Hoek-Brown Criterion Research Articles

Theoretical Optimization Method of Tunnel Lining in Fractured Rock Mass Based on Rock Classification and Hoek–Brown Criterion

Theoretical Optimization Method of Tunnel Lining in Fractured Rock Mass Based on Rock Classification and Hoek–Brown Criterion

Pseudo-dynamic stability analysis of 3D rock slopes considering tensile strength-modified Hoek–Brown failure criterion: Seismic UBLA implementations

The Hoek-Brown (HB) criterion has found extensive application in the stability analysis of intact or jointed rock slopes, in both two-dimensional (2D) and three-dimensional (3D) contexts. In the early days, the assessment of Hoek-Brown slopes rarely incorporated tensile strength until, recently, limited studies examined the tension-controlled toppling failure at the sliding head. Nevertheless, the adapted tensile strength, which is extrapolated from the HB envelope the tensile regime, has been found overestimate experimental results. Thus, for more realistically representing the influence of rock tensile strength, the present study introduces a novel procedure for assessing the seismic stability of 3D HB slopes. In this study, the tensile strength-modified HB envelopes considering cut-off effects are derived in the principal stress space and Mohr space. To assess the slope stability, the obtained HB envelope is employed in a multi-cone mechanism based on upper bound limit analysis (UBLA). Specifically, the counter of this mechanism is determined by 10 optimized dependent rupture angles, highly considering the nonlinearity of the Hoek-Brown envelope compared to the traditional linearization strategies. As an improved seismic stability analysis, the seismic load is characterized by a modified pseudo-dynamic method. Analytical expressions for two stability indicators, i.e., the stability number and safety factor, are derived. Comparisons with numerical simulations demonstrate the rationality of the proposed procedure. Parametric analysis indicates the significant effect of the tension cut-off on the HB rock slope stability and the contour of the sliding surface.

An exact implementation of the generalized Zhang-Zhu criterion for elasto-plastic finite element calculations

The generalized Zhang-Zhu (GZZ) strength criterion was proposed as an extension to the Hoek-Brown criterion and the Mogi criterion. The introduction to mean normal stress results in a non-smooth and non-convex yield surface, which presents a challenge for updating plastic stress. Current research primarily focuses on modified smooth GZZ criteria or approximate solutions, which inevitably lead to increased computational costs or inaccuracies. In this paper, an accurate stress updating algorithm is proposed based on the original GZZ criterion. The algorithm operates entirely in the principal stress space, where numerical singularities at the intersection of yield surfaces are avoided by defining four different types of stress updating. This approach simplifies the GZZ criterion compared to its formulation in general stress space. The return mapping is employed to compute the updated stress and consistent stiffness matrix, facilitating calculations using both finite element implicit and explicit algorithms. Finally, the accuracy of the proposed method is validated using rock true triaxial test data and semi-analytical solutions for stresses and displacement around a circular opening under the GZZ criterion.

Dynamic statistical damage constitutive model based on the Hoek–Brown criterion at high strain rates

Dynamic statistical damage constitutive model based on the Hoek–Brown criterion at high strain rates

A comparative study of three types of strength criteria for rocks

Currently, the unified strength criterion (USC), the three-dimensional Hoek-Brown (3D H-B) criterion and the generalized unified strength theory (GUST) are the three types of typical rock strength criteria. In this paper, a comparative study of the three types of strength criteria is performed for rocks. Based on the nonlinear characteristics of rock strength on meridian and deviatoric planes, the USC can predict rock strength under triaxial stress state. The USC is composed of two failure functions on meridian and deviatoric planes. The failure surface of the USC in principal stress space satisfies smoothness and convexity. The predicted strength for five types of rock under the true triaxial tests were compared among the USC, the GUST, and the 3D H-B criterion. The results indicate that the USC can effectively reflect the influence of the intermediate principal stress on rock strength and accurately predict rock strength under both triaxial tension (σ1=σ2>σ3) and triaxial compression (σ1>σ2=σ3). Additionally, the conventional triaxial tests were conducted on other eight types of rock to measure the strength on meridian plane. The predicted strengths for the eight types of rock on meridian plane were compared between the USC and the original H-B, which suggests that the USC is suitable for various types of rock and provides higher accuracy.

