An Empirical Failure Criterion for Intact Rocks

The parameter m _i is an important rock property parameter required for use of the Hoek–Brown failure criterion. The conventional method for determining m _i is to fit a series of triaxial compression test data. In the absence of laboratory test data, guideline charts have been provided by Hoek to estimate the m _i value. In the conventional Hoek–Brown failure criterion, the m _i value is a constant for a given rock. It is observed that using a constant m _i may not fit the triaxial compression test data well for some rocks. In this paper, a negative exponent empirical model is proposed to express m _i as a function of confinement, and this exercise leads us to a new empirical failure criterion for intact rocks. Triaxial compression test data of various rocks are used to fit parameters of this model. It is seen that the new empirical failure criterion fits the test data better than the conventional Hoek–Brown failure criterion for intact rocks. The conventional Hoek–Brown criterion fits the test data well in the high-confinement region but fails to match data well in the low-confinement and tension regions. In particular, it overestimates the uniaxial compressive strength (UCS) and the uniaxial tensile strength of rocks. On the other hand, curves fitted by the proposed empirical failure criterion match test data very well, and the estimated UCS and tensile strength agree well with test data.     

岩体经验强度准则估算岩基强度参数和变形模量的方法

本文结合岩体CSIR和NGI分类体系,介绍了由岩体经验强度准则和岩体分类指标RMR和Q估算地基岩体强度参数和变形模量的方法,从而为快速、经济地进行岩基设计提供了重要途径。     

Application of the Christensen Failure Criterion to Intact Rock

The Christensen criterion, originally introduced in materials science, has a simple mathematical form and uniaxial tensile and compressive strength as the only parameters, making it an attractive candidate for rock engineering purposes. In this study, the applicability of the criterion to rock materials is examined. Explicit equations for application of the criterion under biaxial, triaxial compression, triaxial extension, and polyaxial states of stresses are derived. A comprehensive strength data set including the results of tests on synthetic rock, chert dyke, Carrara marble and Westerly granite is utilized to examine the accuracy of the Christensen criterion to the failure of rock material. The two surprising findings about the Christensen criterion are the zero values of tensile strength and the very low slopes of the failure envelope obtained from fitting analyses for chert dyke and Westerly granite. It is shown that the two problems are interrelated and the values of tensile strength tend to zero to produce higher slopes. It is then mathematically proven that the maximum initial slope of the Christensen failure envelope is limited to 4 in triaxial compression and 2.5 in triaxial extension which is considerably lower than the slope of experimental data. The accuracy of the Christensen criterion was found to be significantly lower than the well-established Hoek–Brown criterion. The circular π-plane representations and brittle-to-ductile transition limits from the Christensen criterion are also inconsistent with the observed behavior of rocks.     

Prediction of underground cavity roof collapse using the Hoek–Brown failure criterion

Preventing roof collapse in underground cavities is a challenge to geotechnical engineering. In this study, three independent methods have been used to evaluate the roof collapse of underground rectangular cavities for a range of geometries and rock properties. The rock mass strength has been described by the Hoek–Brown failure criterion. The results of the analysis allow for prediction of roof collapse and to determine whether the failure surface that develops in the rock mass remains localised or extends through the full depth of cover. This is of significance if there are overlying cavities and when estimating surface subsidence.     

Convergence-control approach for rock tunnels reinforced by grouted bolts,using the homogenization concept

The scope of this paper is to introduce a method for the analysis of rock tunnels reinforced by grouted bolts, based on the convergence-control approach. The analytical formulations presented in this paper refers to an elasto-plastic behavior of the rock mass and the latest Hoek and Brown yield criterion (Version 2002). In order to model the reinforced plastic zone, the equivalent material approach was taken into account such that the apparent strength of the rock mass is improved as a consequence of the bolting effect. The general design guides and examples presented are intended to facilitate the comprehension and application of the proposed analytical solution in practice.     

Pseudo-dynamic stability of rock slope considering Hoek–Brown strength criterion

Rock slopes with planar joints or weak structural planes are vulnerable in nature, especially suffering from the natural hazards, instabilities of slopes are more prone to occur. Therefore, concerning to the influence of earthquakes, this paper performs a new procedure to evaluate slope stability in a geomaterial governed by Hoek–Brown strength criterion. A rotational failure mechanism determined by 21 dependent angle variables is introduced to respect the Hoek–Brown strength criterion. The earthquake load is characterized by a modified pseudo-dynamic method that does not violate the zero boundary condition and considers the damping properties of geomaterials. A slice approach is adopted to calculate the earthquake-induced inertial force work rate. The stability number of rock slope is considered to measure the safety. The stability number is formulated as a classical optimization problem controlled by 21 dependent angle variables and a time variable which need to be optimized by the genetic algorithm toolbox. Comparisons with the literature are made to prove rationality and accuracy of the proposed procedure. Parametric study is carried out to reveal the influence of dynamic properties. For engineering application, stability charts are provided for a quick assessment of slope safety.

