Hoek-Brown Strength Research Articles

Stability charts for rock mass slopes based on the Hoek-Brown strength reduction technique

This study presents stability charts for estimating the stability of rock mass slopes that are suitable for the generalized Hoek-Brown (GHB) failure criterion based on the strength reduction technique. First, in order for implementing the GHB strength reduction, the estimation for the local Mohr-Coulomb (MC) friction angle and cohesive strength from the GHB criterion was incorporated into the strength reduction process in ABAQUS 6.10. Second, stability charts for calculating the FOS of rock slopes with a specific slope angle of β=45° and D=0 are proposed based on this method. Third, the disturbance weighting factors fD and slope angle weighting factor fβ are used to quantify the effect of the rock mass disturbance factor and slope angle on the rock slope stability evaluations. Combined with the stability charts based on β=45° and D=0, the weighting factor fD and fβ can be used to calculate the FOS of rock slopes with various slope angles under different blast damage and stress relief conditions. Finally, the stability charts were applied to two real case studies to illustrate the applicability and validity. The results for the two cases demonstrated good agreement with those from the other charts and the FEM software ABAQUS 6.10. The proposed charts provide a simple and straightforward way to estimate the FOS of the rock mass slopes directly from the GHB parameters.

Relationship of box counting of fractured rock mass with Hoek-Brown parameters using particle flow simulation

Influenced by various mining activities, fractures in rock masses have different densities, set numbers and lengths, which induce different mechanical properties and failure modes of rock masses. Therefore, precisely expressing the failure criterion of the fractured rock influenced by coal mining is significant for the support design, safety assessment and disaster prevention of underground mining engineering subjected to multiple mining activities. By adopting PFC2D particle flow simulation software, this study investigated the propagation and fractal evolution laws of the micro cracks occurring in two typical kinds of rocks under uniaxial compressive condition. Furthermore, it calculated compressive strengths of the rocks with different confining pressures and box-counting dimensions. Moreover, the quantitative relation between the box-counting dimension of the rocks and the empirical parameters m and s in Hoek-Brown strength criterion was established. Results showed that with the increase of the strain, the box-counting dimension of the rocks first increased slowly at the beginning and then exhibited an exponential increase approximately. In the case of small strains of same value, the box-counting dimensions of hard rocks were smaller than those of weak rocks, while the former increased rapidly and were larger than the latter under large strain. The results also presented that there was a negative correlation between the parameters m and s in Hoek-Brown strength criterion and the box-counting dimension of the rocks suffering from variable mining activities. In other words, as the box-counting dimensions increased, the parameters m and s decreased linearly, and their relationship could be described using first order polynomial function.

Convergence of synthetic rock mass modelling and the Hoek–Brown strength criterion

Synthetic rock mass modelling is based on estimating the mass scale properties of rock directly from properties of the intact material and observed jointing geometry and strength. This technique explicitly represents characteristics of the rock mass that are implicit in the widely applied, empirical Hoek–Brown strength criterion. We demonstrate here that the two approaches converge in mass strength and the strength envelope when the necessary conditions for the empirical method are satisfied. However, the synthetic rock mass approach is advantageous whenever the necessary conditions of the empirical method are not satisfied. As an example, we use the synthetic rock mass method to describe an anisotropic strength envelope for a rock mass with four independent joint sets, where the orientation-specific strength has a significant impact on rock mass stability. In addition we show that rock mass strength is controlled strongly by joint persistence and the resultant connectivity of discontinuities, and only weakly by joint density and intensity.

Theoretical Solutions of a Circular Tunnel with the Influence of the Out-of-Plane Stress Based on the Generalized Hoek–Brown Failure Criterion

AbstractThe effect of out-of-plane stress on theoretical solutions of stress, displacement, and plastic radius for a circular tunnel were studied by using the generalized Hoek–Brown (HB) failure criterion and nonassociated flow rule in elastic–brittle–plastic rock masses. A technique of using equivalent Mohr–Coulomb and generalized HB strength parameters was adopted to compare results of the developed method and published results. The presented solution was validated by existing theory. The results of the developed method were found to be larger than those previously published, which were determined through programming.