3D seismic bearing capacity of rectangular foundations near rock slopes using upper bound method

The analysis of foundation's bearing capacity is generally conducted under the assumption of plane strain. However, the damage of rectangular foundation usually presents obvious three-dimensional (3D) effect. This article considers this 3D effect, which, for the first time, provides a theoretical framework for assessing the bearing capacity of rectangular foundations placed on rock slopes under seismic condition. In order to apply the kinematic method of limit analysis, a new 3D kinematically admissible collapse mechanism is first constructed. Owing to the point-to-point discretized technique, this 3D mechanism can avoid complex surface integration and effectively reduce spatial coordinate iterative calculations while maintaining high accuracy. The pseudo-static method is adopted to simulate the effects of earthquakes. Generalized multi‑tangential technique is employed to derive the Mohr-Coulomb constants from Hoek-Brown criterion. The comparison between the present results and results of previous literatures proves the validity of the theoretical framework in this paper. Shape factor and reduction factor are introduced in the parametric study, showing that the 3D effect is more obvious when the rectangular foundation aspect ratio is reduced. As the rectangular foundation location moves away from the slope, its bearing capacity gradually converges towards that of the horizontal foundation. The critical collapse mechanism is also explored, demonstrating that the collapse extent expands with increasing internal friction angle of the rock.

Estimation of parameters for 3D geomechanical modeling from triaxial test results

Triaxial testing serves as a fundamental method for evaluating the elastic and strength properties of rocks, crucial for developing accurate 3D geomechanical models. This paper presents a novel method for determining strength parameters by incorporating the dependence of uniaxial compressive strength (UCS) on P-wave velocity into the Hoek-Brown criterion. Additionally, a new approach is introduced to process triaxial test data efficiently using Python libraries such as SciPy, NumPy, Matplotlib, and Pandas. Furthermore, the paper addresses challenges in determining elastic parameters through triaxial testing. A Python script is developed to automate the calculation of elastic modulus and Poisson's ratio, overcoming subjectivity in selecting the linear portion of stress-strain curves. The script optimally identifies the linear region by minimizing the fit error with appropriate constraints, ensuring a more objective and standardized approach. The proposed methodologies are demonstrated using limestone specimens from Central Asian gas fields. These innovations offer faster, more reliable results, reducing error and enhancing the comparability of analyses in geomechanics, with potential applications across various geological settings.

Modeling Brittle-to-Ductile Transitions in Rock Masses: Integrating the Geological Strength Index with the Hoek–Brown Criterion

Many studies focus on brittle–ductile transition stress in intact rocks; however, in real life, we deal with rock mass which contains many discontinuities. To fill this gap, this research focuses on the brittle–ductile transition stress of rock mass by considering the influence of different Geological Strength Index (GSI) values on the brittle–ductile transition stress of rock mass. In other words, the Hoek–Brown failure criteria for rock mass were reformulated mathematically including the ductility parameter (d), which is defined as the ratio of differential stress to minor stress. Then, the results were analyzed and plotted between σ3*σc and GSI, considering different (d) and Hoek–Brown material constant (mi) values. The brittle–ductile transition stress, σ3*, was determined by intersecting the Hoek–Brown failure envelope with Mogi’s line, with ductility parameters d ranging from 3.4 (silicate rocks) to 5.0 (carbonate rocks). Numerical solutions were derived for σ3*σc as a function of GSI using Matlab, and the results were fitted with an exponential model. The analysis revealed an exponential relationship between σ3*σc and GSI for values above 32, with accuracy better than 3%. Increased ductility reduces rock mass strength, with higher d values leading to lower σ3*σc. The diminishing returns in confinement strength at higher GSI values suggest that rock masses with higher GSI can sustain more confinement but with reduced effectiveness as GSI increases. These findings provide a framework for predicting brittle–ductile transitions in rock engineering.

