Hoek-Brown Criterion Research Articles

Mechanical Properties and Strength Characteristics of Rock–Coal–Rock Assemblages under Different Peripheral Pressures

To investigate the deformation and damage characteristics of internal coal bodies of small pillars under different pressures, rock–coal–rock assemblage samples were subjected to the conventional triaxial compression test to analyze the mechanical behavior characteristics under different pressures. The results showed that, with the increase in peripheral pressure, the peak strength and modulus of elasticity of the assemblage specimens increased, the range of fracture compaction stage gradually decreased, and the specimen was gradually transformed from brittle to ductile. With an increase in peripheral pressure, the residual strength gradually increased, and the strength decay coefficient gradually decreased. The strength decay coefficient decreased the most at 0–10 MPa, and this decrease slowed down after exceeding 15 MPa. When the peripheral pressure was 0 MPa, the damage degree of the coal pillar was larger. With the increase in peripheral pressure, the number of cracks in the coal column increased, the damage degree increased more, and mixed damage characteristics of tension–shear were found. Based on the Hoek–Brown criterion, the strength criterion applicable to the specimen of rock–coal–rock combination was obtained through numerical fitting iteration, which provides an experimental and theoretical basis for realizing the stability control of small coal columns.

A Stope Mining Design with Consideration of Hanging Wall When Transitioning from Open Pit Mining to Underground Mining for Sepon Gold Mine Deposit, Laos

This study investigates the transition from surface to underground mining at the Sepon Gold mine. The stability of surface slopes is assessed prior to commencing underground operations. Stope mining is adopted based on ore body characteristics and geological features. Finite element numerical analysis, employing the Generalized Hoek–Brown criterion, evaluates slope stability for surface slopes and opening stopes. The pit design comprises a 70° slope angle, 20 m height, and 10–15 m safety berm, meeting stability requirements with a factor of safety of 2.46. Underground mining design includes a 62° ore body dip, a 50 m thick crown pillar to prevent surface subsidence, and 5 m wide, 5 m high stopes. Sill pillars are left for support after each level of excavation. Rock bolts are employed in specific stope areas where necessary, with continuous monitoring for surface deformation. This study analyzes the influence of stope sizes on the pit wall and pit bottom stability, identifying slope failures near the hanging wall close to the pit bottom during underground mining. A slight increase in the displacement zone is observed on the upper crest of the top footwall. Overall, the findings demonstrate successful stability in the transition from surface to underground mining at the Sepon Gold mine.

Investigation of anisotropic strength criteria for layered rock mass

Layered rock mass is a type of engineering rock mass with sound mechanical anisotropy, which is generally unfavorable to the stability of underground works. To investigate the strength anisotropy of layered rock, the Mohr-Coulomb and Hoek-Brown criteria are introduced to establish the two transverse isotropic strength criteria based on Jaeger's single weak plane theory and maximum axial strain theory, and parameter determination methods. Furthermore, the sensitivity of strength parameters (K1, K2, and K3) that are used to characterize the anisotropy strength of non-sliding failure involved in the strength criteria and confining pressure are investigated. The results demonstrate that strength parameters K1 and K2 affect the strength of layered rock samples at all bedding angles except for the bedding angle of 90° and the angle range that can cause the shear sliding failure along the bedding plane. The strength of samples at any bedding angle decreases with increasing K1, whereas the opposite is for K2. Except for bedding angles of 0° and 90° and the bedding angle range that can cause the shear sliding along the bedding plane, K3 has an impact on the strength of rock samples with other bedding angles that the specimens' strength increases with increase of K3. In addition, the strength of the rock sample increases as confining pressure rises. Furthermore, the uniaxial and triaxial tests of chlorite schist samples were carried out to verify and evaluate the strength criteria proposed in the paper. It shows that the predicted strength is in good agreement with the experimental results. To test the applicability of the strength criterion, the strength data of several types of rock in the literature are compared. Finally, a comparison is made between the fitting effects of the two strength criteria and other available criteria for layered rocks.