A Design-Oriented Combined Model (7 MPa to 190 MPa) for FRP-Confined Circular Short Columns

This study addresses a design oriented combined model to predict the ultimate strengths and ultimate strains in an extensive range of unconfined strength (7 to 190 MPa) for the axially loaded fiber-reinforced polymer (FRP)-wrapped circular short columns. Modified Hoek-Brown strength criterion, which was previously extended to FRP-confined concrete from 7 to 108 MPa, is revisited and verified. An empirical strength model beyond 108 MPa encompassing ultra-high strength concrete (UHSC) and ultra-high performance concrete (UHPC) data, as well as empirical strain models, are defined to accomplish the design oriented combined model. This article especially focuses on the verification of the proposed strain models. The assessment performances of those models for carbon FRP (CFRP) and glass FRP (GFRP) confinement are compared with specific models in the current literature. Strength and strain predictions for UHSC and UHPC are integrated into the design oriented combined model as well. The assessments on this model agree with the experimental results in high accuracy.

Estimation of rock block size distribution for determination of Geological Strength Index (GSI) using discrete fracture networks (DFNs)

Rock mass classification provides fundamental data for a numerical stability analysis of rock structures. Among rock mass classification systems, the RMR and Q systems often are used for rock support system selection and the Geological Strength Index (GSI) system for estimating rock mass strength and deformation parameters. Moreover, the GSI system is the only rock mass classification that is directly linked to engineering design parameters such as the Mohr–Coulomb or Hoek–Brown strength parameters or the rock mass modulus. However, the original application of the GSI system requires long-term experience and a careful approach because of the fact that its use is a subjective decision. A quantitative approach to assist a less experienced engineer in assigning representative GSI values was presented. It employed the rock block volume and joint conditions as quantitative characterisation factors. Their approach is founded on the linkage between descriptive geological terms and measurable field parameters, such as joint spacing and joint roughness. In this study, a discrete fracture network (DFN) model incorporated with stochastic simulation is applied to characterise rock block size distribution for determination of the GSI. The fracture frequency obtained from the core logging data is analysed and provided to the DFN model as input data. Realisation of the DFN and its verification are conducted to establish the joint systems corresponding to the original fracture frequency. As a result, the stochastic simulation can successfully provide the information on the rock block size distribution to the procedure of the GSI determination.

An elastoplastic model of collapse pressure for deep coal seam drilling based on Hoek–Brown criterion related to drilling fluid loss to reservoir

Abstract Too high borehole pressure can lead to the drilling fluid loss to the coalbed methane (CBM) reservoir, and then cause reservoir damage. In CBM rich regions of China, the drilling fluid loss caused by higher borehole pressure is a particularly prominent problem because the target coal seam is deep and soft. Therefore, selecting an appropriate drilling fluid density or borehole pressure is the most effective way to both avoid drilling fluid loss and maintain wellbore stability in these regions, which means that the collapse pressure must be calculated accurately enough. In this paper, the elasto-plastic characteristic of the deep coal seam under high confining pressures at X region in Qinshui basin is determined based on the conventional triaxial compression test, and the mechanical parameters related to solve the collapse pressure are obtained. Then, selecting the Hoek–Brown strength criterion as the limit equilibrium condition, an elasto-plastic model of collapse pressure for deep coal seam drilling is derived to relieve the problem of drilling fluid loss to formation, considering the radius of plastic zone, the geological strength index (GSI), the non-uniform in-situ stress coefficient and the unconfined compression strength of the intact rock. Taking CBM X–X1 well in X region as an example, the elasto-plastic results of the collapse pressure are calculated and compared with elastic results, and the relations between the collapse pressure and the influence factors are analyzed. The results show that the elasto-plastic results of the collapse pressure decrease by 6.64 % than the elastic results calculated by Hoek–Brown criterion when the radius of plastic zone is 1.5 times of borehole radius; the elasto-plastic results of the collapse pressure decrease with the increasing of the radius of plastic zone and the non-uniform stress coefficient; the collapse pressure presents the variation trend of the minus power function as the GSI increasing.

Displacement-based seismic design of a shallow strip footing positioned near the edge of a rock slope

This paper presents a displacement-based dynamic design of shallow strip footings located close to the edge of homogeneous and isotropic rock slopes. The analysis is conducted using the “generalized tangential” technique in which the Hoek–Brown strength is replaced by an “optimal” tangential Mohr–Coulomb domain within the framework of the kinematic approach of limit analysis theory. In order to better understand the influence of these factors including geometrical features, rock strength parameters and seismic loading, a parametric study is carried out. An expression is derived and used to obtain a least upper bound solution for the bearing capacity of a footing on a rock slope considering a kinematically admissible log-spiral failure mechanism. In addition, a similar expression of the horizontal yield seismic coefficient is derived for a given load acting on the foundation.The assessment of earthquake-induced permanent displacement is performed by a simplified procedure. This requires the knowledge of a ground motion parameter that can be evaluated by site-specific seismic hazard analyses or ground motion prediction equations and that considers the actual sliding surface. The calculated displacement should not be greater than a selected limiting tolerable displacement for the specific case. An example is presented illustrating the application of the design procedure for the bridge foundations.

