### Stochastic Analysis
#### 3.2.3 Effect of \(\rm c_{u}/\gamma H\)  
Figure 12 illustrates the effect of normalized undrained shear strength of the soft soils \(c_{\rm u}/\gamma H\) on failure probabilities. To shift from rupture failure to bond failure mechanisms, an increase in \(c_{\rm u}/\gamma H\) is necessary while keeping all other parameters constant. Although this transition is subtle in Figure 12, it is clear that the data points for rupture failure align  
Figure 11: Highest probability of rupture failure for geosynthetic-reinforced embankment with various \(H/D\) compared to mean factor of Safety (\(1.5\leq F_{\rm s}\leq 2.2\)).  
Figure 10: Probabilities of embankment collapse for geosynthetic-reinforced embankments as a function of normalized embankment height, \(H/D\).  
with slightly higher geosynthetic rupture strengths compared to those for bond failure at the same factor of safety. A modest increase in normalized undrained shear strength of the soft soil, typically between 1.575 and 1.6, can effectively alter the failure mechanism from bond to rupture. Even though the factor of safety remains nearly constant for each undrained shear strength, the maximum probability of failure for rupture mechanisms differs significantly from that of bond mechanisms as the undrained shear strength (or safety factor) increases. This suggests that while both failure types are influenced by the same geosynthetic parameters, rupture failures become less likely with lower undrained shear strength. Additionally, the lower curve in Figure 12 demonstrates a marked decrease in the probability of rupture failure as the undrained shear strength increases, highlighting the effectiveness of greater shear strength in reducing rupture failures.  
Figure 13 examines a factor of safety \(F_{s}\) range of 1.5 to 2.2 and maximum failure probabilities \(P_{t}\) up to 20%. The 0.01% threshold is notably lower than the failure probabilities presented for \(F_{s}\) values between 1.5 and 2.2. This figure specifically illustrates the peak failure probabilities, underscoring that maintaining an \(F_{s}\) within this specified range profoundly impacts overall failure probabilities. These findings underscore the importance of precise and thorough design and analysis to achieve desired safety margins. By employing this detailed probabilistic methodology, scenarios with very low failure likelihoods are also rigorously evaluated, ensuring a strong foundation for embankment design and safety assurance. The analysis's inclusion of various shear strengths of soft soil further highlights its adaptability and precision, making it an invaluable approach for optimizing embankment stability and safety across different geotechnical scenarios.  
Figure 14 reveals that to achieve maximum probabilities of failure within the practical limits, combinations of relatively low shear strength and high factors of safety are necessary. It is noted that for rupture failures in geosynthetic-reinforced embankments, higher failure probabilities may be considered acceptable. These higher probabilities can be attained with lower \(F_{s}\) and increased normalized shear strength compared to the stringent 0.01% failure probability criterion. This differentiation underscores the varying acceptance thresholds for rupture and bond failures, highlighting their distinct impacts on the structural integrity of geosynthetic-reinforced embankments. By recognizing the different probabilistic thresholds for these failure modes, engineers can better tailor their design and safety evaluations to ensure robust and reliable embankment performance across diverse conditions. This approach not only enhances the understanding of failure mechanisms but also supports the development of optimized reinforcement strategies to mitigate potential risks effectively.  
Figure 12: Probabilities of bond and rupture failure for geosynthetic-reinforced embankments as a function of the undrained strength of clay.