Hoek-Brown Model Research Articles

Estimation of the equivalent Mohr–Coulomb parameters using the Hoek–Brown criterion and its application in slope analysis

The linearisation of the Hoek–Brown failure criterion is a commonly encountered problem in rock mechanics. The fact that the minor principal stress used for estimating the equivalent Mohr–Coulomb parameters is the stress at failure cannot be ignored. This article proposes a method for estimating the most likely position of any arbitrary stress point that corresponds to the minimum shear strength on the Hoek–Brown curve. The interval of the minor principal stress at failure, which is employed in the existing equivalent estimation methods, is determined from the actual stress level of the rock mass. The Hoek–Brown failure envelope is divided into several zones, which enables its actual stress gradient variation to better match its curvature. The proposed method is applied to analysis of a slope example. The results show that, as the actual Hoek–Brown strength curve of each material is more finely divided, both the safety factor of the slope and the location of the slip surface based on the equivalent Mohr–Coulomb model agree more closely with those obtained from the Hoek–Brown model.

Characterizing and Modeling the Mechanical Properties of the Cement Mortar Modified with Fly Ash for Various Water‐to‐Cement Ratios and Curing Times

Despite many research studies on the effect of the fly ash content (FA) on the mechanical behavior of the cement mortar, there has not been an extensive study investigating the effect of FA, curing time (t), and water‐to‐cement ratio (w/c) on the compressive (σc), tensile (σt), and flexural (σf) strengths of cement mortar. Therefore, this study investigates the subject which could be beneficial for the building and construction field. In this study, more than 1000 data on the mechanical properties of the cement mortar modified with different percentages of fly ash varying from 5% to 75% (by dry weight of the cement) were collected from the literature. The statistical analysis and modeling were performed on the collected data. The w/c of the cement mortar ranged from 0.20% to 0.80%, and the compressive, split tensile, and flexural strengths of cement mortar modified with fly ash and cured up to 90 days ranged from 15 MPa to 88 MPa, 0.4 MPa to 5 MPa, and 1 MPa to 10 MPa, respectively. The Vipulanandan model was also used and compared with the Hoek–Brown model to correlate the mechanical properties of cement mortar modified with fly ash. The results of this study showed that there is a good relationship between the compressive strength (σc) and w/c, curing time, and fly ash content. The compressive, split tensile, and flexural strengths of cement mortar quantified well as a function of w/c, fly ash content, and curing time using a nonlinear relationship.

Experimental Study of the Triaxial Strength Properties of Hollow Cylindrical Granite Specimens Under Coupled External and Internal Confining Stresses

High geostresses and stress gradients are the predominant stress conditions in deep excavation-disturbed rock masses. The aim of this study is to determine the triaxial compressive strength properties of hollow cylindrical granite specimens under a radially non-uniform confining stress field with different radial stress gradients determined by coupled external and internal confining stresses. Triaxial compression testing of hollow cylindrical rock specimens was performed to investigate the influence of the radial stress gradient, external confining stress and specimen length-to-diameter (L/D) ratio on the triaxial compressive strength. The experimental results and regressed failure criteria indicate that the triaxial compressive strengths of the hollow cylindrical granite specimens increase with the external confining stresses, but decrease with an increase in the radial stress gradients. The calculated goodness of fit (R2) and root-mean-squared error suggest that the nonlinear failure criterion based on the Hoek–Brown model is more accurate than the linear failure criterion based on the Mohr–Coulomb model for determining the influences of the external confining stress and radial stress gradient on the triaxial compressive strength. In addition, the triaxial compressive strength increases with a decreasing L/D ratio due to the strengthening end effect of the hollow cylindrical granite specimens and the change in the failure pattern of these specimens from shear to slabbing.

Influence of the Constitutive Model for Shotcrete on the Predicted Structural Behavior of the Shotcrete Shell of a Deep Tunnel.