A Photogrammetric Approach for Stability Analysis of Weathered Rock Slopes

The geological strength index (GSI) system is dependent on the rock block volumes and the joint surface conditions. The weathering degree of rock slopes and their strength properties also depend on these characteristics. This study thus focuses on the use of the GSI system and the Hoek–Brown strength criterion to estimate the engineering parameters for weathered rock masses. Photogrammetric methods based on 3D surface models are used to obtain reliable data on the joint sets in rock slopes, instead of general site investigation using labor-intensive techniques. Photogrammetric surveys were conducted on weathered rock slopes in Gold Coast, Australia to obtain the joint spacing, orientation and roughness. The 3D models are then used to estimate the block sizes the joint roughness coefficients (JRC). The block volumes and JRC values were then used to estimate GSI values. Then parametric studies using the finite element method is conducted to investigate the stability of the slope using the GSI values.

The Three-Dimensional (3D) Numerical Stability Analysis of Hyttemalmen Open-Pit

Abstract The purpose of this paper was to perform the 3D numerical calculations allowing slope stability analysis of Hyttemalmen open pit (location Kirkenes, Finnmark Province, Norway). After a ramp rock slide, which took place in December 2010, as well as some other small-scale rock slope stability problems, it proved necessary to perform a serious stability analyses. The Hyttemalmen open pit was designed with a depth up to 100 m, a bench height of 24 m and a ramp width of 10 m. The rock formation in the iron mining district of Kirkenes is called the Bjornevaten Group. This is the most structurally complicated area connected with tectonic process such as folding, faults and metamorphosis. The Bjornevaten Group is a volcano-sedimentary sequence. Rock slope stability depends on the mechanical properties of the rock, hydro-geological conditions, slope topography, joint set systems and seismic activity. However, rock slope stability is mainly connected with joint sets. Joints, or general discontinuities, are regarded as weak planes within rock which have strength reducing consequences with regard to rock strength. Discontinuities within the rock mass lead to very low tensile strength. Several simulations were performed utilising the RocLab (2007) software to estimate the gneiss cohesion for slopes of different height. The RocLab code is dedicated to estimate rock mass strength using the Hoek-Brown failure criterion. Utilising both the GSI index and the Hoek-Brown strength criterion the equivalent Mohr-Coulomb parameters (cohesion and angle of internal friction) can be calculated. The results of 3D numerical calculations (with FLA3D code) show that it is necessary to redesign the slope-bench system in the Hyttemalmen open pit. Changing slope inclination for lower stages is recommended. The minimum factor of safety should be equal 1.3. At the final planned stage of excavation, the factor of safety drops to 1.06 with failure surface ranging through all of the slopes. In the case of a slope angle 70° for lower stages, FS = 1.26, which is not enough to provide slope stability. Another series of calculations were therefore performed taking water table lowering into consideration, which increases the global safety factor. It was finally evaluated, that for a water table level of 72 m the factor of safety equals 1.3, which is enough to assure global open-pit stability.

Hoek-Brown 강도기준식 및 암질강도지수를 이용한 고압 유체 지하저장 공동의 융기에 대한 안정성 평가

Abstract A simple analytical approach for stability assessment of underground storage caverns against ground uplift of overburden rock above the rock caverns for high pressurized fluid such as compressed air energy storage (CAES) and compressed natural gas (CNG) was developed. In the developed approach, we assumed that failure plane of the overburden is straight upward to ground surface, and factor of safety can be calculated from a limit equilibrium analysis in terms of this cylindrical shape failure model. The frictional resisting force on the failure plane was estimated by Hoek-Brown strength criterion which replaces with Mohr-Coulomb criterion such that both intact rock strength and rock mass conditions can be considered in the current approach. We carried out a parametric sensitivity analysis of strength parameters under various rock mass conditions and demonstrated that the factor of safety againt ground uplift was more sensitive to Mohr-Coulomb strength criterion rather than Hoek-Brown criterion.

