Hoek Brown Research Articles

Range of confining pressures for the Hoek–Brown criterion

Engineering in weak rock using modern, computer-based design techniques is notoriously challenging because the laboratory tests and classification methods routinely carried out on such materials yield rock parameters. Such parameters are not directly compatible with the most widely used constitutive model of those computer-based methods – that is, the Mohr–Coulomb failure envelope. There is no direct relationship between Mohr–Coulomb (linear) and Hoek–Brown (non-linear) parameters, and marrying those two failure envelopes is notoriously difficult. When formulating a failure criterion for rock materials, based on rock and rock mass parameters, a correlation with the Mohr–Coulomb failure criterion has been proposed, which can be obtained from the computer program RocLab. One of the essential input parameters for such correlation is called the maximum confining pressure. This parameter directly affects the correlated shear strength parameters on the Mohr–Coulomb failure envelope and hence choosing it with care is the key to producing a safe, yet economic design. The present paper discusses the use of several methods for determining the maximum confining pressure to derive Mohr–Coulomb parameters for weak rock in slope and retaining wall applications. The suitability of each method considered is discussed against the results of back analyses, and design recommendations are proposed.

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.

Application of quantitative risk assessment in wellbore stability analysis

Abstract Elastic and strength parameters, together with pore pressure and in-situ stresses are key parameters required to be known for determination of safe mud weight window (MWW) in vertical wellbores. Estimation of these parameters, however, is subjected to wide uncertainties mainly due to lack of adequate calibration information including lab and field test data. While there are literatures on the applications of probabilistic and risk analysis on wellbore stability evaluation, limited numbers of publications report on the impact of the chosen failure criteria in estimation of safe MWW under uncertain condition. In this study, data corresponding to a wellbore located in south part of Iran was analyzed using quantitative risk assessment to consider the effect of uncertainty on estimation of safe MWW using different failure criteria. The results indicated that Mogi–Coulomb and Hoek–Brown are more robust against the uncertainty of input parameters and mud weight used for this wellbore could have slightly been increased to reduce the shear failure of the borehole wall. The uncertainty in the input data might also be very critical for casing design when only a simple margin together with pore and fracture pressures are used to select the grade of the casing against burst or collapse loads. It was also noted based on sensitivity analysis that the maximum horizontal stress is the most effective parameter in estimation of MWW. This emphasizes the importance of a reliable estimation of in-situ stresses for safe drilling. The results presented here are based on a single case study, and further studies are still required to get any ultimate conclusion.

Bearing capacity of circular footings over rock mass by using axisymmetric quasi lower bound finite element limit analysis

The ultimate bearing capacity of a circular footing, placed over rock mass, is evaluated by using the lower bound theorem of the limit analysis in conjunction with finite elements and nonlinear optimization. The generalized Hoek–Brown (HB) failure criterion, but by keeping a constant value of the exponent, α=0.5, was used. The failure criterion was smoothened both in the meridian and π planes. The nonlinear optimization was carried out by employing an interior point method based on the logarithmic barrier function. The results for the obtained bearing capacity were presented in a non-dimensional form for different values of GSI, mi, σci/(γb) and q/σci. Failure patterns were also examined for a few cases. For validating the results, computations were also performed for a strip footing as well. The results obtained from the analysis compare well with the data reported in literature. Since the equilibrium conditions are precisely satisfied only at the centroids of the elements, not everywhere in the domain, the obtained lower bound solution will be approximate not true.

Theoretical solution for displacement and stress in strain-softening surrounding rock under hydraulic-mechanical coupling

This study focuses on the stress and displacement of a circular opening that is excavated in a strain-softening rock mass under hydraulic-mechanical coupling. It follows the generalized Hoek-Brown (H-B) failure criterion. Moreover, an improved numerical method and stepwise procedure are proposed. This method considers the deterioration of the strength, deformation, and dilation angle. It also incorporates the hydraulic-mechanical coupling and the variation of elastic strain in the plastic region. Several examples are conducted to demonstrate the validity and accuracy of the proposed solution through MATLAB programming and FLAC software. Parametric studies are also conducted to highlight the influence of hydraulic–mechanical coupling on stress and displacement. Results show that in this case, stress confinement is lower and tunnel convergences are higher than the corresponding stresses and displacements obtained when those factors are not considered. The displacement and plastic radius are also larger than those obtained when hydraulic-mechanical coupling is not considered.