Seismic stability charts for three-dimensional rock slopes using Hoek–Brown criterion

Seismic excitation serves as a crucial trigger for slope collapse in seismically active regions. This study employs the non-linear shear strength reduction (SSR) technique in conjunction with the pseudo-static method to analyse the seismic stability of three-dimensional (3D) rock slopes following the generalised Hoek–Brown (GHB) criterion. Considering the non-linear characteristics of the GHB envelope, the instantaneous Mohr–Coulomb (MC) parameters for rock masses are identified by locating the tangent at the intersection of the GHB envelope with the shear and normal stress planes. These parameters are subsequently incorporated into the non-linear SSR analysis for seismic stability based on the MC criterion. The non-linear SSR technique is then validated by comparing the calculated factor of safety (FOS) with those from other methods and used to assess the impact of geometric and GHB parameters on the FOS of 3D rock slopes. Finally, a set of seismic stability charts for 3D rock slopes in GHB media has been developed and applied to evaluate the seismic stability of two practical rock slope cases following the GHB failure envelope.

Roof stability analysis for a shallow-buried cylindrical tunnel in Hoek-Brown rock strata

Roof stability of shallow-buried cylindrical tunnels in rock strata following the generalized Hoek-Brown criterion (GHB) is analyzed in this work. Based on the limit analysis method, a three-dimensional (3D) shallow-buried collapse mechanism for tunnel roofs is developed to derive various stability measures. Optimization codes are programmed to capture the optimal stability measure solutions. The effects of the 3D geometric characteristics, the rock mass strength parameters, and the buried depth on tunnel roof stability are investigated by a parametric analysis. The critical buried depth ratios corresponding to the transition between the shallow-buried and deep-buried tunnel roof collapse mechanisms are also presented. Thereafter a series of stability charts of the required supporting pressure and the factor of safety (FoS) is proposed for design purposes. This study provides a theoretical basis for design and engineering of tunnels with limited buried depth.

Modeling spatial variability of mechanical parameters of layered rock masses and its application in slope optimization at the open-pit mine

Mining engineering with layered structures often obtains extensive geological data to ensure reliability of mining operations. Effectively utilizing these data to characterize the spatial distribution of mechanical parameters in layered rock mass is crucial for slope stability analysis. This study focuses on the Heishan open-pit mine as a case study, leveraging the distinctive characteristics of mining engineering geological data. A three-dimensional fracture network was then generated based on the distribution of fractures in the rocks, and the representative elementary volume of the rock mass was determined by calculating the volumetric joint count (Jv). Statistical characteristics of various geological data are analyzed, and a geological statistical method combining the Kriging method and inverse distance power method is applied to establish a spatial distribution block model for geological strength index (GSI) and rock mechanical properties. Subsequently, a spatial variability model of the mechanical parameters of the layered rock mass was generated based on the Hoek–Brown criterion. Furthermore, the heterogeneous mechanical model is implemented in numerical software for stability analysis and slope angle optimization. The findings indicate that each engineering zone of layered slopes exhibits unique mechanical parameters while demonstrating significant layered distribution patterns at a macro level. Considering spatial variability, the mechanical model significantly impacts slope stability and slope angle optimization outcomes. Our research methodology facilitates accurate quantification of the heterogeneity of mechanical parameters in layered slopes without incurring excessive additional survey costs, enabling more rational explanations of slope landslides and the provision of suitable slope angle optimization designs.