Numerical Simulation of Crack Growth and Coalescence in Rock-Like Materials Containing Multiple Pre-existing Flaws

A novel meshless numerical method, called general particle dynamics (GPD), is proposed to simulate samples of rock-like brittle heterogeneous material containing four preexisting flaws under uniaxial compressive loads. Numerical simulations are conducted to investigate the initiation, growth, and coalescence of cracks using a GPD code. An elasto-brittle damage model based on an extension of the Hoek–Brown strength criterion is applied to reflect crack initiation, growth, and coalescence and the macrofailure of the rock-like material. The preexisting flaws are simulated by empty particles. The particle is killed when its stresses satisfy the Hoek–Brown strength criterion, and the growth path of cracks is captured through the sequence of such damaged particles. A statistical approach is applied to model the heterogeneity of the rock-like material. It is found from the numerical results that samples containing four preexisting flaws may produce five types of cracks at or near the tips of preexisting flaws including wing, coplanar or quasi-coplanar secondary, oblique secondary, out-of-plane tensile, and out-of-plane shear cracks. Four coalescence modes are observed from the numerical results: tensile (T), compression (C), shear (S), and mixed tension/shear (TS). A higher load is required to induce crack coalescence in the shear mode (S) than the tensile (T) or mixed (TS) mode. It is concluded from the numerical results that crack coalescence occurs following the weakest coalescence path among all possible paths between any two flaws. The numerical results are in good agreement with reported experimental observations.

Studying the effect of non‐spherical micro‐particles on Hoek–Brown strength parameter mi using numerical true triaxial compressive tests

SUMMARYThe strength parameter mi in the Hoek–Brown strength criterion is empirical and was developed by trial and error. To better understand the fundamental relationship between mi and the physical characteristics of intact rock, this paper presents a systematic study of mi by representing intact rock as a densely packed cemented particle material and simulating its mechanical behavior using particle flow modeling. Specifically, the three‐dimensional particle flow code (PFC3D) was used to conduct numerical true triaxial compression tests on intact rock and to investigate the effect of non‐spherical micro‐particle parameters on mi. To generate numerical intact rock specimens containing non‐spherical micro‐particles, a new genesis process was proposed, and a specific loop algorithm was used based on the efficiency of the process and the acceptability of generated specimens. Four main parameters—number, aspect ratio, size, and shape—of non‐spherical micro‐particles were studied, and the results indicated that they all have great effect on mi. The strength parameter mi increases when the number, aspect ratio, or size is larger or the shape becomes more irregular, mainly as a result of the higher level of interlocking between particles. This confirms the observations from engineering experience and laboratory experiments. To simulate the right strength parameter mi, it is important to use appropriate non‐spherical micro‐particles by controlling these four parameters. This is further demonstrated by the simulation of two widely studied rocks, Lac du Bonnet granite and Carrara marble. Copyright © 2014 John Wiley & Sons, Ltd.

Modification of Hoek–Brown Criterion and Its Application Based on Fuzzy Synthetic Evaluation

The importance of mechanical parameters of rock mass is presented, and some methods to determine these mechanical parameters are introduced, whereas its accuracy is questionable and controversial. As there is difficulty in determining the disturbance factor D relayed on the Hoek–Brown strength criteria, fuzzy comprehensive evaluation is employed. Considering the factors influencing on disturbance and relevant engineering experience, the calculation of the value of D is 0.9379, which is more corresponded to the actual situation. Additionally, the rationality, that the bounds characterized by slip surface for dividing disturbed and undisturbed zone, was discussed and it overcome deficiency of other expressions. Relayed on mechanical parameters of rock mass from Liberia Bone Iron Mine, calculation of the factor D was presented by means of the modified parameters and other methods respectively. On the basis of the comparative analysis, the results showed that the former was more suitable to mirror the disturbance degree in-suit. What’s more, it verified the reasonability and significance of partition of disturbed and non-disturbed zone and provided important reference in the future research.

Mechanical properties of deep-buried marble material under loading and unloading tests

The mechanical properties are essentially different when rock material is subjected to loading or unloading conditions. In this study, loading and unloading tests with various confining pressures are conducted to investigate the mechanical properties of marble material samples taken from the deep diversion tunnels of Jinping II Hydropower Station. The stress-strain relationship, failure characteristics and strength criterion are compared and analyzed based on the experiment results. The results show: in the loading and unloading test, peak strength, lateral strain, axial strain and plastic deformation increase significantly as the confining pressure increases. Lateral strain increased significantly and obvious lateral dilatancy can be observed to the change of confining pressure; The fracture mode is mainly the single shear fracture for the triaxial compression test and post-peak test, angle between the failure surface and the ends of the rock material becomes smaller as the confining pressure increases. Hoek-Brown strength criterion reflects the strength characteristics of marble material under two different unloading conditions, and has some supplementary effects to the rock material of mechanical field.