Study on Rock Failure Criterion Based on Elastic Strain Energy Density

Uniaxial and five conventional triaxial compression tests were conducted on sandstone to obtain the evolution laws of the input energy density, elastic strain energy density, and dissipative energy density. The input and dissipative energy densities increased with increasing axial strain; the elastic strain energy density increased with increasing axial strain at the pre-peak stage and decreased after the peak. According to the linear change rule between the peak elastic strain energy density and confining pressure, the energy density failure criterion of sandstone was established, and the criterion has high precision and few parameters, and the parameters have clear physical meaning. Moreover, the expression of the energy density failure criterion was similar to the classical Hoek-Brown criterion, but its adaptability was more extensive. The strength calculation results for seven different rocks under different confining pressures calculated using the energy density failure criterion were consistent with the experimental values, and the calculation error was smaller than that of the Mohr–Coulomb criterion and Drucker–Prager criterion, verifying the accuracy and applicability of the criterion.

A quantitative analysis procedure for solving safety factor of tunnel preliminary support considering the equivalence between Hoek–Brown and Mohr–Coulomb criteria

There is a development trend from qualitative towards quantitative analysis for tunnel stability. The widely accepted convergence-confinement method (CCM) provides a quantitative framework for tunnel stability analysis, while the conventional CCM based on closed-form functions is essentially an ideal model without considering the tunnel section shape and rock weight in plastic zone. Herein, based on the CCM framework, a quantitative analysis procedure formulated through numerical simulations for more practical situations was systematically proposed to calculate the safety factor of spray-anchor support system in non-circular tunnel. Besides, similarities and differences between the most widely used Hoek-Brown (HB) and equivalent Mohr-Coulomb (EMC) criteria were considered for practical reference. The proposed method was utilized to conduct quantitative analysis of LDPs, mechanical responses of surrounding rock and safety factors for four typical tunnels, and the consistency between tunnels based on HB criterion (TBHBC) and EMC criterion (TBEMCC) was analyzed. The results indicated that the maximum radial convergence of LDP, plastic zone area, and safety factor of TBEMCC were 34.54%, 37.29%, and 47.71% lower than those of TBHBC respectively. The reasons for differences between results were revealed. The spatial effect induced by the face excavation of TBEMCC was weakened as compared to TBHBC, leading to a drop of LDP. Consequently, the safety factor was underestimated. The method systematically provided an effective tool for quantitative stability analysis and design optimization of support system for an underground excavation.

Experimental Study on Permeability and Deformation Characteristics of Bedding Shale under Triaxial Shear-Seepage Coupling

The bedding structure of shale is generated during the deposition and formation, which results in shales with prominent anisotropic characteristics. It depends on stability, control of oil and gas storage, and deep exploitation. In addition, the mechanical and permeability parts of bedding shale are very complex when it is under deep underground space with coupled high stress and high seepage. In this study, the black bedding shale was used as the research object, and a series of triaxial shear-seepage coupling tests were carried out. Firstly, the triaxial shear stress-shear strain curves and permeability-shear stress curves of different bedding shales under other triaxial shear-seepage coupling conditions were obtained. Secondly, the failure characteristics and shear deformation characteristics of shale under the shear-seepage coupling effect were explored. The shear stress threshold and permeability evolution law at each stage of shear failure were discussed. Thirdly, the shear strength, failure mode, and mechanism parameters of the black bedding shale under different normal stress and seepage pressure were studied. Fourthly, the linear M-C criterion, Ramamurthy criterion, and Hoek-Brown criterion characterize the variation of damage strength of shale with bedding orientation under triaxial shear-seepage coupling. Those results provide an experimental basis for exploring the anisotropic mechanical characteristics and failure mechanism of bedding shale under shear-seepage coupling.

Critical strain method incorporating calculation of equivalent H-B strength parameters for back analysis during tunnel excavation

