## 4 Conclusions  
This study presents a novel probabilistic stability analysis of geosynthetic-reinforced embankments using Monte Carlo simulations and a random adaptive finite element limit analysis (RAFELA) implemented through Optum G2. The innovative aspect of this research lies in its comprehensive examination of two primary failure mechanisms--bond failure, where critical slip surfaces extend beyond the reinforced zone, and rupture mechanisms, where slip surfaces intersect the reinforcement layers. By investigating the transition to composite mechanisms for steep slopes, this study provides new insights into the probabilistic behavior of these systems under varying conditions. Key findings from the analysis reveal several novel contributions: Firstly, the research identifies critical thresholds for reinforcement tensile strength, embankment height, and undrained shear strength required to ensure bond failure mechanisms. For tensile strengths above these thresholds, the factor of safety (\(F_{s}\)) and probability of failure remain unchanged. Secondly, it is shown that increasing reinforcement length while maintaining or reducing  
Figure 14: Highest probability of rupture failure for geosynthetic-reinforced embankment with various \(c_{u}/\gamma H\) compared to mean factor of safety (\(1.5\leq F_{s}\leq 2.2\)). Figure 13: Effect of probabilities for geosynthetic-reinforced embankments as a function of normalized undrained shear strength. tensile strength can shift the failure mechanism from bond to rupture, without affecting the \(F_{s}\). This finding is crucial for optimizing reinforced embankment designs to balance safety and performance effectively. This study also introduces probabilistic stability charts that offer conservative maximum failure probabilities and establish a link between conventional safety factors and failure probabilities. These charts serve as practical tools for geotechnical engineers, providing valuable insights and preliminary probabilistic design guidelines for geosynthetic-reinforced embankments. Furthermore, the analysis underscores the minimal impact of soil spatial variability on the reliability of probabilistic models, highlighting the critical role of proper construction practices. In conclusion, this research advances the understanding of reinforcement effects on embankment stability by integrating RAFELA with Monte Carlo simulations, offering new perspectives on failure mechanisms and probabilistic design. These contributions enhance the safety and efficiency of geotechnical engineering practices and provide a robust framework for the preliminary design of geosynthetic-reinforced embankments. [PERSON] mainly completed experimental data analysis, manuscript writing, and figure production. [PERSON], [PERSON] ([PERSON]), [PERSON], [PERSON] ([PERSON]), [PERSON] and [PERSON] mainly completed the numerical calculation and manuscript verification. All authors have read and agreed to the published version of the manuscript. The authors express their gratitude for the financial support received from the key R&D program of Gansu Province (No. 23 YFGA0028), the Key Research and Development Project of Lanzhou Jiaotong University (LZJTU-ZDYF2305), the Science and Technology Research and Development Plan of China Railway Co., Ltd. (2022-Major Project-07), Tianyou Postdoctoral Science Foundation Project (TYBSH_KJ_202307), and Lanzhou Jiaotong University-Southwest Jiaotong University Joint Innovation Fund project (LH2023012). Not applicable. All data, models, or code that support the findings in this study are available from the corresponding author upon reasonable request. Author [PERSON] was employed by the Seventh Branch, Zhong Qi Jiao Zhou Construction Co., Ltd. The authors declare that this study received funding from China Railway Co., Ltd. The funder was not involved in the study design, collection, analysis, interpretation of data, the writing of this article or the decision to submit it for publication.