Upper bound solution for ultimate bearing capacity with a modified Hoek–Brown failure criterion

Conventional calculations of ultimate bearing capacity are formulated in terms of a linear Mohr–Coulomb (MC) failure criterion. However, experimental data shows that the strength envelops of almost all types of rocks are nonlinear over the wide range of normal stresses. In this paper, the strength envelope of rock masses is considered to follow a modified Hoek–Brown failure criterion that is a nonlinear failure criterion. Two different kinds of techniques are used to develop the ultimate bearing capacity in the framework of limit analysis in plasticity.The first technique is the generalized tangential technique proposed by the authors. Based on a multi-wedge translation failure mechanism, a generalized tangential technique is used to formulate the bearing capacity problem as a classical optimization problem where the objective function, which is to be minimized with respects to the parameters of failure mechanism and the location of tangency point, corresponds to the dissipated power. The minimum solution is obtained by optimization. Using the technique, the effects of rock weight under the base of the footing and surcharge load can be considered. The second technique, “tangential” line technique, was originally used to analyze slope stability with a nonlinear failure criterion. In order to assess the validity of the proposed method, the “tangential” line technique is extended to evaluate the bearing capacity factor with the nonlinear failure criterion, where the effects of self-weight and surcharge load on the bearing capacity cannot considered. The second technique, however, has to utilize the previously calculated ultimate bearing capacity factor with a linear MC failure criterion. Numerical results are compared and presented for practical use in rock engineering.     

Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto-plastic rock masses

We present a novel methodology for estimation of equivalent Mohr–Coulomb strength parameters that can be used for design of supported tunnels in elasto-plastic rock masses satisfying the non-linear empirical Hoek–Brown failure criterion. We work with a general adimensional formulation of the Hoek–Brown failure criterion in the space of normalized Lambe's variables for plane stress, and we perform linearization considering the stress field in the plastic region around the tunnel. The procedure is validated using analytical solutions to a series of benchmark test cases. Numerical solutions are also employed to validate the procedure in cases for which analytical solutions are not available. Results indicate that the stress field in the plastic region around the tunnel, as well as the linearization method employed and the quality of the rock mass, has a significant impact on computed estimates of equivalent Mohr–Coulomb strength parameters. Results of numerical analyses also show that our proposed linearization method can be employed to estimate loads and moments on the tunnel support system. We recommend the equating model responses (EMR) method to compute equivalent Mohr–Coulomb strength parameters when the tunnel support pressure is accurately known, and we further show that our newly introduced linearization method can be employed as an alternative to the best fitting in the existing stress range (BFe) and best fitting in an artificial stress range (BFa) methods, providing performance estimates that are generally better than estimates of the BFe and BFa methods when differences with the response of the Hoek–Brown rock mass are of engineering significance (say more than 10%).     

Equivalent Mohr–Coulomb and generalized Hoek–Brown strength parameters for supported axisymmetric tunnels in plastic or brittle rock

The evaluation of equivalent Mohr–Coulomb (M–C) strength parameters to the prototype Hoek–Brown (H–B) ones for tunnels has been tackled in different ways for many years. The extension of the H–B criterion to the generalized one has made the challenge even greater. Most of the latest methods did not account for the effect of the support pressure and none gave formulae for equivalent parameters of supported or brittle rock. Here, an almost exact explicit solution for the evaluation of the critical pressure, of a tunnel in a rock mass satisfying the generalized H–B criterion, is initially investigated. Then, formulae are derived for the evaluation of equivalent parameters, of either elastoplastic or elastic–brittle plastic rock. They are based either on a best fitting procedure of the two envelopes or on the equation of selected responses of the models. Supported tunnels in equivalent M–C rock masses are then validated against those excavated in the prototype H–B rock masses.     

Stability charts for rock slopes based on the Hoek–Brown failure criterion

This paper uses numerical limit analysis to produce stability charts for rock slopes. These charts have been produced using the most recent version of the Hoek–Brown failure criterion. The applicability of this criterion is suited to isotropic and homogeneous intact rock, or heavily jointed rock masses. The rigorous limit analysis results were found to bracket the true slope stability number to within ±9% or better, and the difference in safety factor between bound solutions and limit equilibrium analyses using the same Hoek–Brown failure criterion is less than 4%. The accuracy of using equivalent Mohr–Coulomb parameters to estimate the stability number has also been investigated. For steep slopes, it was found that using equivalent parameters produces poor estimates of safety factors and predictions of failure surface shapes. The reason for this lies in how these equivalent parameters are estimated, which is largely to do with estimating a suitable minor principal stress range. In order to obtain better equivalent parameter solutions, this paper proposes new equations for estimating the minor principal stress for steep and gentle slopes, which can be used to determine equivalent Mohr–Coulomb parameters.     

