The probabilistic nature of appropriate geologic variables can be included in analyzing the stability of potential slope failure modes. Random variables in the two-dimensional plane shear analysis of a specified structural discontinuity include the shear strength and waviness angle of the discontinuity and, in some cases, the rock mass density. Estimated probability density functions that describe these random variables are combined by Fourier analysis to produce an estimate of the safety factor probability density function, which appears to approximate a gamma distribution. The probability that sliding will occur along the specified structure is equal to the area under this density function where the safety factor is less than one.
Slope failures in discontinuous rock masses are controlled by geologic structure, because displacements occur along zones or surfaces of weakness. Natural variabilities and measurement uncertainties associated with field mapping and laboratory testing prescribe that properties of geologic structure be treated in a statistical manner. They can be quantitatively described and then included in stability analyses to evaluate the risk of slope failures. Plane shear failure is characterized by a potential failure mass capable of sliding along a semi-planar discontinuity that dips flatter than the slope angle. A recently developed probabilistic analysis of the plane shear failure mode is based on Monte Carlo techniques for predicting the probability of sliding (Marek and Savely, 1978). This probability is defined as the area under a simulated safety factor distribution where the safety factor (a random variable) takes on values that are less than one. For a stability analysis that includes the waviness angle of the sliding surface (assumed to be exponentially distributed) and uses a power failure model for shear strength (assumed to be normally distributed), the resulting safety factor is approximately gamma distributed. A Monte Carlo simulation provides only one possible realization of the true probability distribution of the safety factor. Also, a large number of sampling iterations (over 1000) is usually required to provide a reasonable estimate of the probability of sliding, making the associated computational time and costs objectionable and sometimes prohibitive. Therefore, fewer iterations are used, resulting in a poorer estimate of the probability of sliding (Appendix). Fourier analysis provides an alternative to Monte Carlo simulation in predicting the probability distribution of the safety factor. The sum of independent probability densities in space domain is analogous to the product of the Fourier transforms of the densities in frequency domain. An efficient method for estimating the true probability density of the safety factor can be based on discrete Fourier procedures, which take advantage of the computational speed of the fast Fourier transform algorithm.
Plane shear failure in fractured rock slopes is kinematically viable if the average dip of the sliding plane is less than that of the slope face and if the average strike is parallel or nearly parallel to that of the slope face. In addition, the assumption is made in a two dimensional stability analysis that the potential sliding mass is laterally unconstrained ("side-release" assumption).