The aim of the present paper is to investigate the influence of the constitutive model for shotcrete on the predicted displacements and stresses in shotcrete shells of deep tunnels. Previously proposed shotcrete models as well as a new extended damage plasticity model for shotcrete are evaluated in the context of 2D finite element simulations of the excavation of a stretch of a deep tunnel by means of the New Austrian Tunneling Method. Thereby, the behavior of the surrounding rock mass is described by the commonly used Hoek–Brown model. Differences in predicted evolutions of displacements and stresses in the shotcrete shell, resulting from the different shotcrete models, are discussed and simulation results are compared to available in situ measurement data.

Investigation of mine pressure and deformation due to phosphate ore body excavation based on Hoek-Brown model

Phosphate ore body is an important basis for food supplies and the fine phosphorus chemical industry. With underground mining of the Ph3 seam of Xiongjia Gulf Phosphate ore stope in China as a research background, the excavation process is simulated using the Hoek-Brown model in this paper. Hoek-Brown parameter selection method is proposed first, and the influence of the goaf adjacent to this seam and the rock movement caused by underground mining, pillar yield and surface subsidence deformation are then studied. Finally, the feasibility of implementing a room-and-pillar mining system in the gently inclined phosphate ore body is analyzed. It is pointed out that local ground subsidence should cause enough attention.

Effects of strain rate and confining pressure on the compressive behavior of Kuru granite

Understanding the influence of hydrostatic pressure and loading rate on the strength and fracture behavior of rocks is very important for the development of deep drilling technology. This paper presents a systematic study on the mechanical properties and behavior of Kuru Gray granite at confining pressures up to 225 MPa and at strain rates of 10−6 s−1 and 600 s−1. The low strain rate compression tests were carried out with a servo-controlled hydraulic testing machine with a radial confining chamber, and the dynamic tests with a special split Hopkinson pressure bar device with axial and radial confining pressure chambers. The results show that the rock strength increases significantly with strain rate and confining pressure. At confinements below 20 MPa, the strength of the material increases faster at the higher strain rate, but at confinements higher than this, the effect of confining pressure is stronger at the lower strain rate. The strain rate sensitivity increases when even a small confining pressure is applied. However, the rate sensitivity remains rather constant when the confining pressure is increased above 10 MPa. The parameters of the Hoek–Brown model and an alternative power-law model were calibrated for low and high rate data. Also, the fracture behavior of the rock was found to be strongly dependent on strain rate and confining pressure. At the low strain rate, the samples fail by axial splitting in the unconfined tests, whereas the dynamic unconfined tests result in a complete pulverization of the samples. At high confining pressures the fracture behavior is shear fracture for both studied strain rates.

Application of a transversely isotropic brittle rock mass model in roof support design

Accurate modelling of the potential failure modes in the rock mass is an essential task towards a robust design of roof support systems in coal mines. The use of generalised rock mass properties based on averaged properties (e.g. Hoek–Brown model) has been found to limit the capability to reproduce the actual rock mass behaviour which may include a wide range of interacting and complex failure mechanisms such as shear and tension fracturing of the intact rock and shear and separation of pre-existing discontinuities, including re-activation. Recent studies have also shown that traditional models, such as the Mohr–Coulomb, may not accurately describe the behaviour of the intact rock, particularly for stress induced failures where spalling and slabbing are observed. This is mainly due to the cohesion and friction components of the shear strength of the intact rock not being mobilised at the same rate with strain-softening of the former component playing an essential role in the post peak behaviour. In addition, coal measure rocks are often transversely isotropic, both by way of the preferred orientation of clay particles within the finer grained lithology and by bedding textures and bedding partings, and this is often ignored in computer simulations. A newly developed transversely isotropic brittle rock mass model is applied in the simulation of a hypothetical and simple roadway development. A Cohesion Weakening-Friction Strengthening (CWFS) approach is adopted to describe the intact rock where the mobilisation and strain-softening of the two shear strength components are linked to plastic deformation. The impacts of anisotropy and brittle rock on the development of the excavation disturbed zone or height of softening, as often referred to, are investigated and their implication in the roof support design discussed.

