Simplified Probabilistic Transition-Matrix Approach for Progressive Internal Failure in Soil Nail Retaining Walls

A simplified analytical-stochastic procedure was developed to examine progressive internal failure of soil nail retaining walls used in transportation infrastructure. Tensile rupture and pull-out were represented through demand–capacity safety ratios, while friction angle, cohesion, unit weight, and soil–nail bond strength were modeled as random variables. Monte Carlo simulation was used to estimate nail-level failure probabilities. A stage-based transition matrix was then constructed to summarize the evolution of a selected progressive-failure sequence after individual nail levels became non-operative. System states were defined by the number of failed nail levels, and load redistribution was represented through an assumed rule. The resulting probabilities were therefore interpreted as comparative, screening-level measures rather than direct indicators of design safety. Application to a representative wall showed that pull-out governed the probabilistic response for the examined configuration. The probability of avoiding additional nail-level failure remained high during the initial stages but decreased markedly as failures accumulated, illustrating the progressive loss of redundancy within the reinforced retaining system.