Slope stability modeling using limit equilibrium and finite element methods: A case study of the Adama City, Northern Main Ethiopian Rift

Slope failure is a prominent and recurring geohazard in numerous parts of Ethiopia, including Adama City, which is located in the Northern Main Ethiopian Rift (NMER). The city is surrounded by two ridges oriented in the NNE-SSW direction, which are susceptible to slope instability. Thus, this study is aimed at modeling slope stability along these two ridges using Finite Element Method (FEM) and Limit Equilibrium Method (LEM). The modeling was carried out on slopes of multifaceted geometry composed of eluvium soil, pumice, and moderately to highly-weathered ignimbrites. Critical slope sections were identified using satellite imagery and field manifestations such as slope toe condition and slope face tilting. Their geometries were then inferred from detailed geological cross-sections based on field data. Input parameters for the modeling, such as cohesion, friction angle, and elastic modulus, were calculated via Back Analysis using the Hoek-Brown criterion while unit weight and Poisson ratio were determined from empirical equations. For soil formations, the parameters were determined via standard laboratory experiments. The modeling was then carried out under different conditions, including dry, saturated, static, and dynamic conditions. Results from both LEM and FEM models revealed that three of the four analyzed slope segments were unstable under dynamic and saturated conditions, highlighting the influence and importance of precipitation and seismicity as triggering variables. Results from both methods tend to agree when the critical slip surface passes through a single geological material in both models. However, notable differences arise when the slip surface involves multiple geological materials. Under such conditions, LEM tends to yield higher FOS values compared to FEM. The results also showed that all unstable slopes were associated with the NNE-SSW striking fault of the study area, as inferred from failure surfaces generated from both models and field data. The study concluded that unstable slopes pose a serious risk to nearby residents and infrastructure, and as a remedy, it designed and recommended coupled benching and slope flattening.

Probabilistic evaluation method for the stability of large underground cavern considering the uncertainty of rock mass mechanical parameters: A case study of Baihetan underground powerhouse project

For deep large underground cavern projects under complex geological conditions, the determination of rock mass mechanical parameters and stability analysis is fraught with a great deal of uncertainty. This paper proposes a set of methods for estimating rock mass mechanical parameters and probabilistic stability assessment suitable for construction characteristics of large underground caverns. On the one hand, according to the Hoek-Brown criterion and Monte Carlo simulation, a dynamic estimation method for rock mass mechanical parameters is proposed by using Rock Mass Rating (RMR), the uniaxial compressive strength (UCS) of intact rock, and the material constant (mb) as input parameters. On the other hand, considering the uncertainty of rock mass mechanical parameters, a probabilistic evaluation method applicable to the unloading response characteristics of rock mass (including displacement and fracturing zones) around the large cavern is put forward, combining the point estimation method and numerical simulation. The proposed methods were applied to an underground powerhouse, with an excavation span of 34 m and height of 88.7 m, at the Baihetan hydropower station in the southwest of China. Based on extensive field investigations and tests, the probability distributions of key rock mass mechanical parameters were obtained. Furthermore, the high sensitivity of input parameters in the H-B criterion for estimating rock mass mechanical parameters were revealed. Through dynamic simulation, the probability distributions of the displacement and fracturing zones in surrounding rock during the layer-by-layer excavation of the cavern were presented. Comprehensive on-site tests showed that the simulated results were in basic agreement with the actual unloading response behavior during the cavern excavation, validating the correctness and applicability of the proposed methods. The study provides a relatively comprehensive approach for estimating rock mass mechanical parameters and stability evaluation in similar underground engineering projects, and also has guiding significance for predicting and preventing engineering rock mass hazards.

Curvilinear Slope Profile–Based Seismic Stability for Rock Slopes Using Stress Characteristics Method and Generalized Hoek–Brown Criterion

Curvilinear Slope Profile–Based Seismic Stability for Rock Slopes Using Stress Characteristics Method and Generalized Hoek–Brown Criterion

