Rock engineering problems are complex due to inherent variabilities of the geometrical and material properties of the rock mass. The factor of safety method is the commonly used approach of designing an excavation, which depends on the mean or some arbitrarily selected conservative values of the input parameters involved and thereby ignores the inherent variabilities of the parameters. In this study, the Monte Carlo simulation method is used to account for the frictional properties, described by the friction angle and cohesion of the joint planes defining a wedge in a slope stability problem as well as the geometrical properties of the wedge given by the dip and dip direction of the joint planes. Both normal and lognormal distributions are used to describe the variability associated with the frictional properties. Additionally, the bivariate normal distribution is used to characterize the uncertainty of the orientation of the joint planes in a spherical coordinate system. Several analyses with different levels of input parameter variability have been conducted. Although the factor of safety remains invariant with different coefficients of variation, the reliability index varies with the variability of the input parameters and uniquely describes the reliability and the probability of failure of the wedge.
An excavation in a rock mass is a complex structure complicated by the variability of natural materials. Properties of rock may vary spatially, and fractures in a rock mass can exhibit significant variability in orientation, spacing, and strength properties, resulting in an extremely heterogeneous and anisotropic rock mass. Moreover, rock mass may degrade with time and applied loads. Field rock engineering problems are typically solved using deterministic methods. Problems involving earth construction must deal with a high degree of uncertainty associated with load and resistance factors that is commonly compounded by the limited availability of site-specific data. In some cases, these complexities may lead to catastrophic failure with considerable human loss and economic consequences. Traditional deterministic design methods, such as the factor of safety approach, often rely on experience-based knowledge that is not generally applicable to the direct quantification of the underlying safety margins of failure events. The potential effects of the inherent variability of the input parameters using the factor of safety approach are either ignored or at least highly idealized, resulting in a design that may under- or overpredict the influence of the parameter on risk of failure. The need for reliability assessment in practical rock engineering projects is increasingly recognized, particularly where significant safety and economic losses are possible. Consequently, reliability estimation methods are needed to bridge the gap between complex deterministic models and sophisticated uncertainty quantification methods that more realistically assess the potentially high-consequence engineering projects. Each project is unique, and the design depends on the knowledge of rock/soil characteristics and design loads, cost of construction and maintenance, and end use. Probability theory provides a more formal way to quantify the risk associated with a project and results in a potentially significant additional resource in the decision-making process, instead of relying on qualitative engineering judgment alone.