Stability Analysis of an Open Cut Slope in Wardha Valley Coal Field

Abstract Open cast mines are the prime source of coal production in India. Due to an increase in coal demand, deeper surface mines are being planned to ensure better productivity with enhanced safety of these mines. The safe working environment and continuous production of the coal calls for the safe and stable design of cut slopes. The stability analysis of this slope requires in depth geotechnical investigation which are aided by result orientated stability assessment using empirical methods and numerical simulation. In the present study, a slope was examined to understand the mechanism and comparisons were made with field observations. The investigation has involved a 32m high cut slope from an open cast mine in Wardha valley coal field which has been analyzed using a two dimensional numerical simulation. The bench slope consisted of a low strength sandstones, shales and clay sequence. Hoek-Brown strength parameters established and used as input parameters in the model. The results indicate that the slope is critically stable and may lead to failure without warning and needs proper attention.

Pseudo‐static analysis of tunnel face stability using the generalized Hoek–Brown strength criterion

SUMMARYThe stability of a tunnel face driven by a pressurized shield is investigated in the light of a limit analysis kinematic approach formulated in the context of the modified Hoek–Brown strength criterion. The analysis focuses on evaluating the destabilizing effects induced by seismic loading. The pseudo‐static approach is adopted to account for the effects of inertial forces developed in the rock mass by the passage of seismic waves. Taking advantage of the ability to compute analytically, the expressions of the support functions for the modified Hoek–Brown strength criterion, rigorous limit analysis solutions can be derived semi‐analytically for the tunnel face stability. Particular emphasis is given to the description of the Horn failure mechanism and associated lower bound estimates of the limit face supporting pressure. The approach is then applied to investigate the influence of some relevant strength and loading parameters in the tunnel face stability.Copyright © 2013 John Wiley & Sons, Ltd.

Modification of a generalized three-dimensional Hoek–Brown strength criterion

To take account of the influence of the intermediate principle stress, Zhang and Zhu [1,2] proposed a three-dimensional (3D) version of the generalized Hoek–Brown strength criterion. The generalized Zhang–Zhu strength criterion is a true 3D version of the Hoek–Brown criterion, not only inheriting the advantages of the Hoek–Brown strength criterion, but predicts the same strength as the two-dimensional (2D) Hoek–Brown strength criterion at both triaxial compression and extension states. However, the failure surface of the generalized 3D Zhang–Zhu strength criterion is not smooth at either the triaxial compression or extension state and concave at the triaxial extension state, which may have problems with some stress paths and cause inconvenience for numerical applications. In this paper, the reason for the non-smoothness and non-convexity of the generalized 3D Zhang–Zhu strength criterion was first discussed by studying its Lode dependence. Then the criterion was modified by utilizing three different Lode dependences with characteristics of both smoothness and convexity to replace its Lode dependence. Finally the smoothness, convexity and prediction accuracy of the modified criteria were evaluated by applying them to analyze both intact rocks and jointed rock masses. The modified criteria not only keep the advantages of the generalized 3D Zhang–Zhu strength criterion, but solve the non-smoothness and non-convexity problem with no loss of accuracy for strength prediction.

A generalized three-dimensional failure criterion for rock masses

The smooth convex generalized failure function, which represents 1/6 part of envelope in the deviatoric plane, is proposed. The proposed function relies on four shape parameters (Ls, a, b and c), in which two parameters (a and b) are dependent on the others. The parameter Ls is called extension ratio. The proposed failure function could be incorporated with any two-dimensional (2D) failure criteria to make it a three-dimensional (3D) version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion for geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering (1) constant extension ratio and (2) variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio (Ls) that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters (σc and m) are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameter f is introduced. The modified extension ratio is related to f and extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range of f varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression between f and Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures.

Study on Post-Failure Deformation and Residual Strength of Rock Mass Based on Hoek-Brown Criterion

In order to predict the stability of surrounding rock mass in geotechnical engineering, it is important to study the post-failure deformation property and residual strength of rock mass. Based on evolutional behavior of strength parameters, aiming at generalized Hoek-Brown strength criterion, selecting major principal strain as strain softening parameter, this paper presents the method of solving post-failure stress-strain curve . In numerical case, the effect of evolutional law of strength parameters ， and to deformation and residual strength is discussed and we can draw the following conclusions: the greater the residual values of ， are and the smaller the residual value of is, the post-failure strain softening curve falls more gently and the greater the residual strength is.