Hoek-Brown Strength Criterion Research Articles

Deformation Patterns and Failure Mechanisms of Soft-Hard-Interbedded Anti-Inclined Layered Rock Slope in Wolong Open-Pit Coal Mine

Since the beginning of spring 2022, successive landslides have occurred in the eastern pit slope of the Wolong Coal Mine in Qipanjing Town, Otog Banner, Inner Mongolia, which has adversely affected the mine’s production safety. This study aims to reveal the deformation patterns and failure mechanisms of landslides. Firstly, this study establishes the stratigraphic structure of the eastern pit slope of the Wolong Coal Mine using extensive field geological surveys combined with unmanned aerial vehicle photography, drilling, and comprehensive physical exploration techniques. Indoor geotechnical tests and microscopic experiments reveal that rock mass typically exhibits the characteristics of expansibility and water sensitivity. Moreover, the mechanical parameters of the rock mass were determined using a combination of the window sampling method, the Geological Strength Index, and the Hoek–Brown strength criterion estimation theory. Finally, this study consolidates the previously mentioned insights and employs FLAC3D (7.0) software to assess the stress–strain characteristics of the excavated slope. The results indicate that the deformation mode of the Wolong open pit coal mine is the toppling failure of soft-hard-interbedded anti-inclined layered rock slopes. The unloading effect and rock expansion-induced softening lead to stress concentration at the slope corners and more substantial deformation, thereby accelerating upper slope deformation. The deformation and destabilization process of landslides is categorized into four stages: the initial deformation stage, the development stage of lateral shear misalignment, the development stage of horizontal tensile-shear damage, and the slip surface development to the preslip stage. This research offers valuable references and engineering insights for future scientific investigations and the prevention of similar slope-related geological hazards.

Revisiting the face stability of rock tunnels in the Hoek–Brown strength criterion with tension cutoff

AbstractIn this work, the three‐dimensional stability of deep tunnel faces is evaluated in rock masses characterized by the generalized Hoek–Brown (H–B) criterion from the perspective of the limit analysis theorem. Considering that underground engineering is gradually developing towards larger burial depths and larger sizes, and the tensile strength of rocks is usually overestimated, the concept of tension cutoff (T‐C) is introduced to substitute the parabolic form of the H–B envelope in the tensile region with a circular arc. The multitangent technique is used to piecewise approximate the H–B envelope, instead of the conventional linear substitution, accounting for the nonlinear dependence of the shear strength on the pressure. Meanwhile, the earthquake loading is considered by the classic pseudostatic method. The tunnel face stability is measured by a stability number, which is calculated based on a newly developed multicone failure mechanism. Parameter studies indicate that the T‐C has almost no impact on the critical support pressure, while has a significant effect on the critical stability number, especially in the presence of earthquakes. Nearly all stress points back‐calculated from the rupture angles fall within the T‐C region, except for the first stress point associated with the first segment. Interestingly, the influence of the geological strength index (GSI) on the face stability is inverse to that of mi due to the brittleness of rocks. Moreover, the critical stability number is not sensitive to mi, but sensitive with the introduction of earthquake loadings.

Elasto-plastic solution for frost heave force considering Hoek-Brown criterion and freezing temperature gradient in cold region tunnels

The tunnel frost heave force is a primary cause of frost damage in cold region tunnels. To guide the prevention and treatment of tunnel frost damage, an elastic–plastic solution for the frost heave force is proposed based on the nonlinear Hoek-Brown (H-B) strength criterion. The elastic–plastic calculation model for the frost heave force simultaneously considers the variation of the volumetric frost heave strain of the surrounding rock with the freezing temperature and the uneven frost heave linear strain. The elastic–plastic analysis of the tunnel frost heave force and development of plastic damage in rock masses based on the H-B criterion is performed through engineering examples in cold regions. The frost heave force of Dabanshan Tunnel obtained by the H-B criterion elastic–plastic solution is 1.14 MPa, which is basically consistent with the monitoring values. The plastic radius is 5.72 m. But the calculation using the linear strength criterion results in a smaller frost heave force and a larger plastic zone radius. And the corresponding parameters of the H-B criterion directly use the rock mass parameters in the survey report, which is convenient for engineering application. The parameter analysis reveals that the plastic zone and frost heave force gradually increase with decreasing GSI and increasing disturbance parameter D. The degree of disturbance has a significant effect on the frost heave force and the plastic zone radius, and is an influencing factor that cannot be ignored.

