Distribution of the factor of safety, in geotechnical engineering, for independent piecewise linear capacity and demand density functions

In many geotechnical engineering cases, the factor of safety may be defined as the ratio of the capacity, of the geotechnical structure or its support elements, to the pertinent demand. By representing the capacity and the demand as independent piecewise linear random variables, an analytic solution is obtained for the probability density and cumulative distribution functions of the factor of safety. Thus, solutions for the calculation of the mean value, the standard deviation and the minimum and maximum values of the factor of safety, are provided. Application of the developed analytical solutions, to the probabilistic analysis of a published case of rock spalling in a deposition tunnel complex, follows. The methodology allows for the parametric evaluation of the effect of specific design variables to the distribution of the safety factor and to the probability of failure. The closed form solution may be programmed as a computer code that may run easily on a tablet or netbook or even on a smartphone. It proves useful for the probabilistic design of a variety of geotechnical applications, such as foundations, tunneling, mining, underground roof reinforcement, and earth retaining structures, and permits decisions to be taken in terms of risk and reliability.