# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 1 Introduction  
Soil properties vary spatially in the ground as a result of the combined action of physical, chemical and/or biological processes that act at different spatial and/or temporal scales, i.e., they fluctuate over various distances. Traditionally, this soil spatial variability is ignored by assuming a soil profile with uniform soil property values that are usually taken to be the mean. However, characterisation and modelling of spatial variability is essential to achieve a better understanding of the relationships between soil properties and geotechnical soil-structure performance. Much study of this kind in geotechnical engineering is available with differing complexity, ranging from a single random variable approach (e.g., in [1,2]) to many random variable approach (i.e., random field approach, see, for example, [3, 4, 5, 6]). Among the various methods of analysis, the random finite element method (RFEM) is categorised as a level III method (the most sophisticated of its kind, see [7]), which uses a random field to represent soil spatial variability and combines with the finite element method under a Monte Carlo framework to calculate the soil response under various loading conditions. In the case of slope stability, the method automatically seeks out the most critical failure surface without a priori assumption. Therefore, this paper is focused on this level III method. In this kind of analysis, the reliability of the soil structure is targeted for design purposes. Reliability is defined as the probability of a structure to perform a required function under the stated conditions for a specified period of time. It is theoretically defined as the probability of success (i.e., reliability = 1 \(-\) probability of failure) and plays a key role in the cost-effectiveness of systems, in terms of minimising costs and designing reliable engineering works. In a RFEM slope stability analysis, different, but equally likely, random field realisations are generated to gain a probabilistic description of the distribution of the realised factor of safety \(F_{R}\) and failure mechanism. In this case, reliability can be inferred from the computed histogram of \(F_{R}\) (with the probability of failure being found by integrating the area under the normalised histogram for \(F_{R}<1.0\)) (e.g., [8]). Consequence analysis provides quantitative information on the risk and potential hazards that may be caused by the failure of engineering structures (e.g., [9; 10]). With this information, it is possible to improve the original design, incorporate mitigation measures or devise hazard and management strategies to keep the risk at acceptable levels. In slope engineering, the potential failure volume is an important indicator of failure consequence, and therefore it can be important to quantify this volume. Apart from the realised factor of safety (\(F_{R}\), a random variable) based on a Monte Carlo simulation (that is, the distribution of realised factors of safety obtained by analysing the stability of a slope for multiple soil strength realisations based on the same soil strength input statistics), there is one other global factor of safety (\(F_{d}\)) based on the mean undrained shear strength; that is, the factor of safety obtained by carrying out a (traditional) deterministic analysis based only on the mean soil strength. There is a different distribution of \(F_{R}\) for each traditional deterministic \(F_{d}\) and, as mentioned above, the reliability may be assessed from the corresponding distribution of \(F_{R}\). Therefore, each \(F_{d}\) is associated with a particular reliability level, and this provides a way to link traditional and reliability-based designs (i.e., by determining the reliability associated with a traditional \(F_{d}\)).