Mohr–Coulomb and Modified Hoek–Brown Strength Criteria of Layered Sandstone Considering the Unloading Effect and Anisotropy

The Mohr–Coulomb (M-C) and Hoek–Brown (H-B) strength criteria are widely used in various engineering fields, such as mining engineering, tunnel engineering and so on. To investigate the M-C and H-B strength criteria considering the unloading effect and anisotropy, series of triaxial loading (unloading) tests on layered sandstone were conducted. The results revealed that the peak strength was significantly affected by the unloading effect. Moreover, the cohesion and internal friction angle had a significant nonlinear relationship with the bedding angle. Additionally, the M-C and modified H-B strength criteria were established considering the unloading effect and anisotropy. Then, according to the strength criteria established, the peak strength could be estimated theoretically. Furthermore, compared to the M-C strength criteria, the modified H-B strength criteria were more appropriate for accurately estimating the triaxial compressive strength of layered sandstones. The conclusions obtained could provide certain references for the stability control of deep excavation engineering.

Rock Mechanical Properties and Wellbore Stability of Fractured Dolomite Formations.

During the drilling process in the dolomite formation of the Leikoupo formation in western Sichuan, downhole obstruction frequently occurs, which seriously hinders the efficient development of oil and gas resources on site. In view of the problems of borehole stability existing in this type of formation, the geological characteristics, underground complexity, and fracture development of the research block were systematically identified and analyzed, and the physical parameters and mechanical properties of the rock were defined through physical and chemical tests and an understanding of the mechanical properties of the rock. These studies revealed the mechanism of borehole instability and the main factors influencing fractured dolomite formation. Finally, a model of borehole collapse pressure within fractured dolomite formation was established by selecting the appropriate strength criterion as the criterion of borehole instability. The results show that the dolomite fractures of the Leikoupo formation are relatively developed, the formation is dominated by low-angle fractures, and the core integrity is not high. The development of formation fractures has an obvious effect on the mechanical strength of rock, which decreases to different degrees with the decrease in the integrity coefficient of the rock mass, and the anisotropy of the rock's mechanical strength is obvious. The empirical parameters of the generalized Hoek-Brown (H-B) strength criterion were quantified by introducing rock mass acoustic data, and the accuracy of the improved generalized H-B strength criterion was evaluated using triaxial test data. Taking the M-C criterion, the generalized H-B criterion based on acoustic wave, the D-P criterion, and the weak surface strength criterion as wellbore instability criteria, combined with the distribution of wellbore stress, a prediction model of wellbore stability considering the degree of dolomite formation fragmentation was established. The results demonstrate that the generalized H-B criterion model based on acoustic waves can be used to evaluate the wellbore stability of dolomite fracture formation with more than one group of fractures. These research results have certain guiding significance for efficient and safe drilling in fractured dolomite formations.

Mechanical behavior of sandstone during post-peak cyclic loading and unloading under hydromechanical coupling

This paper investigates mechanical behaviours of sandstone during post-peak cyclic loading and unloading subjected to hydromechanical coupling effect, confirming the peak and residual strengths reduction laws of sandstone with water pressure, and revealing the influence of water pressure on the upper limit stress and deformation characteristics of sandstone during post-peak cyclic loading and unloading. Regarding the rock strength, the experimental study confirms that the peak strength σp and residual strength σr decrease as water pressure P increases. Especially, the normalized strength parameters σp/σpk and σr/σre was negatively and linearly correlated with the P/σ3. Moreover, the Hoek-Brown strength criterion can be applied to describe the relationship between effective peak strength and effective confining stress. During post-peak cyclic loading and unloading, both the upper limit stress σp(i) and crack damage threshold stress σcd(i) of each cycle tend to decrease with the increasing cycle number. A hysteresis loop exists among the loading and unloading stress–strain curves, indicating the unloading deformation modulus Eunload is larger than the loading deformation modulus Eload. Based on experimental results, a post-peak strength prediction model related to water pressure and plastic shear strain is established.

