### Stochastic Analysis
#### 3.2.2 Effect of H/D  
Figure 9 presents the impact of normalized embankment height \(H/D\) on failure probabilities. When transitioning from bond failure to rupture failure mechanisms, an increase in \(H/D\) is necessary, assuming all other parameters remain constant. While this transition is subtle in Figure 9, it is evident that the data points for rupture failure correspond to slightly higher geosynthetic rupture strengths than those for bond failure at the same factor of safety. A small increase in normalized embankment height, typically between 1.3 and 1.5, can effectively switch the failure mechanism from bond to rupture. Despite the factor of safety remaining nearly constant for each geosynthetic rupture strength, the maximum probability of failure for rupture mechanisms diverges significantly from that of bond mechanisms as the embankment height (or safety factor) increases. This indicates that while both failure types are influenced by the same geosynthetic parameters, rupture failures become less probable with a larger height of the embankment. Furthermore, the lower curve in Figure 9 highlights a pronounced decline in the probability of rupture failure as the embankment height increases, underscoring the effectiveness of higher embankment height in mitigating rupture failures.  
Figure 10 extends the analysis from Figure 9 to cover a factor of safety \(F_{s}\) range of 1.5 to 2.2 and maximum probability of failure \(P_{t}\) values up to 20%, akin to Figure 7. The 0.01% threshold is significantly lower than the probabilities shown in Figure 10 for \(F_{s}\) values between 1.5 and 2.2. It is important to highlight that the chart displays the maximum probability of failure. Analyzing these results reveals that maintaining a factor of safety within the specified range greatly influences overall failure probabilities. This underscores the critical need for meticulous design and analysis to achieve the desired safety levels. By adopting this comprehensive probabilistic approach, even scenarios with minimal failure likelihoods are thoroughly assessed, ensuring a solid foundation for embankment design and safety evaluation. The inclusion of different embankment heights highlights the adaptability and precision of the probabilistic approach, making it a valuable tool for optimizing embankment stability and safety in diverse geotechnical conditions.  
Figure 9: Probabilities of bond and rupture failure for geosynthetic-reinforced embankments as a function of normalized embankment height.  
Figure 11 illustrates that achieving maximum probabilities of failure within the practical ranges (\(F_{\rm s}\geq 1.5\) and \(P_{\rm f}\leq 0.01\%\)) requires combinations of relatively low embankment height and high \(F_{\rm s}\). Higher probabilities of failure (\(P_{\rm f}\) up to 1%) may be acceptable for rupture failure in geosynthetic-reinforced embankments, achievable with lower \(F_{\rm s}\) and higher normalized embankment height compared to the 0.01% criterion. Consequently, while both rupture and bond failures are significant, the acceptance criteria for their probabilities of failure differ, reflecting their respective impacts on the structural integrity of geosynthetic-reinforced embankments.