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.

Theoretical solutions for a circular opening in an elastic–brittle–plastic rock mass incorporating the out-of-plane stress and seepage force

Seepage force is simplified as seepage volumetric force in the stress field along the radial direction. Out-of-plane stress and seepage force are incorporated, and the theoretical solutions for stress, displacement, and plastic radius of a circular opening for the elastic-brittle-plastic and elastic-plastic rock mass are proposed based on the Mohr–Coulomb (MC) and generalized Hoek-Brown (HB) failure criteria. The presented solution and Wang's solution (2012) are compared, and the corrected version of the proposed method is validated. Numerical examples of the proposed method based on the MC and generalized HB failure criteria reveal that the distributions of stress and displacement in the surrounding rock of the tunnel are significantly influenced by seepage force and out-ofplane stress. Displacement and plastic radius when seepage force and out-of-plane stress are considered are larger than those when the seepage force is not considered; the regulations of stress, however, run opposite. The results of displacement and plastic radius based on the generalized HB failure criterion are larger than those based on the MC failure criterion.

Evaluation of the Impact of Hydrostatic Pressure and Lode Angle on the Strength of the Rock Mass Based on the Hoek–Brown Criterion

Abstract Determination of the global uniaxial compressive strength of rock mass on the basis of the Hoek-Brown failure criterion requires knowledge of the strength parameters: cohesion and the angle of internal friction. In the conventional method for the determination of these parameters given by Balmer, they are expressed by the minimum principal stress. Thus, this method does not allow for the assessment of an impact of hydrostatic pressure and stress path on the value of cohesion, friction angle and global uniaxial compression of rock mass. This problem can be eliminated by using the Hoek-Brown criterion expressed by the invariants of the stress state. The influence of hydrostatic pressure and the Lode angle on the strength parameters of the rock mass was analysed.

Using different rock failure criteria in wellbore stability analysis

The minimum and the maximum allowable drilling mud pressures for two wellbores are calculated analytically by Mohr–Coulomb, Hoek–Brown and Mogi–Coulomb failure criteria and numerically using a Fish program which is implemented in FLAC. The allowable drilling mud pressures that resulted from these methods are compared with each other and then with the field data: two wellbores in two Iranian oilfields are investigated as case studies to assess the results. Finally the criterion that matches closely field data is identified. It is shown that the minimum allowable drilling mud pressure obtained from the Mogi–Coulomb criterion is close to actual data collected during drilling operation. Hence, this criterion is considered more appropriate for wellbore stability analysis than the other two criteria.

A Simple Three-dimensional Failure Criterion for Rocks Based on the Hoek–Brown Criterion

A simple three-dimensional (3D) failure criterion for rocks is proposed in this study. This new failure criterion inherits all the features of the Hoek–Brown (HB) criterion in characterizing the rock strength in triaxial compression, and also accounts for the influence of the intermediate principal stress. The failure envelope surface has non-circular convex sections in the deviatoric stress plane, and is smooth, except in triaxial compression. In particular, the failure function of the proposed criterion has a similar simple expression as that of the HB criterion in terms of the principal stresses. The material parameters can be calibrated from tests in conventional triaxial compression, and predictions using this new criterion generally compare well with polyaxial testing data for a variety of rocks. Comparison of the new 3D failure criterion and two existing criteria demonstrates that the new failure criterion performs better in characterizing the rock strength. A unified expression for the 3D failure criteria is further provided, retaining the features of the classical criteria and recovering several existing ones as specific cases.

Reliability Analysis of Circular Tunnel Face Stability Obeying Hoek–Brown Failure Criterion Considering Different Distribution Types and Correlation Structures

AbstractA reliability analysis is presented of a circular tunnel face driven by a pressurized shield in a highly fractured Hoek–Brown (HB) rock mass. A limit analysis approach is employed to define a limit state function (LSF) based on an advanced rotational mechanism. The objective is to analyze how different reliability methods, different assumptions about distribution types and correlation structures of the random variables, and different tunnel sizes and support pressures influence the computed reliability results. Results indicate that the reliability method has a limited influence on reliability results, hence suggesting that the LSF is not highly nonlinear and that a computationally efficient method can be employed; and that all random variables under consideration are resistance variables, with their assumed distribution types and correlation structures having a significant effect on the reliability results. This emphasizes the importance of an adequate characterization of geotechnical uncertainti...