Two-dimensional face stability analysis in rock masses governed by the Hoek-Brown strength criterion with a new multi-horn mechanism

The face stability problem is a major concern for tunnels excavated in rock masses governed by the Hoek-Brown strength criterion. To provide an accurate prediction for the theoretical solution of the critical face pressure, this study adopts the piecewise linear method (PLM) to account for the nonlinearity of the strength envelope and proposes a new multi-horn rotational mechanism based on the Hoek-Brown strength criterion and the associative flow rule. The analytical solution of critical support pressure is derived from the energy-work balance equation in the framework of the plastic limit theorem; it is formulated as a multivariable nonlinear optimization problem relying on 2m dependent variables (m is the number of segments). Meanwhile, two classic linearized measures, the generalized tangential technique (GTT) and equivalent Mohr-Coulomb parameters method (EMM), are incorporated into the analysis for comparison. Surprisingly, the parametric study indicates a significant improvement in support pressure by up to 13% compared with the GTT, and as expected, the stability of the tunnel face is greatly influenced by the rock strength parameters. The stress distribution on the rupture surface is calculated to gain an intuitive understanding of the failure at the limit state. Although the limit analysis is incapable of calculating the true stress distribution in rock masses, a rough approximation of the stress vector on the rupture surface is permitted. In the end, sets of normalized face pressure are provided in the form of charts for a quick assessment of face stability in rock masses.

Estimating shear strength of high-level pillars supported with cemented backfilling using the Hoek–Brown strength criterion

Deep metal mines are often mined using the high-level pillars with subsequent cementation backfilling (HLSCB) mining method. At the design stage, it is therefore important to have a reasonable method for determining the shear strength of the high-level pillars (i.e. cohesion and internal friction angle) when they are supported by cemented backfilling. In this study, a formula was derived for the upper limit of the confining pressure σ3max on a high-level pillar supported by cemented backfilling in a deep metal mine. A new method of estimating the shear strength of such pillars was then proposed based on the Hoek–Brown failure criterion. Our analysis indicates that the horizontal stress σhh acting on the cemented backfill pillar can be simplified by expressing it as a constant value. A reasonable and effective value for σ3max can then be determined. The value of σ3max predicted using the proposed method is generally less than 3 MPa. Within this range, the shear strength of the high-level pillar is accurately calculated using the equivalent Mohr–Coulomb theory. The proposed method can effectively avoid the calculation of inaccurate shear strength values for the high-level pillars when the original Hoek–Brown criterion is used in the presence of large confining pressures, i.e. the situation in which the cohesion value is too large and the friction angle is too small can effectively be avoided. The proposed method is applied to a deep metal mine in China that is being excavated using the HLSCB method. The shear strength parameters of the high-level pillars obtained using the proposed method were input in the numerical simulations. The numerical results show that the recommended level heights and sizes of the high-level pillars and rooms in the mine are rational.

Experimental and constitutive modeling of the anisotropic mechanical properties of shale subjected to thermal treatment

The physical and mechanical properties of shale exhibit significant anisotropy, impacting the design and construction of high geothermal tunnels and deep resource mining. Laboratory studies on the physical and anisotropic mechanical properties of shale after various thermal treatments showed that rising temperatures caused an exponential decline in shale’s bulk and density, and the uneven expansion of the mineral particles within the rock caused a decrease in its uniaxial compression strength (UCS) and elastic modulus while increasing their anisotropy. The UCS is U-shaped with a change in bedding dip angle, whereas the elastic modulus is U-shaped or shoulder-shaped. Under the assumption that the rock microelement obeyed the Weibull distribution, the microstructure tensor improved Hoek–Brown strength criterion was introduced as the damage basis. Then a thermal–mechanical coupling damage constitutive model considering bedding dip angle and temperature was established. The model was found to be valid via the test results, especially in capturing the compaction and yield softening stages of shale. These findings have reference value for both underground structural engineering and deep resource mining.