A comprehensive method is proposed for the continuous update of the design rockmass parameters during tunnel excavation coupled to novel calculation of the Hoek-Brown strength parameters for given Mohr-Coulomb ones. The method starts with the strength parameters of Mohr-Coulomb or Hoek-Brown criterion used at the design stage of tunnels, and then updates them by using the displacements measured during the excavation of tunnels. The update is performed based on a well established back analysis procedure that determines both cohesion C and internal friction angle φ of the Mohr-Coulomb criterion. Then, these parameters either directly update or are converted to equivalent Hoek-Brown parameters that update the original design parameters of the tunnel. The need arising for conversion of parameters during the back analysis is thus addressed: A method is developed for the evaluation of the equivalent Hoek-Brown parameters for an axisymmetric structure subjected to major principal initial stress p0 and supported by internal pressure pi. Four characteristic cases are analysed theoretically and numerically with a finite differences code and found to be in excellent agreement. The equivalent parameters solution is developed in Matlab source code under a creative commons license and greatly enables the automation of the update procedure.Based on the updated parameters, it is possible to continuously modify the original design of underground support structures, according to the results of back analysis of measured displacements. The modified strength parameters of the H-B criterion can also be used for assessing the stability of the tunnels for the next excavation stage of the tunnels.

Modified minimum principal stress estimation formula based on Hoek–Brown criterion and equivalent Mohr–Coulomb strength parameters

The most critical parameter for determining equivalent values for the Mohr–Coulomb friction angle and cohesion from the nonlinear Hoek–Brown criterion is the upper limit of confining stress. For rock slopes, this value is the maximum value of the minimum principal stress (sigma_{3,max }^{prime }) on the potential failure surface. The existing problems in the existing research are analyzed and summarized. Using the finite element method (FEM), the location of potential failure surfaces for a wide range of slope geometries and rock mass properties are calculated using the strength reduction method, and a corresponding finite element elastic stress analysis was carried in order to determine sigma_{3,max }^{prime } of the failure surface. Through a systematic analysis of 425 different slopes, it is found that slope angle (β) and geological strength index (GSI) have the most significant influence on sigma_{3,max }^{prime } while the influence of intact rock strength and the material constant m_{i} are relatively small. According to the variation of sigma_{3,max }^{prime } with different factors, two new formulas for estimating sigma_{3,max }^{prime } are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity.

Research on the Failure Mechanisms and Strength Characteristics of Deeply Buried Mudstone under the Interaction of Water and Stress

Mudstone is a widely occurring type of rock in deep mining, and it is crucial to understand its failure mechanisms and strength characteristics under the interaction of water and high stress to ensure the stability of deeply buried engineered mudstone. In this study, the composition and the structure of mudstone were obtained, and triaxial tests were conducted on mudstone under five different water contents and four different confining pressures using a triaxial servo press. The variation rules for the peak strength and residual strength were obtained, and the applicability of the strength criteria was analyzed through fitting. The results showed that both the peak strength and the residual strength decreased linearly with increasing water content, with the peak strength decreasing more rapidly. Both the peak strength and the residual strength increased with increasing confining pressure, with the residual strength increasing more rapidly. The decrease in strength was primarily due to the decrease in cohesion, with the cohesion of the peak strength decreasing from 8.40 MPa to 0.94 MPa and the cohesion of the residual strength decreasing from 1.75 MPa to 0.82 MPa. The internal friction angle did not change much, with the internal friction angle of the peak strength decreasing from 41.57° to 37.29° and the internal friction angle of the post-peak strength increasing from 32.35° to 33.28°. For dry and low-water-content mudstone, the peak strength conformed to the Mohr–Coulomb criterion, while for mudstone with a higher water content, the peak strength conformed to the Hoek–Brown criterion. The residual strength conformed to the Hoek–Brown criterion. Under low and medium confining pressures, water played a dominant role in the damage pattern for the fractures produced by the initial damage to the mudstone. Under a high perimeter pressure, water played a guiding role for the fractures produced by the initial damage to the mudstone.

Nonlinear Strength Reduction Method of Rock Mass in Slope Stability Evaluation.

As the strength parameters of rock mass degrade differently during slope instability, different factors should be considered in the strength reduction method. Previous nonlinear reduction methods were essentially implemented based on the Mohr-Coulomb criterion, which was reported not to reflect the nonlinear performance of rock mass. To address this deficiency, in this study, the Hoek-Brown criterion was combined with a nonlinear reduction technique for slope stability evaluation. Firstly, based on the classical definition of safety factors, the relationships that should be satisfied by each parameter of the critical slope were derived. The critical curve of the slope regarding the Hoek-Brown constant mb and the uniaxial compressive strength of rock mass σcmass was then obtained. On the assumption that the slope parameter deterioration conforms to the shortest path theory, the reduction ratio of σcmass to mb was determined. The more objective k-means algorithm was employed to automatically search the potential sliding surface, on which the slope safety factor was calculated as the ratio of sliding resistance to sliding force. Finally, the slopes in published literature were adopted for verification, and the calculated safety factors were compared with those by other methods, which showed better efficacy.