A New Elasto-Viscoplastic Damage Model Combined with the Generalized Hoek–Brown Failure Criterion for Bedded Rock Salt and its Application

According to the requirement of the West–East Gas Transmission Project in China, the solution-mined cavities located in the Jintan bedded salt formation of Jiangsu province will be utilized for natural gas storage. This task is more challenging than conventional salt dome cavern construction and operation due to the heterogeneous bedding layers of the bedded salt formation. A three-dimensional creep damage constitutive model combined with the generalized Hoek–Brown model is exclusively formulated and validated with a series of strength and creep tests for the bedded rock salt. The viscoplastic model, which takes the coupled creep damage and the failure behavior under various stress states into account, enables both the three creep phases and the deformation induced by vicious damage and plastic flow to be calculated. A further geomechanical analysis of the rapid gas withdrawal for the thin-bedded salt cavern was performed by implementing the proposed model in the finite difference software FLAC3D. The volume convergence, the damage and failure propagation of the cavern, as well as the strain rate of the salt around the cavern, were evaluated and discussed in detail. Finally, based on the simulation study, a 7-MPa minimum internal pressure is suggested to ensure the structural stability of the Jintan bedded salt cavern. The results obtained from these investigations provide the necessary input for the design and construction of the cavern project.

Numerical modelling of two stoping methods in two Indian mines using degradation of c and mobilization of φ based on Q-parameters

Two Indian mines are the subject of a comparative study of a strain-softened Hoek–Brown and FLAC 3D modelling, and a novel ‘ c then tan φ’ strain-softening-strain-mobilization approach, using Q-system based input data. This approach is also used with FLAC 3D, using identical stope geometries. The parameters CC and FC, denoting the cohesive component and frictional component of shear strength, are extracted directly from the Q-logging and knowledge of UCS, and are the source of the peak values. Measured deformations, or the strains recorded over the total length of pre-mining installed MPBX, are compared and effectively calibrate the models, in view of the very similar deformations obtained from empirical formulations based on Q using the competence factor approach, as in SRF. The ‘ c then tan φ’ approach appears to give the most realistic match to observations in the mines, including the modelling of a shear band within the back or roof of a stope, rather than at the surface of the stope. The Q-based approach also uses a depth-dependent modulus, and this is perhaps the reason why the strain-softened Hoek–Brown model, without this stiffening with depth, shows ‘global failure’ in a second mine having a wider range of depths within one model, and many openings, since modulus is not increased in standard-method approaches.

Prevention Logic of Rock Burst for Brittle Rock under High Geo-Stress

Rock burst hazard is the main problem of hard rock deep tunnel under high ground stress conditions. The prevention logic of prevention and control of rock burst is proposed by combining local energy release rate index based on the brittle Hoek-Brown model. Stress releasing holes are adopted to lead some energy to release actively and eliminate burst potential of rock burst. Supporting opportunity and parameters were studied by contrasting the magnitude of released energy in FLAC3D numerical software, and then the prevention logic of rock burst is presented. Stress releasing holes relieve stress concentration of the working face, transfer the stress to the deep and reduce the risk of rock burst near the working face. At last, the prevention and control of rock burst for sinping II Hydropower tunnel was analyzed, the results are in good agreement with the actual situations. The proposed method could benefit other deep tunnel projects which have brittle failure.

Laboratory tests and numerical simulations of brittle marble and squeezing schist at Jinping II hydropower station, China

