Abstract Rock mass classification (RMC) systems, including Slope Mass Rating (SMR) and Hazard Index (HI), serve as vital tools for stability assessment of rock slopes. The inherent limitations of discrete rating system in HI, necessitate the development of continuous functions for better accuracy and reliability. Due to discrete nature, slight changes or variations in controlling parameters may not reflect in the final HI value. To combat such limitations of HI, this study induces a set of continuous functions in HI for its sub-factors (f1, f2, f3, and f4), formulated using an exponential decay and linear functions. The introduction of Continuous Hazard Index (CHI) and its stability chart enable users for a more precise assessment of slope stability. Furthermore, the synergistic use of CHI and Continuous Slope Mass Rating (CoSMR) leads to a more realistic vulnerability analysis for engineered slopes. This study evaluates the stability of twenty-five engineered slopes in the Lesser Himalaya along a 62 km stretch of National Highway-5 (NH-5) from Shimla to Narkanda, using both discrete and continuous functions of SMR and HI. The variation between HI and proposed CHI scores falls between –0.12 and 0.18, with an average difference of 0.087. Although the variation is meagre yet consistent variation indicates that the CHI method captures subtle changes in slope conditions more effectively than the discrete HI, offering a more precise in stability assessment. Besides CHI, an Equivalent Probability of Failure (EPoF) stability chart by correlating the equivalent points of probability of failure between CoSMR and CHI has been proposed in the study. The EPoF stability chart provides a more accurate representation of slope stability and a quick reliability assessment on computed stability scores using the mutually inclusive and exclusive stability zones. Field-based validation supports the effectiveness of the proposed charts, ensuring the practical utility of proposed EPoF chart. Furthermore, the remedial measures of CoSMR method are also extended to CHI values using EPoF chart. Research highlights <UnorderedList Mark="Bullet"> <ItemContent> Introduction of Continuous Hazard Index (CHI) using continuous exponential decay and linear functions for sub-factors of Hazard Index (HI), replacing lumped ratings to minimize subjectivity in evaluation. </ItemContent> <ItemContent> Developed novel bivariate CoSMR versus CHI stability chart using an Equivalent Probability of Failure (EPoF), which has been validated by percentages of failed and not failed slopes. </ItemContent> <ItemContent> For critical precipitation ratio (Im/Ith), fixed Icr = 700 mm threshold in conventional HI method has been substituted by site-specific Ith (triggering rainfall threshold), improving hydrological sensitivity evaluation. </ItemContent> </UnorderedList>

Introducing continuous functions-based Hazard Index and Equivalent Probability of Failure stability chart: stability evaluation of engineered slopes in the Himalaya along NH-5 from Shimla to Narkanda, India

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发表日期：

2026年5月23日

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Harsh Varshney,
Tariq Siddique

摘要

Abstract

Rock mass classification (RMC) systems, including Slope Mass Rating (SMR) and Hazard Index (HI), serve as vital tools for stability assessment of rock slopes. The inherent limitations of discrete rating system in HI, necessitate the development of continuous functions for better accuracy and reliability. Due to discrete nature, slight changes or variations in controlling parameters may not reflect in the final HI value. To combat such limitations of HI, this study induces a set of continuous functions in HI for its sub-factors (f₁, f₂, f₃, and f₄), formulated using an exponential decay and linear functions. The introduction of Continuous Hazard Index (CHI) and its stability chart enable users for a more precise assessment of slope stability. Furthermore, the synergistic use of CHI and Continuous Slope Mass Rating (CoSMR) leads to a more realistic vulnerability analysis for engineered slopes. This study evaluates the stability of twenty-five engineered slopes in the Lesser Himalaya along a 62 km stretch of National Highway-5 (NH-5) from Shimla to Narkanda, using both discrete and continuous functions of SMR and HI. The variation between HI and proposed CHI scores falls between –0.12 and 0.18, with an average difference of 0.087. Although the variation is meagre yet consistent variation indicates that the CHI method captures subtle changes in slope conditions more effectively than the discrete HI, offering a more precise in stability assessment. Besides CHI, an Equivalent Probability of Failure (EPoF) stability chart by correlating the equivalent points of probability of failure between CoSMR and CHI has been proposed in the study. The EPoF stability chart provides a more accurate representation of slope stability and a quick reliability assessment on computed stability scores using the mutually inclusive and exclusive stability zones. Field-based validation supports the effectiveness of the proposed charts, ensuring the practical utility of proposed EPoF chart. Furthermore, the remedial measures of CoSMR method are also extended to CHI values using EPoF chart.

Research highlights

Introduction of Continuous Hazard Index (CHI) using continuous exponential decay and linear functions for sub-factors of Hazard Index (HI), replacing lumped ratings to minimize subjectivity in evaluation.

Developed novel bivariate CoSMR versus CHI stability chart using an Equivalent Probability of Failure (EPoF), which has been validated by percentages of failed and not failed slopes.

For critical precipitation ratio (I_m/I_th), fixed I_cr = 700 mm threshold in conventional HI method has been substituted by site-specific I_th (triggering rainfall threshold), improving hydrological sensitivity evaluation.