Predictive analysis of stress regime and possible squeezing deformation for super-long water conveyance tunnels in Pakistan

The prediction of the stress field of deep-buried tunnels is a fundamental problem for scientists and engineers. In this study, the authors put forward a systematic solution for this problem. Databases from the World Stress Map and the Crustal Stress of China, and previous research findings can offer prediction of stress orientations in an engineering area. At the same time, the Andersonian theory can be used to analyze the possible stress orientation of a region. With limited in-situ stress measurements, the Hoek–Brown Criterion can be used to estimate the strength of rock mass in an area of interest by utilizing the geotechnical investigation data, and the modified Sheorey’s model can subsequently be employed to predict the areas’ stress profile, without stress data, by taking the existing in-situ stress measurements as input parameters. In this paper, a case study was used to demonstrate the application of this systematic solution. The planned Kohala hydropower plant is located on the western edge of Qinghai–Tibet Plateau. Three hydro-fracturing stress measurement campaigns indicated that the stress state of the area is SH>Sh>SV or SH>SV>Sh. The measured orientation of SH is NEE (N70.3°–89°E), and the regional orientation of SH from WSM is NE, which implies that the stress orientation of shallow crust may be affected by landforms. The modified Sheorey model was utilized to predict the stress profile along the water sewage tunnel for the plant. Prediction results show that the maximum and minimum horizontal principal stresses of the points with the greatest burial depth were up to 56.70 and 40.14MPa, respectively, and the stresses of areas with a burial depth of greater than 500m were higher. Based on the predicted stress data, large deformations of the rock mass surrounding water conveyance tunnels were analyzed. Results showed that the large deformations will occur when the burial depth exceeds 300m. When the burial depth is beyond 800m, serious squeezing deformations will occur in the surrounding rock masses, thus requiring more attention in the design and construction. Based on the application efficiency in this case study, this prediction method proposed in this paper functions accurately.

일반화된 Hoek-Brown 파괴조건식에 내포된 접선점착력과 접선마찰각의 상관성

The generalized Hoek-Brown (H-B) function provides a unique failure condition for a jointed rock mass, in which the strength parameters of rock mass are deduced from the intact values by use of the GSI value. Since it is actually the only failure criterion which accounts for the rock mass conditions in a systematic manner, the generalized H-B criterion finds many applications to the various rock engineering projects. Its nonlinear character, however, limits more active usage of this criterion. Accordingly, many attempts have been made to understand the generalized H-B condition in the framework of the M-C function. This study presents the closed-form expression relating the tangential cohesion to the tangential friction angle, which is derived by the non-dimensional stress transformation of the generalized H-B criterion. By use of the derived equation, it is investigated how the relationship between the tangential cohesion and friction angle of the generalized H-B criterion varies with the quality of rock masses. When only the variation of GSI value is considered, it is found that the tangential friction angle decreases with the increase of GSI, while the tangential cohesion increases with GSI value.

A new experimental approach to the pushover analysis of masonry buildings

The present paper proposes a new approach for the pushover analysis of masonry structures: the shear strength of a masonry building is determined using experimental values combined with an analogy with rock masses. The shear strength criterion of Mohr–Coulomb is used, the values for the cohesion and friction angle being obtained by the non-linear criterion of Hoek–Brown for rock masses. The results of the proposed method are in good agreement with the values of the shear strength obtained using the Italian Code (NTC 08) for an LC3 level of knowledge of a masonry structure.

Similarity solution for a spherical or circular opening in elastic-strain softening rock mass

This paper deals with the similarity solution for a spherical or circular opening excavated in elastic-strain softening rock mass compatible with a linear Mohr–Coulomb (M–C) or a nonlinear Hoek–Brown (H–B) yield criterion. A similarity solution for stresses and displacement is presented by replacing the partial differential equations from stress equilibrium, constitutive law, and consistency equations with first-order ordinary differential equations. The Runge–Kutta (R–K) method is used to solve those first-order ordinary differential equations. Some measures are discussed to solve numerical instability problems in the use of R–K driver with adaptive steps. For comparison, the simple numerical stepwise procedure for a spherical opening is also presented by modifying the previous procedure for a circular opening. Three data sets are used to show the accuracy and practical application of the proposed methods. The results show the importance in choosing the solver for the system of ordinary differential equations and the initial values in the similarity solution.

