# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 5 Slope Reliability Assessment--The Two Approaches
### Approach II
Therefore, for \(COV=0.3\), the transformed outputs \(F_{i}\) (the subscript \(i\) denotes \"identical\") are the same, based on Equation (6), if the initial seed for generating the random field is identical (thereby the relative strong and weak spatial distribution of soil properties in the individual realisation of random field is the same for the two cases, only the random field cell values differ by a factor of 2). This is clearly shown in Figure 8, where an analysis based on 40 kPa and 12 kPa gives the same failure volumes as those for an analysis based on 20 kPa and 6 kPa, as indicated by the points sitting in the centre of the circles. The results of analyses based on Approach I with different sets of inputs (Table 1) are also shown in Figure 7. They are in good agreement with the results of Approach II. For each set of input, if the analysis were done in the same way as Approach II to get the whole distribution of \(F_{R}\), the fitted curve would be similar to those shown in Figure 9 (not necessarily the same, because those curves are based on hypothetically generated random numbers). Figure 9 shows the changing of probability of failure as \(F_{d}\) increases, as represented by the areas under the corresponding pdf curves to the left of 1.0 (the probability of failure can also be read directly from the cdf curves shown in Figure 10). Those curves represent the fitted curves of the realised \(F_{R}\). The curve corresponding to \(F_{d}=1.0\) is hypothetically created based on a mean of 0.9 and a standard

deviation of 0.2 (that is, \(\mu_{F_{R}}=0.9\) and \(\sigma_{F_{R}}=0.2\); note that the value of \(\mu_{F_{R}}=0.9\) is assumed here to represent a smaller mean \(F_{R}\) than \(F_{d}=1.0\) that was observed in a typical RFEM analysis), and those corresponding to 1.1, 1.2, 1.3, 1.4 and 1.5 are created by scaling up the \(F_{R}\) for \(F_{d}=1.0\) by factors of \(\alpha_{1}=1.1\), \(\alpha_{2}=1.2\), \(\alpha_{3}=1.3\), \(\alpha_{4}=1.4\) and \(\alpha_{5}=1.5\), respectively (cf. Equation (6)). By doing so, \(\mu_{F_{R}}\) and \(\sigma_{F_{R}}\), for various \(F_{d}\), are also scaled up by the same factors. Therefore, the relative position of \(\mu_{F_{R}}\) and \(F_{d}\) does not change; this is in accordance with Equation (6). Figure 8: Comparison of the two approaches in terms of potential sliding volumes. Figure 7: Comparison of Approach I and Approach II for a 45\({}^{\circ}\) slope (\(\theta_{v}=1.0\) m and \(\theta_{h}=1.0\) m). Figure 10: cdf of realised \(F_{R}\) showing the decrease of probability of failure as \(F_{d}\) increases. \begin{table}
\begin{tabular}{c c} \hline \(F_{d}\) & **Statistics** \\ \hline \(F_{d}=1.0\) & \(\mu=16.3\times 1.0\) kPa, \(\sigma=4.9\times 1.0\) kPa \\ \(F_{d}=1.1\) & \(\mu=16.3\times 1.1\) kPa, \(\sigma=4.9\times 1.1\) kPa \\ \(F_{d}=1.2\) & \(\mu=16.3\times 1.2\) kPa, \(\sigma=4.9\times 1.2\) kPa \\ \(F_{d}=1.3\) & \(\mu=16.3\times 1.3\) kPa, \(\sigma=4.9\times 1.3\) kPa \\ \(F_{d}=1.4\) & \(\mu=16.3\times 1.4\) kPa, \(\sigma=4.9\times 1.4\) kPa \\ \hline \end{tabular}
\end{table} 
Table 1: Input datasets for Approach I.

Figure 9: pdf of realised \(F_{R}\) showing the decrease of probability of failure as \(F_{d}\) increases. ### Equivalence and Differences between the Two Approaches

In essence, these two methods are equivalent to each other. Approach II is a unified approach, regardless of the specific input provided that COV is constant. This can be explained by Equation (6). In fact, once a distribution of \(F_{R}\) is obtained from a base case, for example, \(F_{d}=1.3\), the distributions of \(F_{R}\) for other values (i.e., any value) of \(F_{d}\) can be inferred. This is shown in Figure 11. Not only the distribution can be inferred, but also the probability of failure \(p_{f}\) for any \(F_{d}\). As shown in Figure 11, the shaded area below the pdf to the left side of \(1.0\), for a given \(F_{d}=1.8\), is equal to the shaded area below the pdf to the left side of \(F_{d}^{\text{base}}/F_{d}^{\text{any given}}=1.3/1.8\).