An improved three-dimensional extension of Hoek–Brown criterion for rocks

The Hoek–Brown (H–B) criterion has found widespread application in numerous rock engineering projects. However, its efficacy is compromised by an underestimation of rock strength due to its neglect of the influence of the intermediate principal stress (σ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{2}$$\\end{document}). Experimental evidence underscores the significant impact of σ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{2}$$\\end{document}. Consequently, there exists an imperative to formulate a three-dimensional (3D) criterion. In this study, a new deviatoric function with two additional parameters (k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document} and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A$$\\end{document}) is developed firstly, which ensure compliance with the prerequisites of smoothness and convexity. In addition, the parameters k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document} and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A$$\\end{document} are bonded with the weakening effect of the Lode angle (θσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta_{\\sigma }$$\\end{document}) and the strengthening effect of the mean stress (σm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{m}$$\\end{document}), respectively. Then a new 3D strength criterion for rocks is proposed by combining this new deviatoric function and the triaxial compression meridian function of the original H–B criterion. Four distinct sets of test data encompassing various rock types are employed to validate the proposed criterion. The results demonstrate that the proposed criterion adeptly captures the strength characteristics of the four rock types, providing a good depiction of failure surfaces within the 3D principal stress space. Comparative analyses involve the utilization of several existing 3D H–B criteria for strength predictions. The proposed criterion exhibits superior fitting performance for all the selected rocks.

Meso/Microscale Parameter Correlation Model of Sandstone Based on the Hoek–Brown Criterion and Parallel Bond Model

Meso/Microscale Parameter Correlation Model of Sandstone Based on the Hoek–Brown Criterion and Parallel Bond Model

Determining the parameters for the overlying stratum caving zones during re-peated mining of pillars

Purpose. The research aims to determine the parameters for the overlying stratum caving zones above the mined-out space during repeated mining of pillars at the Zhomart Mine of the Zhaman-Aybat field for the purpose of predicting hazardous zones of influence of mining operations. Methods. Research includes an analysis of the results of previous in-situ studies conducted using seismic surveys, as well as modeling with the Examine2D software to determine the stress-strain state of the overlying stratum mass. The elastic and strength characteristics of an anisotropic mass are used for modeling, taking into account the generalized Hoek-Brown criterion with the Geological Strength Index (GSI) based on a geomechanical description of the mass quality. Comparative analysis of modeling results with seismic survey results is conducted to confirm the effectiveness of the developed methodology. Findings. The research provides an opportunity to determine the overlying stratum caving parameters, such as the caving arch height and the condition for complete undermining of the overlying stratum at the Zhaman-Aybat field at different spans of the mined-out space (from 50 to 350 with a step of 50 m, reaching the maximum span of 370 m). Originality. . It has been determined that the caving arch height depends on the outcrop span and increases exponentially (hcav = 16.473 e0.008Le). In addition, the condition of the earth’s surface complete undermining has been identified depending on the depth of the site to be gotten. Practical implications.The research results can be used to develop a normative document for calculating the earth’s surface shear during the repeated mining of pillars at the Zhomart Mine of the Zhaman-Aybat field. The data obtained will also be useful in planning repeated mining and predicting the earth’s surface shear to avoid negative impacts of mining operations on surface structures.

Enhanced Hoek-Brown (H-B) criterion for rocks exposed to chemical corrosion

Underground constructions often encounter water environments, where water–rock interaction can increase porosity, thereby weakening engineering rocks. Correspondingly, the failure criterion for chemically corroded rocks becomes essential in the stability analysis and design of such structures. This study enhances the applicability of the Hoek-Brown (H-B) criterion for engineering structures operating in chemically corrosive conditions by introducing a kinetic porosity-dependent instantaneous mi (KPIM). A multiscale experimental investigation, including nuclear magnetic resonance (NMR), X-ray diffraction (XRD), scanning electron microscopy (SEM), pH and ion chromatography analysis, and triaxial compression tests, is employed to quantify pore structural changes and their linkage with the strength responses of limestone under coupled chemical-mechanical (C-M) conditions. By employing ion chromatography and NMR analysis, along with incorporating the principles of free-face dissolution theory accounting for both congruent and incongruent dissolution, a kinetic chemical corrosion model is developed. This model aims to calculate the kinetic porosity alterations within rocks exposed to varying H+ concentrations and durations. Subsequently, utilizing the generalized mixture rule (GMR), the kinetic porosity-dependent mi is formulated. Evaluation of the KPIM-enhanced H-B criterion using compression test data from 5 types of rocks demonstrated a high level of consistency between the criterion and the experimental results, with a coefficient of determination greater than 0.96, a mean absolute percentage error less than 4.84%, and a root-mean-square deviation less than 5.95 MPa. Finally, the physical significance of the porosity-dependent instantaneous mi is clarified: it serves as an indicator of a rock’s capacity to leverage the confining pressure effect.

