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

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

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.

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.

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.

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

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

Study of slope stability analysis method based on equivalent Mohr‐Coulomb strength parameters of three‐dimensional stress state

AbstractMost rock slope stability analysis methods are based on the Hoke–Brown criterion under a two‐dimensional state of stress, which somewhat ignores the effect of intermediate principal stress. In this paper, by introducing the improved three‐dimensional H‐B criterion into the Meridian plane, the tangent line of a point on the H‐B strength envelope is regarded as its instantaneous equivalent M‐C parameter. Based on this, a formula for solving the equivalent M‐C strength parameters under a three‐dimensional stress state is established to describe the shear strength parameters of rock mass, considering different stress states. Taking three kinds of rocks as examples, the influence of the intermediate principal stress on their strength parameters is analyzed. On this basis, the realization scheme of the nonlinear strength reduction method is established, the stability of the slope is studied, and an example verifies the feasibility of this method. This method fully considers the nonlinear characteristics of rock slope strength parameters under a three‐dimensional stress state and provides a solid basis for slope stability analysis.

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.

Rock mass strength variability for probabilistic open-pit slope stability analysis

ABSTRACT As part of reliability-based design acceptance criteria, probabilistic slope stability analysis is increasingly being used for open-pit slope design. This analysis evaluates the mean factor of safety, probability of failure, and coefficient of variation for the resulting probability density function of factor of safety values. Estimating rock mass strength variability is crucial. Hoek–Brown criteria are commonly used strength parameters, as are equivalent Mohr–Coulomb parameters (calculated from Hoek–Brown), particularly for probabilistic slope stability analysis. This article describes these two strength criteria when considering univariate and bivariate distributions of the unconfined compressive strength and rock material constant. Results demonstrate differences in the variability of the equivalent Mohr–Coulomb parameters under different dependence considerations than the Hoek–Brown parameters, potentially affecting the calculated probability of failure and factor of safety results. Furthermore, they highlight an inherent correlation between Mohr–Coulomb parameters that derives from the algorithm used to calculate them from Hoek–Brown criteria. This inherent correlation is important to obtain mean factors of safety, probabilities of failure, and coefficients of variation that are consistent with the variability in Hoek–Brown parameters estimated by the practitioner and are, therefore, key to informed implementation of reliability-based design acceptance criteria.

Investigation on the Mechanical Characteristics of the Excavation of a Double-Line Highway Tunnel Underpass Existing Railway Tunnel under the Influence of Dynamic and Static Load

Research on the excavation mechanical properties of underpass tunnels has already had certain results, but only a few of them consider the effects of dynamic and static loads on the excavation mechanical properties of underground tunnels at the same time; particularly, there is a lack of research investigating double-line highway tunnels with angled underpasses of existing railway tunnels. In this paper, based on the tunnel project of the new double-line Shiqian Highway Tunnel passing under the Hurong Railway with an oblique angle, based on the method of over-advance geological prediction and investigations into the palm face surrounding the rock, the rock degradation caused by dynamic and static loads is quantified using the perturbation system. Additionally, the mechanical parameters of the rock under the influence of dynamic and static load coupling in the influence area of the cross-tunneling project are determined using the Hoek–Brown criterion, and the mechanical characteristics of the excavation of a tunnel under the double-lane highway tunnel passing under the existing railroad are constructed with the mechanical characteristics of the double-lane highway tunnel, taking into consideration the influence of the dynamic and static load coupling in a three-dimensional model. The results show that, in line with the new tunnel rock movement law for the top of the arch sinking, the bottom plate bulging, the side wall outward movement, the height and width of the arch, and the bottom plate arch show an increase with the tunnel excavation, while the side wall rock displacement effect is smaller; the left and right line tunnel disturbed area of the rule of change is similar; the existing tunnel bottom plate displacement is larger than the top plate and the left and right side wall, under the influence of the excavation time step. Typical profile point displacement is mainly determined by the distance from the excavation surface; von Mises stress extremes are observed in the top plate and side walls of the existing tunnel, which occur in the tunnel structure, and there are unloading and pressure-bearing zones in the bottom plate; the new tunnel has the same rock disturbance angle under the four calculation conditions and, based on the displacement control criterion, the excavation method is preferred and the upper and lower step blasting excavation method is recommended.