A benchmark study of different numerical methods for predicting rock failure

Nowadays, with many available numerical methods developed in rock mechanics, researchers have always focused on the parameter calibration of the numerical model but ignored the predictive capability of these methods. A comparative study of nine commonly used numerical methods was performed for predicting rock failure through international cooperation organized by the Discontinuous Deformation Analysis (DDA) commission of the International Society for Rock Mechanics (ISRM). Two steps of numerical modelling were conducted including a calibration procedure from given experimental results and a numerical prediction for benchmark tests with these calibrated parameters for three types of rocks. Through the comparison between different numerical and experimental results, the inherent weaknesses and strengths of different numerical methods in terms of predicting rock failure were identified and analysed. The influence of human intervention in terms of parameter selection is even more significant than the choice of different numerical methods. Some potential factors (i.e., different boundary conditions, heterogeneity of rock material, strength parameters, particle packing, and failure criteria) that may occur in numerical and physical tests were further discussed. Through the comparison of different failure criteria, we found the selection of rock failure criterion might be the major factor that affected the predictive capability of numerical methods, and the nonlinear failure model (the Hoek–Brown criterion) showed the superiority in the prediction of rock fracturing subjected to complex stress conditions. This comparative work also enlightens the significance of a high-quality calibration process and the future advancement of rock failure criteria.

Estimating Mohr–Coulomb Strength Parameters from the Hoek–Brown Criterion for Rock Slopes Undergoing Earthquake

The direct application of the Hoek–Brown failure criterion to practical slope engineering is still an urgent problem. The slope geometries and earthquake effect need to be considered in the determination of linear Mohr–Coulomb (MC) strength parameters from the Hoek–Brown criteria for slope stability analysis. This study adopted the tangential method to construct a three-dimensional (3D) rotational failure mechanism using the Hoek–Brown failure criterion for homogeneous rock slopes undergoing earthquake. The quasi-static method was employed to treat the seismic action as an external seismic force in the work–energy equation of the limit analysis theory. Based on the numerical optimization, the least upper-bound solutions and equivalent MC strength parameters were derived with respect to different strength parameters and seismic loads. The influences of nonlinear strengths, geometric parameters and earthquake load on the equivalent MC strength parameters were thoroughly investigated. The results suggested that the nonlinear parameters have different influences on the equivalent MC parameters for general steep slopes and vertical slopes. The effects of nonlinear parameters on the equivalent MC parameters become obvious for vertical slopes. The disturbance factor D affects the equivalent MC parameters only for very steep slopes in fractured rock masses. Additionally, the effect of slope inclination on the equivalent MC parameters becomes obvious for slopes in fractured hard rock masses. The 3D effect of the rock slope on the equivalent MC parameters was found to be slight. Moreover, the impact of earthquakes on the approximate MC parameters becomes weaker for steeper rock slopes. The tables of approximate MC strength parameters were given for various slopes with different nonlinear strength parameters. The presented tables can provide certain references for practical slope engineering.

Fracture angle of Hoek–Brown materials

The fracture angle (defined as the angle between the normal vector of the fracture plane and the first principal stress) of Mohr–Coulomb is 45°+ ϕ/2, which plays a crucial role in soil mechanics. However, for the non-linear Hoek–Brown criterion, the fracture angle cannot be derived as easily as Mohr–Coulomb. Based on the physical evidence that there is no parallel plastic strain in the shear zone, the fracture angle of Hoek–Brown materials, is derived as tan−11+m2s+mσ3/σc. The experimental data of marble and granite and numerical results demonstrate that the obtained theoretical fracture angles agree well with the experimental and numerical fracture angles.