     

Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

This paper presents a probabilistic analysis to compute the probability density function of the bearing capacity of a strip footing resting on a spatially varying rock mass. The rock is assumed to follow the generalised Hoek–Brown failure criterion. The uniaxial compressive strength of the intact rock (σc) was considered as a random field and the geological strength index was modelled as a random variable. The uncertainty propagation methodology employed in the analysis is the sparse polynomial chaos expansion. A global sensitivity analysis based on Sobol indices was performed. Some numerical results were presented and discussed.     

A comparison of predicted and actual tunnel behaviour in the İstanbul Metro,Turkey

The empirical method proposed by Hoek and Brown is combined with rock mass classifications to predict the deformation behaviour of İstanbul Metro. The rock mass was assessed using both Bieniawski's RMR₈₉ and its strength was estimated from the formula of Hoek and Brown for the RMR₈₉ system. The applicability and validity of the proposed procedure has been checked by comparing the predictions with actual observations. It is found that the predictions agree well with observations in the intact rock. Although there are some differences between predictions and observations in the faulted rock, these probably reflect the inevitable inaccuracies in empirical approaches, and differences in behaviour of the actual support system that is assumed in modelling. In addition, anisotropy features like faults which are not accounted properly in the Hoek et al. equation that assumes isotropic behaviour.     

An empirical relation for parameter mi in the Hoek–Brown criterion of anisotropic intact rocks with consideration of the minor principal stress and stress-to-weak-plane angle

Acta Geotechnica - The parameter mi accounts for the anisotropy of rock strength, and the accurate determination of mi is a primary requirement of the Hoek–Brown (H–B) strength...     

A generalized three-dimensional Hoek–Brown strength criterion

Summary  Although the Hoek–Brown strength criterion has been widely used in rock mechanics and rock engineering, it does not take account of the influence of the intermediate principal stress. Much evidence, however, has been accumulating to indicate that the intermediate principal stress does influence the rock strength in many instances. Therefore, researchers have developed three-dimensional (3D) versions of the Hoek–Brown strength criterion. In this paper, three existing 3D versions of the Hoek–Brown strength criterion are reviewed and evaluated. The evaluation shows that all of the three 3D versions of the Hoek–Brown strength criterion have limitations. To address the limitations, a generalized 3D Hoek–Brown criterion is proposed by modifying the generalized Hoek–Brown strength criterion. The proposed 3D criterion not only inherits the advantages of the Hoek–Brown strength criterion but can take account of the influence of the intermediate principal stress. At a 2D stress state (triaxial or biaxial), the proposed 3D criterion will simply reduce to the form of the generalized Hoek–Brown strength criterion. To validate the proposed 3D strength criterion, polyaxial or true triaxial compression test data of intact rocks and jointed rock masses has been collected from the published literature. Predictions of the proposed generalized 3D Hoek–Brown strength criterion are in good agreement with the test data for a range of different rock types. The difference of the proposed generalized 3D Hoek–Brown strength criterion from and its advantages over the existing 3D versions of the Hoek–Brown strength criterion are also discussed. It should be noted that the proposed 3D criterion is empirical in nature because it is an extension of the 2D Hoek–Brown strength criterion, which is empirical. Because of the non-convexity of the yield surface for a biaxial stress state, the proposed 3D criterion may have problems with some stress paths. Correspondence: L. Zhang, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, Arizona 85721, USA     

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 ?) 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.     

Considerations on strength of intact sedimentary rocks

This study presents the results of laboratory testing of sedimentary rocks under point loading as well as in uniaxial and triaxial compression. From the statistical analysis of the data, different conversion factors relating uniaxial compressive and point loading strength were determined for soft to strong rocks. Additionally, the material constant m_i, an input parameter for the Hoek and Brown failure criterion, was also estimated for different limestone samples by analysing the results from a series of triaxial compression tests under different confining pressures. The uniaxial compressive strength (UCS) of intact rocks, as estimated from the point load index using conversion factors, together with the Hoek–Brown constant m_i, and the Geological Strength Index (GSI) constitute the parameters for the calculation of the strength and deformability of rock masses.     

