The probability of instability for 2-dimensional plane shear failure of a slope is a function of the shear strength, orientation, and length of the fracture forming the potential failure mode. The probability that a fracture has a dip within a given range (PD) and also a sufficient length in that orientation to reach from the toe to the top of the slope (PL), can be calculated from dip and length distributions obtained from field mapping. Using a rigid block analysis, the probability of sliding can be calculated by Monte Carlo sampling of the shear strength and fracture roughness distributions to determine the distribution of safety factors. The area of the safety factor distribution less than 1 is the probability of sliding (Ps).
Slope instability from rock sliding along a single failure plane can be analyzed to determine the probability of slope failure. Variability in estimates of rock mass properties and rock strength measurements implies the probabilistic nature of geologic phenomenon. A design based on average values or on a single value does not account for this variability. A high safety factor might be calculated by using average values for the geologic parameters, but because of the variability shown by the distributions of these parameters, a high probability of failure may also be present (Höeg and Murarka, 1974). In addition, it is difficult to incorporate the traditional safety factor calculation into an economic analysis. The probability of instability, however, can be used with an economic risk analysis to determine an economic optimum design that considers cost impact of failure (Kim and others, 1976).
The simple plane shear failure geometry is analyzed to determine the probability of failure (Figure 1). The geologic structure must occur in an orientation that makes this failure mode viable.
