Study on the Stability Evaluation Index System for Rock Slope–Anchoring Systems

Abstract

The stability of rock slope–anchoring systems is one of the core concerns in protecting the ecological environment and ensuring the safe operation of hydropower, transportation, and construction projects. The stability evaluation index system is a critical factor influencing the accuracy of such assessments. This study establishes a stability evaluation index system for rock slope–anchoring systems by incorporating multi-factor influence mechanisms. The approach involves indicator screening, development of a hierarchical analytical structure, definition of classification criteria, and comparative analysis. The results indicate the following: (1) The proposed index system fully considers the deformation and failure modes of rock slopes, the factors influencing stability, and the safety-related parameters of anchoring structures. (2) It comprehensively captures the multi-factor influence patterns affecting the stability of the rock slope–anchoring system. (3) Compared with traditional empirical and equal-interval grading methods, the grading standards defined by this system are more accurate, better reflect the intrinsic data characteristics, and yield higher classification precision.

1. Introduction

Anchoring technology offers advantages such as ease of construction, low self-weight, and cost-effectiveness [1,2,3,4] and has been widely adopted for slope reinforcement in infrastructure projects related to transportation, mining [5,6], and hydropower [7,8]. This has led to the formation of numerous rock slope–anchoring systems. The stability of these systems is a critical factor directly affecting the operational reliability of major infrastructure and the effectiveness of disaster prevention and mitigation. System instability can result in severe economic losses, environmental hazards, and casualties. Therefore, establishing a reliable and systematic evaluation index system for rock slope–anchoring system stability is of great importance for ensuring accurate stability assessments, safeguarding the safe operation of engineering projects, and protecting both the ecological environment and human safety.

At present, extensive research has been conducted on the evaluation of rock slope stability. The primary methods include the Limit Equilibrium Method (LEM), numerical simulation techniques, machine learning approaches, reliability analysis, and vector sum methods [9]. A summary of the main rock slope stability evaluation methods is presented in

Table 1. Summary of methods used for rock slope stability evaluation.

Stability Evaluation Method	Description	Reference
Limit Equilibrium Method	Slope stability is evaluated by calculating the factor of safety (FoS) using limit equilibrium analysis, which is suitable for simple slope assessments.	Ullah [9] Bi [18] Yan [19]
Numerical Simulation Methods	Methods such as the Finite Element Method (FEM), Finite Difference Method (FDM), and Discrete Element Method (DEM) can be used to analyze the stress–strain behavior of complex slopes.	Ullah [9] Chen [20] Liu [21] Xia [22]
Artificial Intelligence Methods	Machine learning methods predict slope stability based on training data, making them suitable for large-scale data analysis.	Koopialipoor [10]
Limit Analysis Method	Plasticity theory is used to determine the conditions of slope failure, making this method suitable for investigating slope failure mechanisms.	Ullah [9] Wang [23]
Vector Sum Method	This method calculates the sliding direction and stability of slopes and is applicable in complex geological settings.	Ullah [9] Guo [24] Yang [25]
Reliability Theory	Reliability theory is applied to estimate the probability of failure in rock slopes, thereby enabling a probabilistic assessment of slope stability.	Bian [26] Chen [27]
Fuzzy Comprehensive Evaluation Method	The membership degree of each indicator is determined based on fuzzy set theory. According to the weights and membership degrees, fuzzy operations are applied to obtain the membership degree at the criterion level, thereby enabling the evaluation of slope stability.	Xia [28]

. Among these, LEM remains the most widely used approach, while numerical simulations and artificial intelligence techniques are emerging as prominent research trends [9]. Current applications of artificial intelligence in slope stability evaluation primarily utilize machine learning algorithms such as Artificial Neural Networks (ANNs), Support Vector Machines (SVMs), and Random Forests (RFs) [10,11,12,13,14,15,16,17]. A summary of the applications of AI-based methods in slope stability assessment is provided in

Table 2. Summary of the application of artificial intelligence algorithms in slope stability evaluation.

Stability Evaluation Method	Artificial Intelligence Algorithms	Reference
Slope Stability Evaluation Method Based on Artificial Intelligence Algorithms	Artificial Neural Network (ANN), Imperialist Competitive Algorithm (ICA), Genetic Algorithm (GA), Particle Swarm Optimisation (PSO), and Artificial Bee Colony (ABC).	Koopialipoor [10]
	Artificial Neural Network (ANN) combined with an improved Sine Cosine Algorithm (SCA).	Khajehzadeh [11]
	Decision Tree (DT), Random Forest (RF), and AdaBoost.	Asteris [12]
	Support Vector Machine (SVM), Gradient Boosting Regression (GBR), and Bagging methods.	Lin [13]
	AdaBoost, Gradient Boosting Machine (GBM), Bagging, Extremely Randomized Trees (ET), Random Forest (RF), Histogram-Based Gradient Boosting (HistGB), Voting, and Stacking.	Lin [14]
	Single-Valued Neutrosophic Number (SVNN) and Adaptive Neuro-Fuzzy Inference System (ANFIS).	Qin [15]
	Neuro-Fuzzy (NF) systems integrated with Invasive Weed Optimization (IWO) and Elephant Herding Optimization (EHO).	Moayedi [16]
	Multilayer Perceptron (MLP), Support Vector Machine (SVM), k-Nearest Neighbors (k-NN), Decision Tree (DT), and Random Forest (RF).	Ahangari [17]
	Intelligent visual analysis utilizing methods such as Principal Component Analysis (PCA), Kernel PCA, Factor Analysis (FA), Independent Component Analysis (ICA), Non-negative Matrix Factorisation (NMF), and t-distributed Stochastic Neighbor Embedding (t-SNE).	Wang [29]

.

In addition, intelligent slope stability prediction based on visual exploratory data analysis has been developed using six dimensionality reduction techniques: Principal Component Analysis (PCA), Kernel PCA, Factor Analysis (FA), Independent Component Analysis (ICA), Non-negative Matrix Factorisation (NMF), and t-distributed Stochastic Neighbor Embedding (t-SNE) [29]. Furthermore, several researchers have assessed slope stability based on parameters such as the Geological Strength Index (GSI), Slope Mass Rating (SMR), Chinese Slope Mass Rating (CSMR), and Continuous Slope Mass Rating (CoSMR) [30,31].

In the evaluation of the safety of anchoring structures, the main methods currently employed include the analytic hierarchy process (AHP) [32,33,34], numerical simulation [35,36,37], non-destructive testing (NDT) [5,38,39,40,41], monitoring data analysis [42,43,44], and artificial intelligence-based approaches [45,46]. Among these, non-destructive testing remains the most commonly used method for evaluating the safety of anchoring structures. A summary of commonly used non-destructive testing techniques is provided in

Table 3. Summary of non-destructive testing techniques for evaluating anchorage structures.