Elastic–plastic response of a deep tunnel excavated in 3D Hoek–Brown rock mass considering different approaches for obtaining the out-of-plane stress

The difficulty in the 3D analysis of a circular tunnel excavation lies in the selection of a reasonable softening equation for the out-of-plane stress (σz) in the plastic zone. This study developed and compared the semi-analytical solutions of a circular opening considering different approaches for obtaining σz, and the results were verified by the numerical results (1) The elastic approaches (e.g., generalized Hooke's law). cannot accurately reflect the mechanical properties of rock in the plastic zone and the results exhibit large differences compared with numerical results. (2) Plastic approaches demonstrate that the in-site axis stress (q) has a limited effect on the stability of surrounding rock, whereas the elastic approach-based results exhibit severe dependence on q. (3) The excavation responses based on the GZZ strength criterion and Hoek–Brown strength criteria have significant differences, reflecting the obvious effect of σz (σ2) on the stability of rock mass, and an unreasonable criterion may result in inappropriate support measures (4) Different post-peak behaviors of rock mass (e.g., constitutive model, residual strength, dilatancy, etc.). significantly affect the surrounding rock stresses, and the applicability of elastic theories and shallow tunnel experiences need to be further considered in deep tunnel where the plastic zone is widely distributed.

A three-dimensional grain-based model for studying the microscopic fracture behaviour of granite

Grain-based models (GBMs) are widely used to analyse the micromechanical characteristics of granite. However, the classical two-dimensional (2D)-GBM has some shortcomings: poor matching of mineral composition and grain size distribution to the rock, small number of intragranular microcracks, and inability to represent the three-dimensional (3D) structure of the grains. Therefore, we propose a new method to establish a 3D-GBM in the particle flow code (PFC). The PFC3D-GBM parameters are obtained by calibrating the experimental data, such as the elastic modulus, Poisson's ratio, tensile strength, compressive strength, Hoek-Brown (HB) strength criterion, and fracture characteristics. The fracture mode of granite is analysed on the basis of the experimental and simulation results. In addition, the fracture characteristics and microscopic fracture mechanisms of granite are analysed from the perspective of mineralogy. The results reveal that the mineral composition and the grain size distribution of PFC3D-GBM match well with those of the rock, and the model can simulate the macroscopic and microscopic fracture characteristics of granite. This provides an effective way to study the relationship between microscopic parameters and macroscopic behaviours in a 3D model.

The novel strength criterion and the associated constitutive model based on the finite deformation behavior for the rock under the disturbance stress paths

To develop a novel 3D Hoek–Brown strength criterion and an associated constitutive model based on the finite deformation behavior of rock under the disturbance stress paths, a series of mechanical tests are initially conducted under designed disturbance stress paths (hydrostatic stress stage (HSS), initial loading–unloading stage (ILUS), and disturbance stage (DS)). Additionally, low disturbance (LD), medium disturbance (MD), and high disturbance (HD) modes are taken into consideration. Then, the finite deformation theory is employed to study the nonlinear behavior. This nonlinear behavior can be regarded as a nonlinear relationship between the stress and strain of rock under loading, especially the initial compression stage and plastic stage, as well as a complex deformation field, which is described by the finite deformation theory and featured by the mean rotation angle (Θ). Afterwards, the damage variable (DV) regarding Θ is proposed and used to establish a novel strength criterion (NSC) based on the current Hoek–Brown strength criterion. Finally, by considering the NSC, an associated elasto-plastic constitutive model is developed. The results show that the initial confining pressure (ICP) and disturbance mode have a significant effect on the elastic modulus (E). Meanwhile, the ICP has a higher effect on E than the disturbance mode. Then, the evolution of Θ is closely associated with the nonlinear deformation of rock masses because the σ1−Θ curves can be divided into three stages. Those stages can match well with the initial compression stage, elastic stage, and plastic stage, respectively. Regarding Θ as a parameter of the damage variable (DV), the NSC is established. Additionally, it can be observed that the failure envelopes of the NSC scale down gradually with increasing Θ on the deviatoric plane. Meanwhile, as the value of the coefficient of determination (DC) is higher than that of other strength criteria, the NSC may exhibit better agreement with the experimental data. Furthermore, a constitutive model based on the aforementioned strength criterion and the hardening function that takes Θ and the disturbance of rock masses (D) into account is proposed and validated as a solution to predict the nonlinear behavior of rock specimens under the disturbance stress paths.

A pseudo-static approach to the seismic bearing capacity of eccentrically loaded footing lying on rock obeying a generalized Hoek–Brown strength criterion

A pseudo-static approach to the seismic bearing capacity of eccentrically loaded footing lying on rock obeying a generalized Hoek–Brown strength criterion

Criterion of Grouting Pressure in Regional Advance Grouting Treatment to Prevent Water Disaster from Karst Aquifers in Coal Seam Floors.