Bearing capacity of foundations on rock mass using the method of characteristics

The method of stress characteristics has been used for computing the ultimate bearing capacity of strip and circular footings placed on rock mass. The modified Hoek‐and‐Brown failure criterion has been used. Both smooth and rough footing‐rock interfaces have been modeled. The bearing capacity has been expressed in terms of nondimensional factors N_σ0 and N_σ, corresponding to rock mass with (1) γ = 0 and (2) γ ≠ 0, respectively. The numerical results have been presented as a function of different input parameters needed to define the Hoek‐and‐Brown criterion. Slip line patterns and the pressure distribution along the footing base have also been examined. The results are found to compare generally well with the reported solutions.     

Instantaneous Friction Angle and Cohesion of 2-D and 3-D Hoek–Brown Rock Failure Criteria in Terms of Stress Invariants

The Mohr–Coulomb (M–C) failure criterion is one of the most widely used failure criteria in rock mechanics, although it has a number of shortcomings such as neglecting the nonlinear strength observed in rock or the effect of the intermediate principal stress σ ₂. Other failure criteria have been proposed to effectively include in the predictions of failure the non-linear response of rock to confinement or the effects of the intermediate principal stress. The M–C criterion is still widely used, and it is arguably the criterion most used in practice. For example, stability evaluations of shallow rock structures such as slopes and foundations are routinely carried out by estimating a friction angle and a cohesion of the rock mass. To include the dependency of cohesion and friction angle on stresses, efforts are being made to estimate equivalent values of the M–C parameters for the range of stresses applicable to a particular design. The paper suggests a new and convenient approach to find the equivalent friction angle and cohesion from any failure criterion that can be expressed in terms of the Nayak and Zienkiewicz’s stress invariants. To demonstrate the capabilities and application of the methodology, the new approach is applied to two failure criteria: the Hoek–Brown (H–B) criterion and the Hoek–Brown and Willam–Warnke (HB–WW) criterion, 2-D and 3-D failure criteria, respectively. Results from the new method, in terms of equivalent friction and cohesion for the H–B criterion, are exactly the same as the results obtained from Balmer’s theory, which confirms the validity of the new method. The predicted equivalent friction and cohesion for the HB–WW criterion show a dependency on σ ₂, which does not occur for a 2-D failure criterion.     

Seismic bearing capacity of shallow foundations near rock slopes using the generalized Hoek–Brown criterion

The seismic bearing capacity of shallow foundations resting on a modified Hoek–Brown rock mass is investigated within the framework of the kinematic approach of limit analysis theory. The analysis focuses on evaluating the reduction in bearing capacity induced by seismic loading and by the proximity of a rock slope. A pseudo‐static approach is adopted to account for the earthquake effects for the seismic bearing capacity evaluations. At the rock material level, the closed‐form expressions previously obtained for the support functions of the rock failure criterion allow the implementation of different failure mechanisms families, and thus to derive rigorous upper bounds estimates of the load‐bearing capacity in both static and seismic conditions. The effects of geometrical, strength and loading parameters are assessed through a large number of parametric computations. Finally, design tables are presented for practical use in rock engineering. Copyright © 2010 John Wiley & Sons, Ltd.     

Distinct element simulation of ultimate bearing capacity in jointed rock foundations

In this paper, distinct element method numerical modeling is applied to evaluate bearing capacity of strip footing rested on anisotropic discontinuous rock mass. As yet, a little work has been carried out to investigate the effect of joint set orientation on the bearing capacity of rock mass. Generally, the overall behavior of rock mass under loading is very complicated and such analysis should include deformation determination, sliding along discontinuities and failure of rock material. Failure mechanism of rock mass depended on both geometrical parameters of joint sets and strength parameters of rock mass. In this research, it is assumed that rock mass contains one joint set, and therefore the anisotropy in bearing capacity and rock behavior is only due to the existence and orientation of the joint set. In this study, by assuming constant strength parameters and using Mohr–Coulomb failure criterion for the single joint set and nonlinear Hoek–Brown failure criterion for rock material, variation of the bearing capacity values and the type of failure mechanism of rock mass with different joint set dips is investigated. The obtained results indicate that the ultimate bearing capacity of rock mass containing one joint set varies between 27 and 86 % of intact rock.     

A nonlinear deformation modulus based on rock mass classification

Summary This paper describes the development of an empirical nonlinear stress dependent expression for the deformation modulus of rock masses based on rock mass classification. The expression defines the deformation modulus as the ratio of the deviator stress at failure to the major principal strain at failure. The Hoek and Brown failure criterion is used to predict the deviator stress at failure. Research was directed toward developing a failure criterion defining the major principal strain at failure. The expression for the deformation modulus was extended to rock mass conditions through correlations with observed deformations from case history studies and predicted deformations from finite element analyses.