Non-Destructive Testing Technologies	Detection Objective	Reference
Ultrasonic Testing	Assess internal defects and corrosion level of rock bolt	Lama [5] Yu [41]
Fibre Optic Sensing Technology	Real-time monitoring of strain and deformation in rock bolt	Lama [5]
Piezoelectric Sensors	Dynamic response of rock bolt	Lama [5]
Electromagnetic Induction Technology	Evaluate metal loss in rock bolt	Lama [5]
Impact-Echo Method	Detect the integrity of rock bolt	Lama [5] Skipochka [38] Zatar [39]
Acoustic Emission Technology	Identify microcracks and damage in rock bolt	Lama [5]
Vibration Attenuation Measurement	The rock bolt fastening quality	Skipochka [38]

.

Although various methods have been developed to assess slope stability and the safety of anchoring structures, the key challenge lies in establishing a comprehensive and systematic evaluation index system. The indicators proposed thus far are mainly categorized into five aspects: rock mass-related indicators, geometric structure indicators, environmental and dynamic indicators, integrated safety performance indicators, and anchoring system indicators. A summary is presented in

Table 4. Summary of evaluation indicators for slope stability and anchoring structure safety.

Indicator Category	Contents Included	Reference
Geomechanical Indicators	Uniaxial compressive strength, rock mass type, BQ (basic quality index of rock mass), friction angle of geomaterials, cohesion of geomaterials, unit weight of geomaterials, rock density, lithology, geomaterial type, dry density of rock mass, dip, dip direction and friction angle of dominant discontinuities, tensile strength of rock layers and shear strength of joints, depth of tensile fractures	Taheri [47]; Song [48]; Koopialipoor [10]; Khajehzadeh [11]; Asteris [12]; Lin [13]; Lin [14]; Qin [15]; Moayedi [16]; Ahangariet [17]; Wang [29]; Moses [49]; Mazzoccola [50]; Basahel [51]; Zheng [52]; Aladejare [53]
Topographic and Geometric Structure Indicators	Qslope (rock slope quality evaluation), slope angle, slope height, potential slip surface angle, elevation, slope aspect	Song [48]; Azarafza [54]; Ghafour [55]; Koopialipoor [10]; Khajehzadeh [11]; Asteris [12]; Lin [13]; Lin [14]; Qin [15]; Moayedi [16]; Ahangariet [17]; Wang [29]
Environmental and Dynamic Indicators	Peak ground acceleration, horizontal acceleration coefficient, pore water pressure, stream power index (SPI), topographic wetness index, direction of seismic inertial force, variation in water depth	Koopialipoor [10]; Khajehzadeh [11]; Asteris [12]; Lin [13]; Lin [14]; Qin [15]; Moayedi [16]; Mazzoccola [50]; Zheng [52]; Aladejare [53]
Comprehensive Safety Performance Indicators	BQ, Qslope, geological strength index (GSI), slope mass rating (SMR), continuous slope mass rating (CoSMR), Chinese slope mass rating (CSMR), failure approaching index (FAI) of anchoring structures	Song [48]; Azarafza [54]; Sardana [30]; Sardana [31]; Zhu [33]
Anchoring System Indicators	Rock mass integrity, influence of groundwater, properties of anchoring materials, prestress level, installation accuracy, grouting effectiveness, long-term stress variation, environmental impact, support parameters, quality of bolts and accessories, construction management and monitoring level, rock strength, development of joints, rebar type, grouting materials, borehole quality, grouting methods, groundwater flow, temperature variation, static load, dynamic impact, corrosion resistance and tensile strength of materials, grouting quality, anchoring depth, humidity, chemical erosion, stress redistribution, deformation control, rock hardness and integrity, compressive strength of grouting materials, rock mass classification, anchor rod diameter, number of bolts, excavation timing, layer characteristics, anchorage length, bolt type and construction technology, grouting quality, bolt type, installation angle, water–cement ratio of grout body, curing time and properties of grouting materials	Zhu [33]; Wang [34]; Kim [56]; Frenelus [57]; Yi [58]; Hosseini [45]; Han [59]; Luga [7]; Lin [60]; Hosseini [46]

.

Extensive research has been conducted on the evaluation of rock slope stability and anchoring structure safety, resulting in a wide range and variety of proposed evaluation indicators. However, studies focusing on establishing a unified stability evaluation index system for the systems remain limited. Moreover, the classification standards for indicators are still largely based on empirical judgment and equal-interval division, which are non-quantitative methods that may significantly affect the accuracy of stability evaluations. This study, based on a comprehensive literature review and the analytic hierarchy process (AHP), constructs a three-tiered index system comprising 24 primary indicators, 5 secondary indicators, and 1 tertiary indicator. A novel method is proposed for determining classification standards of continuous indicators, based on the influence patterns of stability coefficients in the systems. Finally, a comparative analysis using the K-nearest neighbor (KNN) algorithm is conducted to verify the rationality of the proposed indicator system.

2. Hierarchical Analytic Structure of the Indicator System

In statistics, an indicator is defined as a concept that reflects the quantitative characteristics of a population. The various characteristics of a whole are represented by different indicators that are relatively independent yet interrelated, collectively forming an indicator system.

2.1. Construction of the Hierarchical Analytical Structure for Rock Slope–Anchoring Systems

Based on the characteristics of the rock slope–anchoring system, as well as the influencing factors of slope stability and anchorage structure safety, a three-level hierarchical analytical structure—comprising the criteria layer, sub-criteria layer, and indicator layer—was established for the stability evaluation index system of rock slope–anchoring systems, as illustrated in Figure 1.

2.2. Indicator Selection

2.2.1. Preliminary Screening of Indicators

The stability of the systems is influenced by numerous and complex factors, with diverse failure mechanisms. Consequently, the evaluation indicators vary accordingly. Due to the significant influence of subjective judgment in the selection process, different researchers often adopt different sets of indicators. Based on the characteristics of the systems, existing technical standards [61], and relevant literature, a total of 36 indicators affecting the system’s stability were identified, as summarized in

Table 5. Preliminary indicators for stability evaluation of rock slope–anchoring systems.

Rock Slope Height	Cohesion of Rock Mass	Distribution of Isolated or Overhanging Rock Blocks	Earthquake
Rock Slope Angle	Internal Friction Angle of Rock Mass	Lithology	Strength Reserve Coefficient
Basic Quality Grade of Rock Mass	Included Angle of Discontinuities in Wedge Structure	Drainage System	Visual Condition
Connectivity of Discontinuities	Dip Angle of Adverse Bedding Plane	Rainfall Intensity	Grouting Saturation
Cohesion of Discontinuities	Dip Angle of Favorable Bedding Plane	Rainfall Duration	Corrosion Rate
Internal Friction Angle of Discontinuities	Degree of Weathering of Rock Mass	Groundwater Development	Prestress Loss Rate
P-wave Velocity of Rock Mass	Riverbank Erosion	Unit Weight of Rock	Gully Development
Jointing Degree (Fracturing Degree)	Slope Crest Load	Erosion Characteristics at Slope Angle	Surface Water
Fracture Density	Distance from River Influence	Elastic Modulus	In situ Stress

.