Damage assessment of High-Stress brittle surrounding rock masses in a deep Large-Span High-Sidewall underground Cavern: In situ measurement and numerical simulation

Excavation within high-stress brittle rock masses often encounters severe problem of structural damage in deep underground engineering, necessitating a significant scientific demand for accurately assessing the damage to surrounding rock masses through in-situ measurement and numerical simulations. This study reports detailed findings from an exhaustive investigation on the damage characteristics and underlying mechanisms in the high-stress brittle surrounding rock masses during excavation at the Shuangjiangkou (SJK) underground caverns. These findings were obtained through a comprehensive long-term in situ measurement using multi-point extensometer monitoring, borehole digital optical televiewer observing, and acoustic testing. Based on a deep understanding of the damage and failure mechanisms peculiar to the SJK high-stress brittle surrounding rock masses, a new damage index specifically designed for high-stress brittle rock masses was developed. This index is formulated by incorporating the failure approach concept into a modified Hoek−Brown criterion adapted for high-stress brittle rock masses. Comparison between the numerical predictions and in situ measuring results of the excavation damage zones during layered excavation of the SJK underground powerhouse revealed remarkable concordance. This consistency validates that the newly proposed damage index has good applicability and reliability for quantifying excavation-induced damage in high-stress brittle surrounding rock masses. These research findings are poised to provide valuable insights into the damage identification of brittle surrounding rock masses subjected to high-stress excavation, contributing to optimize excavation design and enhance the safety and sustainability of underground engineering projects.

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.

Rock slope stability assessment based on the critical failure state curve for the generalized Hoek‒Brown criterion

Rock slope stability assessment based on the critical failure state curve for the generalized Hoek‒Brown criterion

Study of longitudinal deformation profiles in high-ground-stress mega-section tunnels based on the Hoek–Brown criterion

In rock tunnel construction, the analysis of rock deformation characteristics is an essential aspect of the overall safety evaluation. In this study, the high-ground-stress section of the Tongzi tunnel was taken as the research object and the longitudinal wave velocities of the disturbed and undisturbed areas of the surrounding rock were measured using on-site drilling to determine the disturbance range of the surrounding rock excavation. The uniaxial and triaxial compressive strengths of the rocks sampled in the field were measured through indoor tests. Through a combination of the field acoustic test and indoor test results, the Hoek–Brown (H-B) failure criterion parameters of the surrounding rock were quantitatively obtained. A numerical model was developed for the analysis, and the following conclusions were drawn: At the pre-deformation stage, the influence of the crown settlement, horizontal displacement, and the tunnel face extrusion deformation were all within the range of one times the tunnel diameter. The location and area of the first tunnel face influenced the extent of the surrounding rock’s pre-deformation. A more pronounced effect on the pre-deformation of the tunnel crown settlement was obtained the closer the first tunnel face was to the tunnel crown. In addition, the closer the first tunnel face was to the arch waist, the more pronounced the impact was on the pre-deformation of the horizontal displacement, and the pre-deformation increased exponentially with the increase in the significance of the excavation area of the first tunnel face. The deformation varied in tunnels that had been excavated using different construction methods. The three-step method employed in this study resulted in the most considerable deformation, followed by the CD method, and the double-side heading method resulted in the smallest deformation. The results show that increasing the number of working faces and using small cross-sectional staggered slot excavation are beneficial to tunnel stability. The field monitoring data are generally consistent with the numerical simulation results, providing a degree of validation for the numerical simulation results. The intrinsic model used in the numerical simulation influences the simulation results. When compared with the Hoek–Brown principal structure model, using the Mohr–Coulomb central structure to simulate the high-ground-stress rock tunneling may result in more considerable elevation arch uplift and tunnel face extrusion deformation. Therefore, it is more reasonable to use the Hoek–Brown principal structure model.

Experimental Study on Mechanical Properties of Anisotropic Slate under Different Water Contents

Slate typically possesses a pronounced layered structure and tends to soften when exposed to water, leading to numerous detrimental effects on the construction of related underground projects. In this study, X-ray diffraction (XRD) analysis was first performed to investigate the mineral composition of the typical slate from Changsha, China. Then, uniaxial and triaxial compression tests under varying bedding angles (i.e., 0°, 30°, 45°, 60°, and 90°) and moisture levels (i.e., dry state, natural state, and saturated state) were conducted to explore the anisotropy characteristics and susceptibility to water-induced softening of the slate. The results reveal that: (1) The exposure of slate to water exacerbates the deterioration of its layered structure, making it more prone to shear failure along the bedding planes. Furthermore, the energy released during shear-slip damage is reduced, which is macroscopically manifested by the decrease in slate brittleness and the increase in plasticity. (2) The slate’s compressive strength, elastic modulus, and cohesion vary in a U-shaped pattern with the increase in bedding angles. However, Poisson’s ratio and internal friction angle are slightly affected by the bedding angle and water content, which do not exhibit a clear variation pattern. (3) In addition, the formulations for strength and stiffness predictions of slate were also discussed in this study. The results show that the modified Hoek–Brown criterion characterizes the uniaxial and triaxial compressive strengths of slate more accurately, and the generalized Hooke’s Law more effectively predicts the elastic modulus.