3D limit analysis of rock slopes based on equivalent linear failure criterion with tension cut-off

Hoek-Brown (HB) failure criterion is widely used to predict the strength of intact or heavily jointed rock mass. For stability analysis of rock slopes governed by the HB failure criterion, the equivalent linearity to Mohr-Coulomb (MC) criterion is often adopted, leading to the well-known equivalent Mohr-Coulomb method (EMCM). Existing studies on EMCM analysis mainly consider the shear strength of rock material, while consideration of the tensile strength is rare. This contradicts the fact that the underlying tensile strength of rock mass has considerable impact on the rock slope stability in real world. In this regard, this paper proposes a limit analysis-based approach that can account for tension in the three-dimensional (3D) stability analysis of HB rock slope. This approach is established on the equivalent linearity of the HB criterion with consideration of tensile strength, known as the equivalent tension cut-off MC method (ETMCM), and using a horn-like 3D mechanism of limit analysis. The safety factor solutions given by the proposed approach are validated by previous studies and numerical results. Parametric studies are conducted to investigate the effect of rock tensile strength on slope stability. Results show that the consideration of tension leads to a more conservative safety factor and a sharper curvature of the failure surface, and these impacts tend to be more obvious with the increases in slope inclination and slope width. Finally, the stability of the HB rock slope under seepage conditions is studied using the proposed approach. The results indicate that the effect of tensile strength is highly remarkable in seepage circumstances.

Influence of the Hoek–Brown failure criterion with tensile strength cut-off on the roof stability in deep rock tunnels

The Hoek–Brown (HB) failure criterion provides reasonable estimates of the peak strength of isotropic rocks for shear failure; however, this is less so for tensile fractures. Hoek and Brown advised against the direct application of the HB criterion during excavation at deep depths and introduced a tension cut-off (TC) based on reliable extension test data. However, its influence on rock structures remains unclear, with no previous studies on safety assessment considering TC. In this study, a stability analysis of deep tunnel roofs was performed using the HB yield function with reduced or eliminated tensile strength. The parabolic form of the HB envelope was terminated using a circular arc in the tensile stress regime. Such a modification allows some part of the failure mechanism to be separated from the stationary rock with a large volumetric strain governed by the TC. The kinematic approach of limit analysis was utilized as the study approach. The computation results revealed that, among the two common tunnel cross-sections, rectangular tunnels were susceptible to the TC, i.e., the stability number with 10% of the theoretical tensile strength was more than twice that of the original envelope. Circular tunnel roofs can be self-supported even with a zero tensile strength (19% increase in the stability number) owing to the addition of a confining stress. For different HB constants, the relative increase in the stability number subjected to the TC was surprisingly consistent. The factor of safety, defined as the shear strength and required supporting pressure, was derived from the envelope with a TC.

Seismic bearing capacity of rock foundations subjected to seepage by a unilateral piece-wise log-spiral failure mechanism

Most of the current studies on the bearing capacity of rock foundations follow the Hoek-Brown (HB) criterion, but all adopt the generalized tangent technique or the multi-tangent approach, which does not reflect the essence of the nonlinear dependence of the Hoek-Brown criterion. Therefore, a modified unilateral failure mechanism is proposed in the paper, using which the bearing capacity of shallow strip footings considering seepage and seismic forces is evaluated. Use of a statics-like approach for seepage condition and a modified pseudo-dynamic method for seismic condition. The comparison with the results of other literature confirms the feasibility and effectiveness of the study in the paper, while the parametric analysis shows that a smaller upper bound solution is yielded based on the failure mechanism proposed in this paper, and the horizontal gradient ratio, the horizontal factor of seismic acceleration, the normalized frequency, the geological strength index, the disturbance coefficient, and the rock type-dependent parameter all have significant effects on the bearing capacity of rock foundations. In addition, the analysis of the working conditions considering both seepage and seismic forces confirmed the dominance of seismic forces. The results of the superposition treatment between different working conditions are inaccurate, which will underestimate the foundation bearing capacity and cause a waste of cost. This paper develops a more accurate method of assessing the bearing capacity of rock foundations and gives a table of coefficients that can be adopted in engineering practice.

A generalized nonlinear three-dimensional failure criterion based on fracture mechanics

Based on fracture mechanics theory and wing crack model, a three-dimensional strength criterion for hard rock was developed in detail in this paper. Although the basic expression is derived from initiation and propagation of a single crack, it can be extended to microcrack cluster so as to reflect the macroscopic failure characteristic. Besides, it can be derived as Hoek–Brown criterion when the intermediate principal stress σ2 is equal to the minimum principal stress σ3. In addition, the opening direction of the microcrack cluster decreases with the increase of the intermediate principal stress coefficient, which could be described by an empirical function and verified by 10 kinds of hard rocks. Rock strength is influenced by the coupled effect of stress level and the opening direction of the microcrack clusters related to the stress level. As the effects of these two factors on the strength are opposite, the intermediate principal stress effect is induced.