2.2.2. Indicator Screening

Indicator selection was based on five key principles: purposefulness, operability, hierarchical structure, completeness with representativeness, and observability. The preliminary indicators identified in Section 2.1 were further screened according to these principles. Field investigations and geological surveys revealed several typical degradation features in the rock slope–anchoring system, including severe weathering of the rock–soil mass, presence of unfavorable structural planes, steep and high slopes, water-related factors, insufficient grouting in anchorage bodies, and damage or failure of the anchoring structures. A total of 24 indicators were ultimately selected for the stability evaluation of the systems, along with their corresponding acquisition methods, as shown in

Table 6. Summary of the stability evaluation indicator system structure and rating levels for rock slope–anchoring systems.

Criterion Layer	Project Layer	Indicator Layer	Unit or Evaluation Grade	Acquisition Method
Stability of Rock Slope–Anchoring System	Slope Geometrical Features	Slope Angle	°	Surveying
		Slope Height	m	Surveying
	Hydro-meteorological Conditions	Drainage System	Excellent, Good, Fair, Poor, Very Poor	Site Investigation
		Historical Maximum Daily Rainfall	mm	Monitoring
		Rainfall Duration	d	Monitoring
		Groundwater Level Status	Ratio of Groundwater Level to Slope Height	Monitoring
	Rock Mass Conditions	Basic Quality Grade of Rock Mass	I, II, III, IV, V	Testing
		Connectivity of Discontinuities	%	Site Investigation
		Cohesion of Discontinuities	KPa	Testing
		Internal Friction Angle of Discontinuities	°	Testing
		Rock Mass Cohesion	KPa	Testing
		Rock Mass Internal Friction Angle	°	Testing
		Included Angle of Structural Planes	°	Site Investigation
		Dip Angle of Counter-Inclined Rock Strata	°	Site Investigation
		Dip Angle of Inclined Rock Strata	°	Site Investigation
		Degree of Weathering	Unweathered, Slightly Weathered, Moderately Weathered, Highly Weathered, Completely Weathered	Site Investigation
		Distribution of Isolated or Suspended Rock Blocks		Site Investigation
		Lithology	Hard, Medium Hard, Soft, Moderately Soft, Very Soft	Testing
	Accidental Factors	Earthquake	I, II, III, IV, V	Monitoring
	Anchoring Structure Parameters	Strength Reserve Coefficient	%	Design
		Visual Condition	I, II, III, IV, V	Site Investigation
		Grouting Saturation	%	Testing
		Corrosion Degree of Anchor Cables	%	Testing
		Prestress Loss Rate	%	Testing

.

Based on the hierarchical analytical structure described in Section 2.1, the stability evaluation index system for the rock slope–anchoring system was constructed, as illustrated in Figure 2.

Since rock slopes exhibit diverse deformation and failure modes, the corresponding evaluation indicators vary. Figure 3 illustrates the hierarchical structure of the stability evaluation index system tailored to different failure mechanisms of rock slope–anchoring systems.

3. Grading Criteria

3.1. Classification of Indicators

The classification methods for qualitative and quantitative indicators differ; therefore, the indicators are first categorized into qualitative and quantitative types. The classification results are shown in

Table 7. Summary of indicator classification.

Type	Indicator
Qualitative Indicators	Drainage System	Degree of Weathering	Lithology
	Rock Mass Quality Grade	Distribution of Isolated or Overhanging Rock Blocks	Visual Condition
Quantitative Indicators	Rock Slope Height	Internal Friction Angle of Rock	Groundwater Development
	Rock Slope Angle	Interfacial Angle of Wedge-shaped Discontinuities	Seismic Activity
	Joint Connectivity Rate	Dip Angle of Adversely Dipping Discontinuities	Strength Reserve Coefficient
	Joint Cohesion	Dip Angle of Favorably Dipping Discontinuities	Grouting Saturation
	Joint Internal Friction Angle	Maximum Daily Historical Rainfall	Corrosion Rate
	Rock Cohesion	Rainfall Duration	Prestress Loss Rate

.

3.2. Determination of Classification Criteria

3.2.1. Determination of Classification Criteria for Qualitative Indicators

Qualitative indicators were classified based on descriptive criteria, as shown in

Table 8. Summary of classification criteria for qualitative indicators.

Indicator	Grading Standard
Drainage System	Excellent, Good, Moderate, Poor, Very Poor
Basic Quality Grade of Rock Mass	I, II, III, IV, V
Degree of Weathering	Unweathered, Slightly Weathered, Moderately Weathered, Strongly Weathered, Completely Weathered
Distribution of Boulders or Overhanging Blocks	No signs of surface loosening; small overhanging blocks (0.01 < volume < 1 m3); some loosened surfaces and small overhanging blocks; several loosened surfaces and small overhanging blocks; potentially detached overhanging blocks (volume > 1 m3)
Lithology	Hard Rock, Moderately Hard Rock, Soft Rock, Rather Soft Rock, Very Soft Rock
Surface Appearance Condition	I, II, III, IV, V

below.

3.2.2. Determination of Grading Standards for Quantitative Indicators

According to the classification of rock slope stability levels specified in the existing standard [61], as shown in

Table 9. Classification table of rock slope stability.

Operating Conditions	Slope Stability Grade
	1	2	3	4	5
Normal Operating Condition	1.30~1.25	1.25~1.20	1.20~1.15	1.15~1.10	1.10~1.05
Unusual Operating Condition I	1.25~1.20	1.20~1.15	1.15~1.10	1.10~1.05
Unusual Operating ConditionII	1.15~1.10	1.10~1.05		1.05~1.00

, rock slopes in hydropower projects are categorized into five levels, with corresponding safety factor requirements for each level. In this study, the evaluation indices are also divided into five levels, as stipulated in the Technical Code for Prestressed Anchoring in Hydraulic Structures (SL-T212-2020) [62]. Technical Specification for Prestressed Anchoring in Hydraulic Engineering, 2007b), the recommended stability safety factors for slopes reinforced with prestressed anchors (rods) should follow the requirements outlined in the Code for Design of Slopes for Hydraulic and Hydroelectric Engineering Projects (SL 386-2007) [61].

In currently adopted slope stability evaluation indicator systems, the classification of continuous indicators is typically based on the equal-width discretisation method, which divides the data into n intervals of equal width in ascending order. In some cases, experts determine classification thresholds based on personal experience. These methods often result in classification intervals that do not correspond clearly to slope stability and heavily rely on subjective judgment without quantitative justification. Furthermore, the equal-width discretisation approach presumes a linear relationship between the indicator and slope safety, which is rarely the case in complex geotechnical systems.