The deep mining of coal mines in North China faces the serious threat of water inrush from karst aquifers in the coal seam floors, and regional advance grouting technology (RAGT) is an effective means to prevent and control such disasters. However, it is difficult to choose the grouting pressure during the implementation of RAGT, and excessive grouting pressure will lead to the splitting of karst fracture and reduce the grouting effect. In this study, based on the Bernoulli equation, the relationship between the ground grouting pressure and critical grouting pressure during grouting is established. Based on the Hoek–Brown (H-B) strength criterion and a fracture mechanics analysis of hydraulic fracturing, a theoretical equation of the critical grouting pressure for fracture splitting during grouting is obtained. The determination methods of the main parameters, such as the length of the fracture, internal friction angle, and H-B constant of the intact rock and geological strength index, and their effects on the critical grouting pressure, are discussed. The results show that the joint influence of the H-B constant and geological strength index of the intact rock is the key factor influencing the critical grouting pressure. The theoretical research results are applied to the Xujiazhuang limestone grouting reinforcement project of the floor of coal seam 11 in the Zhaoguan coal mine. The critical grouting pressure of the aquifer is determined to be 14.54 MPa, which guides the smooth implementation of the project.

Development and verification of three-dimensional equivalent discrete fracture network modelling based on the finite element method

The discrete fracture network (DFN) method is essential for estimating the mechanical properties of naturally fractured rock masses. A three-dimensional (3D) equivalent DFN model was proposed and thoroughly validated for fractured rock masses. In the equivalent DFN modelling, the Monte Carlo method and 3D rock failure process analysis (RFPA3D) software were used. Additionally, the fracture and its spatial distribution were described explicitly, and the strength and deformability of complex fractured rock masses were investigated. A case study was conducted at the dam site area of Lianghekou Hydropower Station's the left bank slope. The ShapeMetriX3D system was used to acquire the geometric parameters and probability distribution of fractures via digital photogrammetry. Additionally, a series of numerical models were created by combining the proposed equivalent DFN method with the fracture geometry information from the case study. Based on the concept of scale effect and anisotropy, the representative element volume (REV) and mechanical parameters of fractured rock masses were determined. The mechanical parameters of slope rock masses obtained by the proposed DFN modelling and generalized Hoek Brown strength criterion were compared. Furthermore, the applicability and reliability of the proposed model were validated using site data and numerical verifications. The proposed 3D equivalent DFN modelling can analyze the scale effect and anisotropy of mechanical properties of fractured rock masses effectively. Finally, the influence of water head on the hydraulic characteristics of fractured rock masses was analyzed based on the proposed model. Results showed that fractures were the main pathway of water.

Multiyield-Surface Implementation of a Simplified Three-Dimensional Hoek–Brown Strength Criterion

This work presents the implementation of a simplified three-dimensional Hoek–Brown strength criterion using a multiyield-surface plasticity approach. The adopted three-dimensional simplification of the strength criterion accounts for the intermediate principal stress influence, maintaining the simplicity in the numerical implementation. This criterion corresponds to a paraboloid surface of circular cross section in the principal stress space. The introduction of multiple yield surfaces allows for a smooth definition of the nonlinear stress–strain behavior of the material. In addition, a kinematic hardening rule is adopted, allowing yield surfaces to translate with loading, enabling a hysteretic and path-dependent stress–strain response.

Three-dimensional tunnel face extrusion and reinforcement effects of underground excavations in deep rock masses