Quantitative estimation method for the excavation-induced weakening effect of rock mass parameters in deep tunnels

The excavation-induced weakening effect of rock mass parameters is detrimental to the stability of deep tunnels. This study presents a quantification method for the weakening effect of rock mass parameters. In this method, the weakening factor is introduced as a quantitative index of the weakening effect of rock mass parameters, which can be calculated by the acoustic P-wave velocity measured in the surrounding rock and the Hoek–Brown criterion. Taking three deep tunnels as examples, the weakening law of rock mass parameters is investigated. The results indicate that the weakening effect is most severe near the excavation outline and gradually decays with increasing distance. Significant differences in the weakening effect can be observed at different parts of the tunnel cross-section and under different excavation methods. The weakening effect can be further exacerbated by multiple excavation disturbances. Among the strength parameters of rock masses, the weakening effect of uniaxial compressive strength is the most obvious, followed by uniaxial tensile strength, cohesive strength, and equivalent internal friction angle. The FLAC3D software is utilized to reproduce different weakening effects and analyze the energy distribution around tunnels. It is found that as the weakening effect intensifies, the accumulated elastic strain energy within the surrounding rocks gradually diminishes, and the peak elastic strain energy is at the junction of damaged and undamaged rock masses.

A new three-dimensional rock strength criterion based on shape function in deviatoric plane

Rock strength criteria are the theoretical grounding of geotechnical design and stability estimation, the Mohr–Coulomb (MC) and Hoek–Brown (HB) criteria are the widely accepted criteria at present, due to their reasonability and unambiguous concept, however they overlook the effect of intermediate principal stress, and contain six singular corners in π plane. Aimed at overcoming those limitations, the MC and normal parabolic criterion (NPC) were improved to their 3D versions that lead to smooth and convex for a wide range of strength parameters. The extended 3D strength criteria coincide with corresponding original forms in the triaxial compression and triaxial extension states, which not only take intermediate principal stress into account, but also provide great convenient in numerical calculation. Multigroup of poly-axial strength datasets gathered from the references are used to check the prediction accuracy of the proposed 3D criteria by the least absolute deviation method. Research proved that the 3D NPC criterion has a relatively larger deviation on poly-axial strength data prediction, but the proposed 3D MC criterion can describe peak strength with low misfit for soft or hard rocks. Peak strength σ1 increases first and then decreases with the increase of σ2, whether increasing or decreasing σ2, both will result in rock failure. Moreover, the 3D MC can fit the poly-axial strength data well for lower or higher values of σ3, which strongly suggests the proposed 3D MC criterion is adequate. Applicability of the proposed strength criterion will be discussed in further research.

Determination of Physical and Geomechanical Properties of Black Shale Hosting Lead and Zinc Ore in Eyingba, Ebonyi

This study was conducted to investigate the physical and geo-mechanical properties such as specific gravity, porosity, uniaxial compressive strength (UCS) etc. of shale, hosting lead and zinc ore in Enyigba, Ebonyi State, Nigeria. The strength properties of the black shale were analyzed because the variability in the strength of rock masses must be properly accounted for, to determine their utility in engineering design. Several tests were carried out on the intact shale samples, and the results were converted to bulk mass strength values using the Hoek–Brown criterion. The results showed that the UCS of the black shale ranged from 38.3 to 43.7 MPa for a geological strength index (GSI) value of 75; and from 54.2 to 60.4 MPa for a GSI value of 80. The specific gravity was 2.89 while the bulk density and porosity values were 290.3 kg/m3 and 15.56% respectively. Also, it was determined that there is a strong correlation between the UCS and specific gravity of the rock, with negligible levels of correlation, both positive and negative, between the UCS and the other mechanical properties examined. In conclusion, the black shale sample from Enyigba is characterized as a medium to hard rock that is suitable for engineering and mining projects.