(1) Patterns of Influence of Multiple Indicators on the Stability of Rock Slope–Anchoring Systems

Based on the indicators selected in Section 2, this section employs numerical simulation methods to investigate the influence of five categories of factors—slope geometry, rock mass conditions, hydrometeorological factors, seismic activity, and anchoring parameters—on the stability of the systems. The slope geometry, rock mass conditions, and anchoring parameters are analyzed using the Discrete Element Method (DEM), while hydrometeorological and seismic effects are evaluated via rigid-body limit equilibrium combined with seepage analysis.

The corresponding numerical analysis model is illustrated in Figure 4 below. Discrete element software is employed to analyze the effects of slope geometry, rock mass characteristics, and anchoring parameters on the stability of the slope–anchoring system. The fundamental parameters of the numerical model are presented in

Table 10. Rock mass parameters used in the discrete element numerical model.

Type	Shear Strength Parameters		Bulk Modulus (Pa)	Shear Modulus (Pa)
	C (Pa)	φ (°)
Rock Mass	1 × 106	23.0	2 × 108	1 × 108

and

Table 11. Anchor element parameters used in the discrete element numerical model.

Type	Shear Strength Parameters of Grouted Body		Stiffness of Grouted Body (Pa/m)	Perimeter of Grouted Body (m)	Cross-Sectional Area (m2)	Young’s Modulus	Yield Strength of Anchor Cable (Pa)
	C (Pa)	Φ (°)
Anchor Cable	1.75 × 106	20	1.12 × 107	1.75 × 106	1.81 × 10−4	2 × 108	4000

. The impact of hydrometeorological conditions and seismic loading on the system’s stability is examined in the finite element analysis numerical simulation software, with the corresponding model shown in Figure 5. The fundamental parameters of the Geostudio numerical model are presented in

Table 12. Parameter values of rock mass in Geostudio mumerical model.

Type	Shear Strength Parameters		Saturated Water Content	Saturated Permeability Coefficient (m/d)	Density (kN/m3)
	C (Pa)	φ (°)
Rock Mass	30 × 106	28.0	0.38	0.579	20

and

Table 13. Parameter values of anchor cable elements in Geostudio.

Type	Pullout Strength (kPa)	Tensile Strength (kN)	Bond Length (m)	Shear Strength (kN)	Bond Diameter (m)	Anchor Cable Spacing (m)	Safety Factor for Pullout Resistance	Safety Factor for Tensile Resistance	Safety Factor for Shear Resistance	Anchor Cable Length	Anchor Cable Angle
Anchor Cable	300	2000	13	0	0.383	4	4.5	2	1	25	45°

.

Based on the results of numerical simulations, the strength reduction method was used to calculate the slope stability coefficient. The relationships between the stability coefficient and various indicators were obtained and fitted to construct mathematical relationships between the stability coefficient of the rock slope–anchoring system and multiple indicators. The analysis results are shown in Figure 6. Specifically, Figure 6a–q present the correlation curves, fitted mathematical expressions, and coefficients of determination between the stability coefficient and the following indicators: slope angle, slope height, cohesiveness of structural plane, internal friction angle of structural plane, cohesion of rock mass, internal friction angle of rock mass, connectivity of structural plane, bedding slope dip angle, dip angle of strata in anti-dip slopes, rainfall duration, groundwater development, rainfall intensity, seismic activity, strength reserve coefficient of anchor materials, corrosion rate, prestress loss rate, and grouting saturation.

As shown in Figure 6, most influencing factors exhibit non-linear relationships with the stability of the systems. The stability of the rock slope–anchoring system exhibits varying sensitivity to different indicators across their respective value ranges. For instance, slight variations in the slope angle within the range of 30–35° can markedly influence stability, as illustrated in Figure 6a. Similar pronounced effects are observed for slope heights within 30–50 m (Figure 6b), structural–plane cohesion exceeding 400 kPa (Figure 6c), structural–plane friction angles between 18° and 30° (Figure 6d), rock mass cohesion in the range of 40–400 kPa (Figure 6e), rock mass friction angles between 15° and 30° (Figure 6f), structural–plane connectivity ratios of 10–20% (Figure 6g), bedding dip angles in anti-dip slopes within 80–90° (Figure 6i), rainfall durations of 1–2 days (Figure 6j), groundwater-level-to-slope-height ratios of 0.3–0.5 (Figure 6k), anchor material strength reserve factors of 1.08–1.6 (Figure 6n), and grouting saturation levels of 10–50% (Figure 6q). Minor changes in these parameter ranges can substantially affect the overall stability of the rock slope–anchoring system. Therefore, the use of equal-interval discretisation to determine grading criteria is inappropriate.

(2) Determination of Grading Criteria

This section proposes a grading criterion determination method based on the influence patterns of stability factors in the systems. The method adopts the stability coefficient of the rock slope–anchoring system as a key quantitative indicator for stability evaluation. It is based on the empirical relationships between each indicator and the stability coefficient, which are expressed through fitted mathematical functions. For example, in the case of the strength reserve coefficient of the anchoring structure, numerical simulations are used to obtain the stability coefficients of the rock slope–anchoring system under varying conditions of this parameter. The data are then fitted using the least squares method to derive a mathematical model describing the relationship between the anchoring structure’s strength reserve coefficient and the system’s stability coefficient. The influence curves and corresponding mathematical models for representative indicators are shown in Figure 7. Based on available data samples, literature, and relevant standards, the range of each indicator is defined. This range, along with the fitted influence curves, is used to determine the corresponding range of stability coefficients. The stability coefficient range is then divided into equal-width intervals to define grading levels. Finally, the grading thresholds for each indicator are obtained by inversely solving the influence curves based on the interval endpoints of the stability coefficient.

This method not only determines the classification standards for continuous indicators but also establishes a one-to-one correspondence between indicator intervals and the stability grades of the rock slope–anchoring system. The implementation steps are as follows:

(1) Based on discrete element or finite element analysis methods, and taking into account the interaction between the anchoring structure and the rock slope, the stability coefficients of the rock slope–anchoring system under different levels of each indicator are analyzed. Figure 8 shows the stability coefficients obtained under various slope angles using the discrete element numerical simulation method.

(2) The data obtained in step (1) are fitted using methods such as least squares fitting, polynomial fitting, and orthogonal fitting, to derive the influence laws of each continuous indicator on the stability coefficient of the rock slope–anchoring system, as illustrated in Figure 9.

