for the factor of safety in underground rock mechanics

In many underground rock mechanics cases, the factor of safety may be defined as the ratio of the capacity, of the rock or its support elements, to the pertinent demand. By representing the capacity and the demand as uniform random variables, an analytic solution is obtained for the probability distribution of the factor of safety and its probability density and cumulative distribution functions. Closed form solutions for the calculation of the mean value, the standard deviation and the minimum and maximum values of the factor of safety, are thus provided. Four relative positions are possible for the demand and capacity density functions, according to their limits. Closed form solutions are provided for the probability of failure for the identified relative positions.Application of the developed analytical solutions for the probabilistic analysis of two cases of underground roofs and four cases of room pillars follows straight forward. This methodology proves also useful for the utilization of field observations or for the extrapolation of any specific design. It allows, finally, for the parametric evaluation of the effect of specific design variables to the distribution of the safety factor.