Four 16.7 km-long tunnels with diameters ranging from 12.4 to 14.6 m are now under construction at Jinping II hydropower station along the Yalong River. The tunnels pass through Triassic rocks below Jinping Mountain. The tunnels are characterized with high overburden, long alignment and complex geological conditions. Brittle failure in marble and squeezing in schist are the primary problems in tunnelling. This paper introduces the studies of laboratory tests on Jinping II marble as well as numerical prediction of excavation damaged zone (EDZ) of tunnel section in brittle marble and determination of reinforced concrete lining thickness for restraining time-dependent deformation in the schist tunnel section. Laboratory tests indicate that Jinping II marble presents a complex brittle-ductile-plastic transition behavior of post-peak response with increasing confining pressure. Such behavior can be described numerically with the Hoek-Brown model. The EDZ was calibrated and predicted using both fast Lagrangian analysis of continua (FLAC) and particle flow code (PFC). The predicted result of EDZ in sections with different qualities of rock mass under various overburden pressures is quite helpful in understanding EDZ characterization and support design. A power-law creep model was used to support the lining design, especially in determining the lining thickness. Field convergence measurement data over 19 months were used to calibrate the creep model properties, followed by a sensibility analysis of reinforced concrete lining thickness that was launched to present the maximum lining compressive stress.

Effects of spatially variable weathered rock properties on tunnel behavior

Design parameters commonly used in numerical modeling for tunnel stability analyses tend to be representative (or average) values of global-scale properties. However, the spatial variability of design parameters, such as geotechnical and geological properties, greatly affects the behavior of tunnels during and after construction as well as their long-term responses. Thus, this study presents a simple but robust procedure for stochastic numerical analyses using the finite difference method (FDM) and explores the effects of spatially variable weathered rock properties on various tunnel behaviors, such as deformation, elastic–plastic interface, ground reaction curve, and failure mechanism. It was found that the inherent spatial variability of stiffness and strength parameters affects the deformation behavior of tunnels and even changes its failure mechanism: Elastic modulus for the Mohr–Coulomb model and geological strength index (GSI) for the Hoek–Brown model play a key role in deformation characteristics. Considering the wide range of spatial variability in in-situ deposits, the accurate estimation of elastic modulus and GSI is very important. The spatial variability of the ground can affect the ground reaction behavior and can bring on an unfavorable ground reaction curve (GRC). It can cause an increase in the tunnel support pressure, and can induce a larger displacement than the homogeneous case. The shear failure mechanism of the tunnel can be significantly affected by a large relative correlation length. It is suggested that we should estimate and consider the variability of rock properties accurately as part of a routine tunnel design framework.

Seismic rock slope stability charts based on limit analysis methods

Earthquake effects are commonly considered in the stability analysis of rock slopes and other earth structures. The standard approach is often based on the conventional limit equilibrium method using equivalent Mohr–Coulomb strength parameters ( c and ϕ) in a slip circle slope stability analysis. The purpose of this paper is to apply the finite element upper and lower bound techniques to this problem with the aim of providing seismic stability charts for rock slopes. Within the limit analysis framework, the pseudo-static method is employed by assuming a range of the seismic coefficients. Based on the latest version of Hoek–Brown failure criterion, seismic rock slope stability charts have been produced. These chart solutions bound the true stability numbers within ±9% or better and are suited to isotropic and homogeneous intact rock or heavily jointed rock masses. A comparison of the stability numbers obtained by bounding methods and the limit equilibrium method has been performed where the later was found to predict unconservative factors of safety for steeper slopes. It was also observed that the stability numbers may increase depending on the material parameters in the Hoek–Brown model. This phenomenon has been further investigated in the paper.

Mechanical behavior of Tien-Liao mudstone in hollow cylinder tests

A series of hollow cylinder tests were conducted to study the stress–strain characteristics and the yield surface of Tein-Liao mudstone, a soft rock widely distributed in southwestern Taiwan. These tests were carried out under different stress paths designed to utilize the versatility of the hollow cylinder testing system. The test results were used to establish the yield surface of the mudstone in the principal stress space. The experimental results were used to examine the accuracy of existing yield criteria, such as the Coulomb criterion, the extended Hoek-Brown model, and the Kim-Lade model. Among the three yield criteria examined, the Kim-Lade model provided the best agreement with the experimental results, although both the extended Hoek-Brown model and the Kim-Lade model are adequate for modeling the yield surface of the mudstone.Key words: rock, yield surface, stress path, hollow cylinder test.