(3) Determination of Indicator Grading Intervals

The first step is to define the range of each indicator, followed by determining the number of grading intervals based on practical evaluation requirements. The range of indicators is established by referring to existing engineering cases, technical specifications, and relevant literature. For example, a review of existing studies shows that the height of most engineering slopes typically ranges from 30 to 100 metres, thus setting the height range of the rock slope–anchoring system to [30, 100]. Let [xa, xb] represent the range of the continuous indicator. Based on this range and the fitted influence curve derived in Step (2), the corresponding stability coefficient range [ya, yb] is defined. As shown in Figure 10, the range of stability coefficients is further divided into five intervals, each corresponding to a specific stability level of the rock slope anchoring system. As shown in Figure 10I–V below: This subsection takes the groundwater development indicator as an example to demonstrate the grading process, as shown in Figure 10.

As illustrated in Figure 10, the indicator research range [xa, xb] is first established, followed by determining the corresponding stability coefficient range [ya, yb] for the rock slope–anchoring system. The number of intervals within [ya, yb] is then decided based on analytical requirements. In this study, five levels are defined according to existing guidelines [61]. Accordingly, the stability coefficient range is evenly divided into five grading intervals: [ya, y2], [y2, y3], [y3, y4], [y4, y5], and [y5, yb], which correspond to stability levels I through V, as shown in Figure 11.

(4) Based on the stability coefficient grading intervals established in Step (3) and the mathematical models derived in Step (2), the threshold values (interval endpoints) for each indicator are calculated via inverse analysis. The resulting indicator grading standards and corresponding intervals are illustrated in Figure 11.

As shown in Figure 11, the threshold values (interval endpoints) of the indicator—namely, x2, x3, x4, and x5—are obtained through inverse calculation using the fitted correlation curve and the stability coefficient intervals defined in Step (3). Together with the boundary values xa and xb of the indicator’s full range, the indicator range is divided into five intervals corresponding to the five stability grades of the rock slope–anchoring system, as denoted by the red Roman numerals in Figure 11.

(3) Classification of Continuous Indicator Intervals

Based on the findings of the influence patterns of each indicator on the stability of the rock slope–anchoring system and the proposed grading standard determination method, the classification intervals for the selected continuous indicators are established.

(1) Geometric Conditions of the Slope

The slope height in hydropower engineering generally ranges from 0 to 100 m, while the slope angle typically falls between 0°and 80°. Based on the correlation curves and mathematical models established for the relationship between slope height, slope angle, and the stability coefficient of the rock slope–anchoring system (as illustrated in Figure 6a,b), the classification intervals for continuous variables are determined using the grading method proposed in this section. The analysis results are summarized in

Table 14. Classification intervals and corresponding safety grades for geometric indicators of the rock slope–anchoring system.

Indicator	Classification Interval	Stability Grade	Decision Attribute
Elevation of the Rock Slope–Anchoring System (m)	[0~3]	V	(Safe)
	[3~7]	IV	(Basically Safe)
	[7~12]	III	(Potential Risk)
	[12~21]	II	(Unsafe)
	[21~100]	I	(Extremely Unsafe)
Slope Angle of the Rock Slope–Anchoring System(°)	[30~31]	V	(Safe)
	[31~32.2]	IV	(Basically Safe)
	[32.2~34]	III	(Potential Risk)
	[34~37]	II	(Unsafe)
	[37~80]	I	(Extremely Unsafe)

.

(2) Hydrometeorological Factors

According to statistical analysis by Wu Renxi [63], there is a certain time lag between rainfall and landslide occurrence. Approximately 45.45% of landslides are triggered by one-day rainfall events, and rainfall within the preceding three days accounts for 90% of all landslide incidents. Therefore, this study defines the rainfall duration range as 0 to 5 days. In addition, landslide occurrence is significantly correlated with rainfall intensity. Based on the classification criteria issued by the China Meteorological Administration (as shown in

Table 15. Classification of rainfall intensity.

Rainfall Intensity (mm/d)	Q < 10	10 < Q < 25	25 < Q < 50	50 < Q < 100	100 < Q
Rainfall Intensity Classification	Light Rain	Moderate Rain	Heavy Rain	Rainstorm	Severe Rainstorm

), the selected range of rainfall intensity in this study is [0–150 mm/day]. The relationship curves between rainfall intensity and duration and the stability coefficient of the rock slope–anchoring system are illustrated in Figure 6i,j.

In addition to rainfall, the development of groundwater is also a critical factor influencing the stability of the rock slope–anchoring system. This section investigates the ratio of groundwater head to slope height as a quantitative indicator of groundwater development, and its relationship with the system’s stability coefficient. The corresponding relationship curve is shown in Figure 6k. Based on the classification methodology for continuous variables proposed in this section, the grading intervals for continuous indicators are determined. The analysis results are summarized in

Table 16. Classification intervals and corresponding safety levels of hydrometeorological and seismic indicators for rock slope–anchoring systems.

Indicator	Classification Interval	Stability Grade	Decision Attribute
Rainfall Duration(d)	[0–0.4]	V	(Safe)
	[0.4~0.9]	IV	(Basically Safe)
	[0.9~1.6]	III	(Potential Risk)
	[1.6~2.7]	II	(Unsafe)
	[2.7~5]	I	(Extremely Unsafe)
Rainfall Intensity (mm/d)	[[0–30]	V	(Safe)
	[30~60]	IV	(Basically Safe)
	[60~90]	III	(Potential Risk)
	[90~120]	II	(Unsafe)
	[120~150]	I	(Extremely Unsafe)
Groundwater Development (Ratio of Groundwater Head to Slope Elevation)	[0–0.06]	V	(Safe)
	[0.06~0.14]	IV	(Basically Safe)
	[0.14~0.24]	III	(Potential Risk)
	[0.24~0.41]	II	(Unsafe)
	[0.41~0.9]	I	(Extremely Unsafe)

.

(3) Seismic Effects

This study employs peak ground acceleration (PGA) as the metric for seismic intensity, offering greater precision than the use of seismic intensity scales alone. According to the code [61], the comprehensive horizontal seismic acceleration coefficient is used to assess the impact of seismic activity on the stability of rock slopes, as shown in

Table 17. Correspondence between seismic intensity and horizontal seismic coefficient.

Basic Seismic Intensity	7 Degrees		8 Degrees		9 Degrees
Peak Ground Acceleration	0.1 g	0.15 g	0.2 g	0.3 g	0.4 g
Comprehensive Horizontal Seismic Coefficient	0.025	0.038	0.05	0.075	0.1

. The relationship curve between seismic activity and the stability coefficient of the rock slope–anchoring system, derived in Section 3.2, is illustrated in Figure 6m.

