# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 5 Slope Reliability Assessment--The Two Approaches
### Approach II
The reason behind this lies in the following expression of the cumulative distribution function of a normal distribution,

\[\Phi(F_{R})=\frac{1}{2}[1+\text{erf}(\frac{F_{R}-\mu_{F_{R}}}{\sigma_{F_{R}} \sqrt{2}})] \tag{8}\]

where \(\text{erf}()\) is the error function. Based on the above equation, and by setting \(F_{R}=1.0\) and changing the values of \(\mu_{F_{R}}\) and \(\sigma_{F_{R}}\) for any given series of values of \(F_{d}\) (which is proportional to the value of \(\mu_{F_{R}}\) and \(\sigma_{F_{R}}\) for the base case \(F_{d}\)), the reliability versus \(F_{d}\) curve can be obtained; this is essentially Approach I. The same results can be obtained by only looking at the distribution of \(F_{R}\) for the base case. That is, in Equation (8), let \(\mu_{F_{R}}\) and \(\sigma_{F_{R}}\) be the values based on the base case, and change the value of \(F_{R}\) to be \(F_{R}=F_{d}^{\text{base}}/F_{d}^{\text{any given}}\). This way, the reliability versus \(F_{d}\) curve can also be obtained; this is essentially Approach II. Note that the series of \(F_{d}\) values can be any prescribed values, for example, \(F_{d}=1.0,1.1,1.2,1.3,1.4,1.5\). In a typical RFEM simulation, one can either prescribe a series of \(F_{d}\) values or by simply taking the values of \(F_{R}\) from the base case and using them as the series of \(F_{d}\) values (or take the values of \(F_{i}\) (Equation (6)) as the prescribed \(F_{d}\)). The results from the base case of \(F_{d}=1.3\) (\(\mu_{F_{R}}=0.9\times 1.3\) and \(\sigma_{F_{R}}=0.2\times 1.3\)) is shown in Figure 12 for the two cases: (1) when prescribing \(F_{d}=1.0,1.1,1.2,1.3,1.4,1.5\) (the discrete points in the figure); (2) when prescribing \(F_{d}=F_{R}\) (i.e., the series of values of \(F_{R}\) from the base case). Figure 11: pdf with equal shaded area showing that \(p_{f}\) for other \(F_{d}\) (e.g., 1.8) can be inferred from one single set of inputs based on \(F_{d}=1.3\) (\(\mu_{F_{R}}=0.9\times 1.3\) and \(\sigma_{F_{R}}=0.2\times 1.3\)). For approach I, if the slope fails for a trial factor (SRF) of 1.0, the realised \(F_{R}\) could be some value smaller than 1.0. For instance, if the plastic iteration count reaches the limit for SRF = 1.0, it might have reached the limit for SRF = 0.9, or some value that is smaller than 1.0. However, it did not really get the \(F_{R}\) for those realisations that fail (\(F_{R}\leq 1.0\)). They were assumed to be 1.0, consequently, \(F_{i}\) was assumed to be \(F_{d}\) (Equation (6)). Actually, this is better illustrated in Figure 8. With Approach I, the figure is plotted by pulling back these points sitting to the right of the vertical lines of \(F_{d}=1.3\), \(F_{d}=1.2\), \(F_{d}=1.1\) and \(F_{d}=1.0\). The calculated reliability is identical for the two approaches. That is, for each given \(F_{d}\), the probability of failure is the ratio of the number of points (realisations) sitting to the right of this, \(F_{d}\), to the total number of points (realisations) in Figure 8. Note that the points in any vertical line comprise all the projective points (from the base case in Approach II) sitting to the right of that line. So, if the increment of \(F_{d}\) is fine enough, e.g., \(\Delta F_{d}=0.01\) or \(\Delta F_{d}=0.001\), instead of the previously used \(\Delta F_{d}=0.1\), the figure plotted using Approach I (that is, in terms of vertical lines, excluding the realisations or points that have already failed for larger \(F_{d}\) under consideration, and only including these new failed realisations or points) would be the same as that produced by Approach II. The linear trend for failure volumes shown in Figure 8 indicates that the volumes are smaller for higher \(F_{d}\)[21]. As \(F_{d}\) increases or the probability of failure decreases (Figure 7), the potential slide volumes becomes smaller. In other words, the risk associated with a higher global factor of safety (\(F_{d}\)) may be relatively low, due to the low probability of failure and a decreased likelihood of large slide volumes (or an increased trend of small slide volumes).