A modified three-dimensional Hoek–Brown criterion for intact rocks and jointed rock masses

Accurate description of the failure strength behaviors of rock materials, including intact rocks and jointed rock masses, is essential for engineering design and construction. First, a novel three-dimensional (3D) version of the Hoek–Brown (HB) criterion for intact rocks is proposed in this paper. A stress weighting factor n is used in this criterion to describe the effects of intermediate and minimum principal stresses. The proposed 3D version is validated using six sets of polyaxial test data, and its prediction effect is compared with that of five other existing 3D criteria. Results show that the proposed criterion exhibits the smallest prediction error for most rock types. The fitted n is closely correlated to both the partial correlation factors of intermediate and minimum principal stresses. Then, an empirical relationship mb(β) between the material parameter mb and joint dip angle β is developed to apply the proposed criterion to jointed rock masses. The prediction performance of the proposed empirical relation and three other existing expressions for the mb of six jointed rock masses at different dip angles is compared, and the proposed relation exhibits the best. The performance of the proposed criterion with empirical relation mb(β) is also verified with nine sets of conventional and true triaxial test data. Results indicate that the predicted strengths are in agreement with the test data. The expression form of the established relation mb(β) can also accurately describe the variation in the mb value with dip direction α.

Bearing capacity of foundations resting on rock masses subjected to Rayleigh waves

This study investigates the ultimate bearing capacity of shallow foundations resting on rock masses under earthquakes using upper-bound limit analysis. A modified pseudodynamic method (MPD) that considers the influence of Rayleigh waves is initially used to characterize the spatiotemporal variation in earthquake acceleration. The strength of rock masses is governed by the generalized Hoek‒Brown criterion and the plastic flow at failure is assumed to respect the associated flow rule. The generalized tangential technique is introduced to provide a linear approximation of the strength envelope for capturing the equivalent Mohr‒Coulomb strength parameters. The classic nonsymmetrical multi-block translational mechanism, which is capable of assessing a rock-foundation system under asymmetrical load, is reconstructed to portray the velocity discontinuity of the plastic failure region. Based upon the energy equilibrium theorem of the limit analysis, an analytical solution of seismic bearing capacity is derived and then formulated as a multivariate optimization problem. Comparisons show good agreement with the literature and validate the correctness and rationality of this work. Parametric study indicates that the seismic foundation capacity obtained by MPD tends to be conservative compared to the pseudostatic method with the same seismic coefficients, and that the period of a Rayleigh wave has only a slight effect on the bearing capacity. The failure region is more sensitive to the disturbance coefficient than it is to the other rock strength parameters.

A numerical algorithm for ground reaction curves of circular openings in strain-softening rock masses satisfying the generalized Hoek-Brown criterion

Based on the self-similarity method, a numerical algorithm for constructing ground reaction curves (GRC) of circular openings in elastic-strain softening rock mass that satisfies the generalized Hoek-Brown failure criterion is proposed. The algorithm is derived from the analysis of the axisymmetric plane strain model, where the far-field and excavation boundary are both axisymmetric stresses. Using the self-similar transformation, the plastic zone of the model was transformed into unit plane ring, which has unit radius at the boundary between the plastic zone and the elastic zone. The unit plane ring was dispersed into multiple concentric rings, for which the Runge-Kutta discrete equations of the stress and deformation were established based on corresponding self-similar governing equations. The equations were solved ring by ring in a successive manner, starting from the outer boundary to the excavation boundary. The comparison with the verified algorithm in literature shows that the results of the different algorithms are in good agreement and the relative errors of the plastic zone radius under various softening parameters are only 1.7%–2.0%. These results verified the correctness and reliability of the proposed new algorithm. Finally, the new algorithm is used to investigate the influence of the Hoek-Brown parameter a on GRC. A discussion of the potential capability of practical applications of this algorithm is also provided.