As shown in Figure 6m, the impact of peak ground acceleration on the stability coefficient of the rock slope–anchoring system exhibits a linear relationship. Therefore, the equal-interval method is appropriate for determining the classification standards of PGA values. Based on the existing code, the PGA value range is defined as [0.1 g–0.4 g]. The equal-interval method is then applied to determine the classification intervals for this continuous variable. The results are summarized in

Table 18. Summary of classification intervals and corresponding safety levels for peak ground acceleration in rock slope–anchoring systems.

Indicator	Classification Interval	Stability Grade	Decision Attribute
Peak Ground Acceleration (g)	[0–0.08]	V	(Safe)
	[0.08~0.16]	IV	(Basically Safe)
	[0.16~0.24]	III	(Potential Risk)
	[0.24~0.32]	II	(Unsafe)
	[0.32~0.4]	I	(Extremely Unsafe)

.

(4) Rock Mass Conditions

According to the findings in Section 3.2, the cohesion and internal friction angle of structural planes, the cohesion and internal friction angle of the rock mass, structural plane connectivity, the dip angle of bedding planes in anti-dip slopes, the dip angle of bedding planes in dip slopes, and the intersection angle of wedge structural planes all significantly influence the stability of the systems. Based on the Handbook of Rock Mechanics Parameters [64], the value ranges of each parameter are summarized in

Table 19. Value ranges of rock mass condition indicators.

Indicator	Value Range
Cohesion of Discontinuities	10 kPa~600 kPa
Internal Friction Angle of Discontinuities	10°~60°
Cohesion of Rock Mass	0 kPa~4000 kPa
Internal Friction Angle of Rock Mass	10°~70°
Connectivity Rate of Discontinuities	10%~100%
Dip Angle of Rock Layers in Dip Slope	0°~90°
Dip Angle of Rock Layers in Anti-dip Slope	0°~90°
Intersection Angle of Discontinuities in Wedge Body	20°~150°

. The results in Section 3.2 indicate the influence patterns of rock mass conditions on the stability coefficient of the systems, as illustrated in Figure 6c–i.

Based on the classification method for continuous variables described in this section, the grading intervals for rock mass condition indicators are determined. The classification results are summarized in

Table 20. Classification ranges and corresponding stability levels of rock mass condition indicators for the rock slope–anchoring system.

Indicator	Classification Interval	Stability Grade	Decision Attribute
Joint cohesion (kPa)	[0–322.41]	I	(Extremely Unsafe)
	[322.41~425.77]	II	(Unsafe)
	[425.77~498.01]	III	(Potential Risk)
	[498.01~554.1]	IV	(Basically Safe)
	[554.1~600]	V	(Safe)
Joint internal friction angle (°)	[10–14.4]	I	(Extremely Unsafe)
	[14.4~20]	II	(Unsafe)
	[20~27.3]	III	(Potential Risk)
	[27.3~38.2]	IV	(Basically Safe)
	[38.2~60]	V	(Safe)
Joint connectivity rate (%)	[10–11.4]	V	(Safe)
	[11.4~13.3]	IV	(Basically Safe)
	[13.3~15.9]	III	(Potential Risk)
	[15.9~20.3]	II	(Unsafe)
	[20.3~100]	I	(Extremely Unsafe)
Dip angle of bedding planes in dip direction (°)	[69.56–88] * [88–90]	V	(Safe)
	[60.35–69.56] *	IV	(Basically Safe)
	[51.87~60.35] * [0–3.6]	III	(Potential Risk)
	[42.65~51.87] * [3.6–9.34]	II	(Unsafe)
	[24.68–42.65] * [9.34–24.68]	I	(Extremely Unsafe)
Dip angle of bedding planes in opposite direction (°)	[87.08–90] *	V	(Safe)
	[83.68~87.08] *	IV	(Basically Safe)
	[79.49–83.68] *	III	(Potential Risk)
	[73.75–79.49] * [6.05–19] *	II	(Unsafe)
	[19–35.13]	I	(Extremely Unsafe)
Rock mass cohesion (kPa)	[0–90.6] *	I	(Extremely Unsafe)
	[90.6~207.4] *	II	(Unsafe)
	[207.4–372] *	III	(Potential Risk)
	[372–653.4] *	IV	(Basically Safe)
	[653.4–4000] *	V	(Safe)
Rock mass internal friction angle (°)	[10–14.74] *	I	(Extremely Unsafe)
	[14.74~20.7] *	II	(Unsafe)
	[20.7–28.75] *	III	(Potential Risk)
	[28.75–41.19] *	IV	(Basically Safe)
	[41.19–70] *	V	(Safe)
Included angle of wedge-shaped structural planes (°)	[20–28.22]	V	(Safe)
	[28.22~38.7]	IV	(Basically Safe)
	[38.7–53.18]	III	(Potential Risk)
	[53.18–76.77]	II	(Unsafe)
	[76.77–150]	I	(Extremely Unsafe)

Indicators marked with * in the table represent “the larger, the better”-type data, while those without * represent “the smaller, the better”-type data.

. Indicators marked with * in the table represent “the larger, the better”-type data, while those without * represent “the smaller, the better”-type data. For the “larger-the-better” indicators, a greater numerical value indicates a more stable state of the system, whereas for the “smaller-the-better” indicators, a smaller numerical value reflects a more stable state of the system.

(5) Anchorage Structure Parameters

According to the Ministry of Water Resources of the People’s Republic of China [61] and its detailed provisions, the strength reserve factor of prestressed anchorage structures under design tension load is recommended to be between 1.54 and 1.67. Both domestic and international anchorage engineering projects commonly adopt 60% to 65% of the standard tensile strength of the anchorage material as the allowable design stress. In this study, the selected range of the strength reserve factor is 1–2.

Based on the findings in Section 3.2, when the grouting saturation is 10%, the stability coefficient of the rock slope–anchorage system is 1.01. Therefore, the grouting saturation range selected for this study is 10–100%. According to the mathematical model of the influence of corrosion rate on the stability coefficient of the rock slope–anchorage system shown in Figure 1 of Section 3.2, the safety coefficient is 1.0 when the corrosion rate reaches 15%. Thus, the corrosion rate of anchor cables in this study is set within the range of 0–15%.

Based on the findings in Section 3.2, the influence curves and mathematical relationship models of anchorage structure parameter indicators on the stability coefficient of the rock slope–anchorage system are shown in Figure 6n–q. Applying the continuous indicator classification method proposed in this section, the classification standards for anchorage structure parameters are summarized in

Table 21. Classification intervals and corresponding safety levels of anchorage structure parameter indicators.