Study on Numerical Simulation of Large-Diameter Borehole Pressure Relief in Deep High-Gas Soft Coal Seams.

The deep highly gassy soft coal seam has the characteristics of high ground stress, high gas pressure, and low permeability. In the process of coal roadway excavation, there are problems such as frequent gas concentration exceeding the limit and easy induction of gas dynamic disasters. To investigate the pressure relief and disaster reduction efficiency of large-diameter boreholes in a deep high-gas soft coal seam, the 8002 high-gas working face of the Wuyang coal mine was taken as the engineering background to study the deformation law of large-diameter boreholes in deep high-gas soft coal seams. A coupled damage-stress-seepage model for pressure relief of large-diameter boreholes in gas-bearing coal seams was constructed based on the Hoek-Brown criterion, the correlation between the damage area and the gas pressure distribution in the gas-bearing coal seam after the pressure relief of boreholes of different apertures was analyzed, and the pressure relief efficiency of different technical parameters "three flower holes" in the roadway head was determined. The law of stress transfer, gas migration, and energy release in the coal seam after pressure relief of a large-diameter borehole under different initial gas pressures was revealed, and the power function equations of the damage range and borehole diameter, maximum stress at the roadway head, and driving distance after pressure relief of a gas-bearing coal seam were determined. Results showed that under the confining pressure of the 8002 working face roadway in the Wuyang coal mine, the pressure relief effect of 250 mm aperture is better, the drilling plastic zone is "butterfly" or "X″-type distribution, and the plastic zone range is positively correlated with the aperture size. Under the arrangement of "three flower holes", the plastic zone is larger and the pressure relief effect is better when the hole spacing is 1.4 m. With the increase of initial gas pressure, the vertical stress above the borehole increases and the pressure relief efficiency decreases. According to the vertical stress distribution within 200 h of borehole pressure relief, the pressure relief process is divided into a coal damage and failure stage, stress balance stage, and hole collapse stability stage. The research results provide a theoretical basis for the prevention and control of coal rock gas dynamic disasters by large-diameter drilling in a deep high-gas soft coal seam.

Analytical Solution for Seismic Stability of 3D Rock Slope Reinforced with Prestressed Anchor Cables

Currently, the study of analytical solutions for the seismic stability of slopes under anchorage conditions is one of the hottest subjects in engineering. In this paper, an analytical solution for the seismic safety of the three-dimensional (3D) two-stage rock slope reinforced with prestressed anchor cables governed by the nonlinear Hoek–Brown criterion was deduced, in which the analyses of seismicity were performed by the latest modified pseudo-dynamic method. This method supplements the consideration of the damping effect of the rock medium on seismic waves, which is more in line with the real seismic situation. A mathematical geometric model was developed for calculating the external forces work and internal energy dissipation acting on 3D rotating rock masses reinforced with prestressed anchor cables, in which the seismic work rate was calculated using a new layer-by-layer superposition summation method. The analytical solution of the safety factor could be collated as an explicit function of several variables, and then, the optimal value was obtained by the Particle Swarm Optimization (PSO) algorithm. To corroborate the accuracy of new analytical solutions, the results were contrasted with those of the pertinent literature. The results of the two comparisons were very close. Ultimately, the sensitivity analyses and coupling effects of seismic pseudo-dynamic factors and prestressing anchorage factors were carried out. It was found that even small seismic intensities had a large effect on the stability of rock slopes with developed joints. Increasing the number of steps and prestressing anchors can effectively improve the stability of rock slopes under seismic effects. The conclusions have significant implications for the anchorage design of the 3D two-stage rock slope as seismic events occur.