A simplified approach to directly consider intact rock anisotropy in Hoek–Brown failure criterion

Many rock types have naturally occurring inherent anisotropic planes, such as bedding planes, foliation, or flow structures. Such characteristic induces directional features and anisotropy in rocks' strength and deformational properties. The Hoek–Brown (H–B) failure criterion is an empirical strength criterion widely applied to rock mechanics and engineering. A direct modification to H–B failure criterion to account for rock anisotropy is considered as the base of the research. Such modification introduced a new definition of the anisotropy as direct parameter named the anisotropic parameter (Kβ). However, the computation of this parameter takes much experimental work and cannot be calculated in a simple way. The aim of this paper is to study the trend of the relation between the degree of anisotropy (Rc) and the minimum value of anisotropic parameter (Kmin), and to predict the Kmin directly from the uniaxial compression tests instead of triaxial tests, and also to decrease the amount of experimental work.

Simplified Method for Estimating the Hoek-Brown Constant for Intact Rocks

The constant mi is a fundamental parameter required for the Hoek-Brown (HB) criterion to estimate the strength of rock materials. To calculate mi values, triaxial tests need to be carried out; however, triaxial tests are time-consuming and expensive. In this research, a simplified method is proposed that can estimate mi values using information on rock types and the uniaxial compressive strength (UCS) of intact rocks. To evaluate the reliability of the proposed method, mi values estimated from the proposed method are used in the HB criterion to predict intact rock strength. The predicted intact rock strength is then tested against experimental intact rock strength using existing triaxial tests and tests in the authors’ laboratory. Results from the comparison show that mi values calculated from the proposed method can be used reliably in the HB criterion for estimating intact rock strength when triaxial test data are not available.

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.

Three-dimensional numerical analysis for rock slope stability using shear strength reduction method

Existing numerical modeling of three-dimensional (3D) slopes is performed mainly by using the shear strength reduction (SSR) technique based on the linear Mohr–Coulomb (MC) criterion, whereas the nonlinear failure criterion for rock slope stability is seldom used in slope modeling. However, it is known that rock mass strength is a nonlinear stress function and that, therefore, the linear MC criterion does not agree with the rock mass failure envelope very well. In this research, a nonlinear SSR technique is proposed that can use the Hoek–Brown (HB) criterion to represent the nonlinear behavior of a rock mass in the FLAC3D program to analyze 3D slope stability. Extensive case studies are carried out to investigate the influence of the convergence criterion and boundary conditions on the 3D slope modeling. Results show that the convergence criterion used in the 3D model plays an important role, not only in terms of calculation of the factor of safety (FOS), but also in terms of the shape of the failure surface. The case studies also demonstrate that the value of the FOS for a given slope will be significantly influenced by the boundary condition when the slope angle is less than 50°.

Practical application of failure criteria in determining safe mud weight windows in drilling operations

Wellbore instability is reported frequently as one of the most significant incidents during drilling operations. Analysis of wellbore instability includes estimation of formation mechanical properties and the state of in situ stresses. In this analysis, the only controllable parameter during drilling operation is the mud weight. If the mud weight is larger than anticipated, the mud will invade into the formation, causing tensile failure of the formation. On the other hand, a lower mud weight can result in shear failures of rock, which is known as borehole breakouts. To predict the potential for failures around the wellbore during drilling, one should use a failure criterion to compare the rock strength against induced tangential stresses around the wellbore at a given mud pressure. The Mohr–Coulomb failure criterion is one of the commonly accepted criteria for estimation of rock strength at a given state of stress. However, the use of other criteria has been debated in the literature. In this paper, Mohr–Coulomb, Hoek–Brown and Mogi–Coulomb failure criteria were used to estimate the potential rock failure around a wellbore located in an onshore field of Iran. The log based analysis was used to estimate rock mechanical properties of formations and state of stresses. The results indicated that amongst different failure criteria, the Mohr–Coulomb criterion underestimates the highest mud pressure required to avoid breakouts around the wellbore. It also predicts a lower fracture gradient pressure. In addition, it was found that the results obtained from Mogi–Coulomb criterion yield a better comparison with breakouts observed from the caliper logs than that of Hoek–Brown criterion. It was concluded that the Mogi–Coulomb criterion is a better failure criterion as it considers the effect of the intermediate principal stress component in the failure analysis.