Indicator	Classification Interval	Stability Grade	Decision Attribute
Strength Reserve Factor	[1–1.15] *	I	(Extremely Unsafe)
	[1.15~1.35] *	II	(Unsafe)
	[1.35–1.61] *	III	(Potential Risk)
	[1.61–2.02] *	IV	(Basically Safe)
	[2.02–3] *	V	(Safe)
Grouting Saturation (%)	[10–20] *	I	(Extremely Unsafe)
	[20~30] *	II	(Unsafe)
	[30–42] *	III	(Potential Risk)
	[42–56] *	IV	(Basically Safe)
	[56–100] *	V	(Safe)
Corrosion Rate (%)	[0–5.8]	V	(Safe)
	[5.8~9.2]	IV	(Basically Safe)
	[9.2–11.6]	III	(Potential Risk)
	[11.6–13.5]	II	(Unsafe)
	[13.5–15]	I	(Extremely Unsafe)
Prestress Loss Rate (%)	[[0–20]	V	(Safe)
	[20~40]	IV	(Basically Safe)
	[[40–60]	III	(Potential Risk)
	[[60–80]	II	(Unsafe)
	[[80–100]	I	(Extremely Unsafe)

Indicators marked with * in the table represent “the larger, the better”-type data, while those without * represent “the smaller, the better”-type data.

below.

A summary of the grading criteria for the stability evaluation index system of the rock slope–anchoring system is presented in

Table 22. Summary of grading criteria for evaluation indicators.

Indicator	Grading Criteria	Indicator	Grading Criteria	Indicator	Grading Criteria
Slope Height A1 (m)	[0~3]	Included angle of wedge-shaped structural planes B7 (°)	[20~28.22]	Rainfall Duration C3 (d)	[0~0.4]
	[3~7]		[28.22~38.7]		[0.4~0.9]
	[7~12]		[38.7–53.18]		[0.9~1.6]
	[12~21]		[53.18–76.77]		[1.6~2.7]
	[21~100]		[76.77–150]		[2.7~5]
Slope Angle A2 (°)	[30~31]	Dip angle of bedding planes in opposite direction B8 (°)	[87.08~90]	Groundwater Development C4(Ratio of Groundwater Head to Slope Elevation)	[0~0.06]
	[31~32.2]		[83.68~87.08] *		[0.06~0.14]
	[32.2~34]		[79.49~83.68] *		[0.14~0.24]
	[34~37]		[73.75~79.49] * [6.05~19] * [19~35.13]		[0.24~0.41]
	[37~80]		[0~6.05] * [35.13~58.14]		[0.41~0.9]
Basic Quality Grade of Rock Mass B1	V	Dip angle of bedding planes in dip direction B9 (°)	[69.56~88] * [88~90]	Seismic Activity D1 (peak ground acceleration)	[0~0.08 g]
	IV		[60.35~69.56] *		[0.08 g~0.16 g]
	III		[51.87~60.35] * [0~3.6]		[0.16 g~0.24 g]
	II		[42.65~51.87] * [3.6~9.34]		[0.24 g~0.32 g]
	I		[24.68~42.65] * [9.34~24.68]		[0.32 g~0.4 g]
Discontinuity Connectivity Rate B2 (%)	[10~11.4]	Degree of Weathering B10	Unweathered	Strength Reserve Factor E1	[1~1.15] *
	[11.4~13.3]		Slightly Weathered		[1.15~1.35] *
	[13.3~15.9]		Moderately Weathered		[1.35~1.61] *
	[15.9~20.3]		Strongly Weathered		[1.61~2.02] *
	[20.3~100]		Completely Weathered		[2.02~3] *
Cohesion of Discontinuities B3 (KPa)	[0–322.41]	Distribution of Boulders or Overhanging Blocks B11	No signs of surface loosening	Surface Appearance Condition E2	V
	[322.41~425.77]		Small overhanging blocks (0.01 < volume < 1 m3)		IV
	[425.77~498.01]		Some loosened surfaces and small overhanging blocks		III
	[498.01~554.1]		Several loosened surfaces and small overhanging blocks		II
	[554.1~600]		Potentially detached overhanging blocks (volume > 1 m3)		I
Internal Friction Angle of Discontinuities B4 (°)	[10~14.4]	Lithology B12	Hard Rock	Grouting Saturation E3 (%)	[10~20] *
	[14.4~20]		Moderately Hard Rock		[20~30] *
	[20~27.3]		Soft Rock		[30~42] *
	[27.3~38.2]		Rather Soft Rock		[42~56] *
	[38.2~60]		Very Soft Rock		[56~100] *
Rock Mass Cohesion B5 (KPa)	[0~90.6] *	Drainage System C1	Excellent	Corrosion Rate E4 (%)	[0~5.8]
	[90.6~207.4] *		Good		[5.8~9.2]
	[207.4~372] *		Moderate		[9.2~11.6]
	[372~653.4] *		Poor		[11.6~13.5]
	[653.4~4000] *		Very Poor		[13.5~15]
Internal Friction Angle of Rock Mass B6 (°)	[10~14.74] *	Rainfall Intensity C2 (mm/d)	[0~30]	Prestress Loss Rate E5 (%)	[0~20]
	[14.74~20.7] *		[30~60]		[20~40]
	[20.7~28.75] *		[60~90]		[40~60]
	[28.75~41.19] *		[90~120]		[60~80]
	[41.19~70] *		[120~150]		[80~100]

Indicators marked with * in the table represent “the larger, the better”-type data, while those without * represent “the smaller, the better”-type data.

.

4. Validation of Classification Criteria

Section 3 of this study proposes a method for determining classification criteria based on the influence patterns of the stability coefficient of the rock slope–anchoring system, and the classification thresholds of the evaluation indicators are established accordingly. In this section, the reasonableness of the classification criteria is validated using the K-nearest neighbors (KNN) algorithm.

4.1. Basic Principle of the KNN Algorithm

KNN classifies samples by measuring the distances between different feature values. The fundamental principle is that if the majority of the k nearest neighbors of a sample in the feature space belong to a certain category, the sample is assigned to the same category. In the KNN algorithm, all selected neighbors are objects with known and correctly labeled classifications. The classification decision is made solely based on the category of the nearest neighbor(s). The basic concept is illustrated in Figure 12.

As shown in Figure 12, different colors represent different categories. When K = 3, since two of the three nearest neighbor points are blue triangles (i.e., 2/3 of the majority cases), the green quadrilateral is classified into the same category as the blue triangle.

4.2. Evaluation of the KNN Model

The performance of a classifier can be evaluated based on the classification loss of its predictions. Generally, a better classifier yields a smaller loss value. The classification loss function quantifies the predictive accuracy of the model—higher accuracy indicates better model performance and a stronger ability of the samples to represent intrinsic data characteristics. Based on this principle, the K-Nearest Neighbors (KNN) algorithm can be used to validate the rationality of different discretization methods for defining the interval boundaries of evaluation indices. By calculating the classification accuracy of the KNN algorithm, the validity of the interval partitioning can be assessed, thereby verifying the rationality of the grading standard determination method.

4.3. Implementation Procedure of the Rationality Test

(1) Identify the indicators to be classified and their corresponding value ranges.