The design experience and the shallow tunnel theory, which emphasize the plane strain assumption, two-dimensional (2D) strength theory, and indoor or back analysis-based parameters of rock mass, should not be directly adopted for deep tunnel design . This paper reports the development of a digital acquisition technique to quickly, automatically, and accurately obtain the in-situ strength parameters (e.g., GSI and D ), followed by the establishment of a three-dimensional (3D) numerical model for deep tunneling for a typical tunnel in western China to study the 3D and nonlinear spatial effects and mechanisms during excavation with incorporation of the 3D Hoek-Brown (HB) strength criterion. The reliability and correctness of the developed methods were verified by the field monitoring data , and the results of our study showed that: (1) The deformations based on 2D strength criteria (e.g., HB and MC) and 3D criterion (e.g. GZZ) are close for shallow tunnel, but significantly different for deep tunnel because of the widely distributed plastic zone in deep tunneling, thereby indicating the obvious effect of intermediate principal stress ( σ 2 ) on rock deformation ; (2) The face extrusion deformation exhibits obvious nonlinear trends with an increasing buried depth, and the σ 3 of the core rock reflects a tensile value in deep tunnel, thereby indicating the presence of an extremely unstable state. Reinforcement could increase the overall stress inside the core rock and reduce the stress unloading effect caused by the strong disturbances of deep excavations , thereby promoting a gradual change from tensile stress to compressive stress ; (3) A uniform relationship was found between pre-extrusion and pre-convergence deformations, indicating that the pre-convergence deformation of the surrounding rock could be controlled by constraining the pre-extrusion deformation of the core rock. This digital technique and such 3D HB-based (e.g., GZZ) numerical modeling may be promising for conducting the 3D forward and dynamic analyses of tunnel stability. Additionally, the core rock extrusion control can be treated as an active method to render the rock itself a natural support material and to realize its strength potential, thereby making a valuable contribution to the construction of deep and ultra-deep rock tunnels. • A digital acquisition method for in situ rock mass parameters was proposed and applied to deep tunneling analysis. • The reliability of 3D forward numerical analysis based on the 3D Hoek-Brown strength criterion was verified by the field monitoring data. • The 3D extrusion effects and mechanical mechanism of deep tunnel face extrusion were studied. • The reinforcement effects of the core rock mass under different GSIs were examined. • A uniform relationship was found between the pre-extrusion and pre-convergence of the core rock mass.

Quantifying strength and permeability of fractured rock mass using 3D bonded block numerical model

In this research, a 3D bonded block model (BBM) is applied to numerically quantify the strength and permeability of fractured rock mass. The reliability of the BBM is initially verified through comparisons between conceptual models and theoretical and laboratory results. A series of rock mass models incorporating complex fracture networks is then constructed, and the strengths are quantified accordingly. The rock mass is weakened when low fracture stiffness and friction angle are assigned, and steep fractures are simulated. In both cases the low strength is attributed to the decreasing frictional resistance of the initial fractures. The rock mass is further weakened when more persistent fractures are simulated, in which the weakening effects of the fractures are amplified. The numerical models are compared with the widely used Hoek-Brown strength criterion. The strengths resulting from the two methods converge only when steep and less persistent fractures are simulated. Numerical simulations also suggest that in general, the permeability is positively related to the strength for rock masses that are impermeable in the pre-peak loading stage. In such cases additional flow paths are formed and attributed to failed rock bridges. The research works provide new options to quantify the strength and permeability of fractured rock mass, and offer opportunities to examine relations between these parameters.

Experimental Study on the Anisotropy of Layered Rock Mass under Triaxial Conditions

Weak and hard inhomogeneous rock formations are typically encountered during tunnel excavations. The physical and mechanical properties and geological conditions of these rock formations vary significantly; thus, it is crucial to investigate the mechanical characteristics of deep bedded composite rock formations. Three-dimensional (3D) scanning and 3D printing were used to prepare composite rock specimens to simulate natural rock laminae. Triaxial compression tests were conducted to determine the influence of the bedding angle, rock composition, and confining pressure on the mechanical properties of the composite rock specimens. The anisotropic strength characteristics and the damage patterns of the composite rock specimens were analyzed under different confining pressures, and the failure mechanism during triaxial loading was revealed. The results show that the damage of the composite rock specimens with a bedding structure depends on the bedding dip angle and the rock formation. The stress-strain curves and peak strengths of the composite rock specimens have anisotropic characteristics corresponding to their failure modes. As the bedding dip angle increases, the peak strength of the three groups of specimens first decreases and then increases under different confining pressure levels. The compressive strength has a nonlinear relationship with the confining pressure, and the difference between the compressive strengths of specimens with different inclination angles decreases as the confining pressure increases. The Hoek–Brown strength criterion is a good predictor of the nonlinear increase in peak strength of the composite rock specimens under different confining pressures. The specimen with a β = 60°dip angle shows the most significant increase in the strength difference with increasing confining pressure. The results can be used as a reference for testing and analyzing the anisotropic mechanical properties of bedded rock masses.

Pseudo-dynamic stability of rock slope considering Hoek–Brown strength criterion

Pseudo-dynamic stability of rock slope considering Hoek–Brown strength criterion