### Stochastic Analysis
#### 3.2.1 Effect of R/\(\gamma\)H\({}^{2}\)  
The influences of the normalized geosynthetic tensile strength \(R/\gamma H^{2}\) on the failure probabilities are shown in Figure 6. Switching from rupture to bond failure mechanisms necessitates the use of greater geosynthetic rupture strength, provided all other parameters remain unchanged. In Figure 6, it is not immediately apparent, but the data points for rupture failure are associated with slightly greater geosynthetic rupture strengths compared to those for bond failure at the same safety factor. A small adjustment in normalized geosynthetic rupture strength, ranging from 0.5 to 0.6, is enough to shift the failure mode from bond to rupture. Although the safety factor remains virtually constant for each geosynthetic rupture strength, the disparity in the maximum probability of failure between rupture and bond mechanisms increases with greater geosynthetic rupture strengths (or higher safety factors). Furthermore, the lower curve illustrates a rapid decline in the probability of rupture failure as the geosynthetic rupture strength increases. Typically, the embankments are designed with a factor of safety \(F_{s}\) of at least 1.35 [19]. To better address this practical design range, additional analyses were performed using 1000 Monte Carlo simulations per scenario to precisely estimate the probability of failure \(P_{\text{f}}\) for cases with very low failure probabilities, down to 0.02%, which will be discussed in detail later. [PERSON] et al. (2008) indicated that a \(P_{\text{f}}\) of 0.01% is associated with an earth dam designed with a factor of safety of 1.5, reflecting a higher-than-average engineering standard [29]. However, this 0.01% threshold is considerably lower than the probabilities depicted in Figure 7 for \(F_{s}\) values ranging from 1.5 to 2.2. It is crucial to emphasize that the chart represents the maximum probability of failure. In examining these results, it becomes evident that maintaining a factor of safety within the specified range significantly impacts the overall failure probabilities, underscoring the importance of rigorous design and analysis to achieve desired safety levels. This detailed probabilistic approach ensures that even scenarios with minimal failure likelihoods are thoroughly evaluated, providing a robust foundation for embankment design and safety assessment. Figure 6: Probabilities of bond and rupture failure for geosynthetic-reinforced embankments as a function of geosynthetic rupture strength \(R/\gamma H^{2}\). Figure 7: Effect of \(F_{s}\) probabilities of geosynthetic-reinforced embankments as a function of reinforcement rupture strength \(R/\gamma H^{2}\). Extended analysis of geosynthetic-reinforced embankments' stability and failure mechanisms reveals critical insights into the relationship between the \(F_{s}\) and \(P_{\mathrm{f}}\). Figure 8 extends the analysis from Figure 7, covering a range of \(F_{s}\) from 1.5 to 2.2 and maximum \(P_{\mathrm{f}}\) values up to 20%. This broader scope allows for a detailed examination of the practical and theoretical implications of varying safety factors and failure probabilities in reinforced embankment design. Practical guidelines for geosynthetic-reinforced embankments, as delineated in the lower shaded area of Figure 8, are based on [PERSON] et al. (2008)'s recommendations, which suggest maintaining \(F_{s}\) values of at least 1.5 and \(P_{\mathrm{f}}\) values of no more than 0.01% (2019). Achieving these parameters necessitates specific combinations of normalized reinforcement tensile strength \(R/\gamma H^{2}\) and high \(F_{s}\). For instance, maintaining a low \(P_{\mathrm{f}}\) requires relatively low normalized reinforcement tensile strength alongside higher \(F_{s}\) values. Conversely, for scenarios where the probability of rupture failure is considered, the \(P_{\mathrm{f}}\) can be higher, as these systems, akin to other high-strength redundant soil-structure systems, can redistribute loads if a single reinforcement fails.