(2) Apply different classification methods to the same indicators and value ranges to define grading criteria and divide classification intervals.

(3) Based on the defined intervals, discretize the continuous indicators in the sample data to construct a fully discrete dataset.

(4) Apply the KNN algorithm to classify the discrete dataset and compute the classification accuracy, which is then used to assess model performance.

(5) A better-performing KNN model (i.e., higher classification accuracy) indicates that the corresponding classification method more effectively captures the intrinsic characteristics of the data. Lower information loss after discretization leads to higher classification precision.

4.4. Comparative Analysis of Grading Standard Determination Methods

A comparative analysis is conducted between the commonly used equal-width classification method and the classification method proposed in this study, which is based on the influencing patterns of the stability of the systems, in order to validate the feasibility and rationality of the proposed approach. Nine continuous indicators were selected for classification: slope height, slope angle, cohesion, internal friction angle, daily rainfall, strength reserve coefficient, grouting saturation, anchor corrosion rate, and prestress loss rate. These were used to establish classification criteria and assess the reasonableness of the methods. Due to the lack of sufficient empirical data on anchoring parameters and the stability of the systems, this study employed numerical simulation to generate sample data. The numerical model is shown in Figure 4 and Figure 5, and the parameters are listed in Table 10, Table 11, Table 12 and Table 13. By varying the values of the selected nine indicators, 40 sets of sample data were generated. Based on the generated sample data, classification criteria were determined using both methods. Classification intervals were constructed, and corresponding decision information tables were generated. A KNN-based computation program was implemented. A feature matrix (40 × 9) was constructed from the nine indicator variables for the 40 samples in the decision information table, and a label vector (n = 40) was created from the corresponding stability states. The hyperparameter K was selected via grid search with cross-validation, Euclidean distance was adopted as the distance metric, and the resulting model was then used to perform KNN classification.

The accuracy of the KNN classifier was calculated, where higher accuracy indicates that the decision information table generated by the classification method is more precise, and the resulting classification intervals are more accurate and reasonable. The calculation formula for classification accuracy is shown in Equation (1). Based on the sample data derived from different classification intervals, the classification accuracy using the KNN algorithm was computed. The results of the calculation are presented in

Table 23. Summary of KNN classification accuracy for different grading methods.

Grading Standard Determination Method	Accuracy
Grading Standard Determination Method Based on Stability Influence Patterns	75.3%
Equal Interval Grading Standard Determination Method	65.2%
Difference	10.1%

.

$$Accuracy={\displaystyle \frac{TP+TN}{TP+FN+FP+TN}}$$

(1)

where TP (True Positive) refers to the number of samples that are actually positive and predicted as positive, TN (True Negative) denotes the number of samples that are actually negative and predicted as negative, FP (False Positive) represents the number of samples that are actually negative but predicted as positive, and FN (False Negative) indicates the number of samples that are actually positive but predicted as negative.

As shown in Table 23, the classification accuracy of the grading standard determined by the method based on the stability coefficient influence patterns is approximately 10% higher than that of the equal interval grading method. This indicates that the proposed grading standard determination method, which is based on the influence patterns of the stability coefficient for the rock slope–anchoring system, more effectively captures the intrinsic characteristics of the data. It minimizes the loss of feature information caused by data discretization and thereby improves classification accuracy. This method outperforms the equal interval grading approach. The grading standard determination method for continuous variables, based on the stability coefficient influence patterns of the rock slope–anchoring system, is both reasonable and feasible.

5. Conclusions

(1) The proposed evaluation index system fully considers the deformation and failure modes of rock slopes, the factors influencing stability, and the safety-related parameters of anchoring structures. It provides a systematic and applicable evaluation framework for assessing the stability of the systems under complex geological and engineering conditions and offers a structured and quantitative basis for decision making in slope–anchoring engineering.

(2) The system thoroughly accounts for the multifactorial influence mechanisms affecting the stability of the rock slope–anchoring system.

(3) Compared with traditional equal-interval methods for determining grading standards, the proposed method yields more accurate classification thresholds. It better reflects the intrinsic characteristics of the data and minimizes the loss of feature information after discretization, thus achieving higher classification accuracy.

6. Discussion

The stability of rock slope–anchoring systems is a critical issue in geotechnical engineering, particularly in complex geological environments where the interaction between rock mass structure and anchoring support is strongly coupled. This study proposes a stability evaluation index system for such slopes developed through indicator screening, analytic hierarchy process (AHP) structuring, classification standard formulation, and comparative analysis.

The main innovations of this study are reflected in the following aspects.

First, the proposed index system accounts for the deformation and failure modes of rock slopes, key stability-influencing factors, and anchoring structure safety. It overcomes the limitations of conventional evaluation methods that rely heavily on single indicators or expert subjectivity. By incorporating multidimensional factors such as geological conditions, structural features, environmental influences, and anchoring parameters, the system provides a more comprehensive and realistic representation of the coupled effects that govern the stability of the systems. Second, the system employs the analytic hierarchy process (AHP) to logically decompose the indicator hierarchy and establishes grading standards based on the quantitative influence patterns of multiple factors on slope stability. Compared to traditional equal-interval or experience-based methods, the proposed grading approach better captures the intrinsic characteristics of the data, thereby significantly improving the accuracy and sensitivity of the classification results.

Although this study has made notable progress in both theoretical and practical aspects, certain limitations remain. Due to the difficulty of obtaining extensive real-world engineering data, numerical simulation was primarily used to analyze the influence of various indicators on stability. While simulation offers high controllability and repeatability, it cannot fully capture the stochastic nature and boundary uncertainties inherent in real projects. As a result, some of the grading thresholds may deviate from actual conditions and require further refinement through large-scale empirical data validation. In addition, the indicators exhibit pronounced local sensitivity to random deviations in the initial data, reflecting the high responsiveness of the rock slope–anchoring system to variations in geotechnical parameters such as cohesion, internal friction angle, and stress state. Although this sensitivity may amplify measurement errors, it enables the KNN method to capture subtle yet meaningful variations in rock mass properties. This characteristic should not be regarded as a deficiency; rather, it reflects the model’s reliance on high-precision input data and underscores the importance of improving data acquisition accuracy.

Future work may focus on (1) enriching the database of real-world engineering cases to strengthen empirical validation; (2) incorporating machine learning algorithms to optimize and refine the indicator set; (3) conducting extensive application studies under diverse geological conditions to enhance the universality, adaptability, and robustness of the evaluation system; (4) conducting a systematic sensitivity analysis and uncertainty quantification to further constrain the impact of data perturbations on the evaluation results and explore the feasibility of incorporating noise-reduction strategies into the KNN framework; and (5) integrating the KNN method, physical model experiments, and field monitoring to develop a comprehensive evaluation index system for the stability of rock slope–anchoring systems.