Stability analysis of vertical boreholes using the Mogi–Coulomb failure criterion

A main aspect of wellbore stability analysis is the selection of an appropriate rock failure criterion. The most commonly used criterion for brittle failure of rocks is the Mohr–Coulomb criterion. This criterion involves only the maximum and minimum principal stresses, σ₁ and σ₃, and therefore assumes that the intermediate stress σ₂ has no influence on rock strength. As the Mohr–Coulomb criterion ignores the strengthening effect of the intermediate stress, it is expected to be too conservative in estimating the critical mud weight required to maintain wellbore stability. Recently, Al-Ajmi and Zimmerman [Relationship between the parameters of the Mogi and Coulomb failure criterion. Int J Rock Mech Min Sci 2005;42(3):431–39.] developed the Mogi–Coulomb failure criterion, and showed that it is reasonably accurate in modelling polyaxial failure data from a variety of rocks. We then develop a model for the stability of vertical boreholes, using linear elasticity theory to calculate the stresses, and the fully-polyaxial Mogi–Coulomb criterion to predict failure. Our model leads to easily computed expressions for the critical mud weight required to maintain wellbore stability.     

Strength of massive to moderately jointed hard rock masses

The Hoek-Brown(HB) failure criterion and the geological strength index(GSI) were developed for the estimation of rock mass strength in jointed and blocky ground where rock mass failure is dominated by sliding along open joints and rotation of rock blocks. In massive, veined and moderately jointed rock in which rock blocks cannot form without failure of intact rock, the approach to obtain HB parameters must be modified. Typical situations when these modifications are required include the design of pillars,excavation and cavern stability, strainburst potential assessment, and tunnel support in deep underground conditions(around s1/s ci 0.15, where s1 is the major principal compressive stress and s ciis the unconfined compressive strength of the homogeneous rock) in hard brittle rocks with GSI ! 65. In this article, the strength of massive to moderately jointed hard rock masses is investigated, and an approach is presented to estimate the rock mass strength envelope using laboratory data from uniaxial and triaxial compressive strength tests without reliance on the HB-GSI equations. The data from tests on specimens obtained from massive to moderately jointed heterogeneous(veined) rock masses are used to obtain the rock and rock mass strengths at confining stress ranges that are relevant for deep tunnelling and mining;and a methodology is presented for this purpose from laboratory data alone. By directly obtaining the equivalent HB rock mass strength envelope for massive to moderately jointed rock from laboratory tests,the HB-GSI rock mass strength estimation approach is complemented for conditions where the GSIequations are not applicable. Guidance is also provided on how to apply the proposed approach when laboratory test data are not or not yet available.     

Application of critical plane approach to the prediction of strength anisotropy in transversely isotropic rock masses

A formulation describing the strength anisotropy of transversely isotropic rock masses subjected to a three-dimensional stress state is proposed based on the critical plane approach. It is assumed that the initiation of cracking is governed by the Hoek–Brown failure criterion, and the anisotropy of the strength is described through the orientation dependence of the strength parameters m and s. Using direct optimization of failure function, the direction of potential failure plane, on which the failure function reaches maximum, is determined. True triaxial compression tests as well as conventional triaxial tests are simulated in order to verify the performance of the proposed formulation.     

A simple procedure for ground response curve of circular tunnel in elastic-strain softening rock masses

This paper presents a simple procedure for the ground response curve of a circular tunnel excavated in elastic-strain softening rock mass compatible with a linear Mohr–Coulomb or a nonlinear Hoek–Brown yield criterion. The numerical stepwise procedure proposed by Brown et al. [Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E., (1983). Ground response curves for rock tunnels. J. Geotech. Eng. ASCE 109, 15–39] is modified by including the effects of elastic strain increments and variable dilatancy within the plastic region. The accuracy and practical application of the proposed procedure are shown through some examples. Four different combinations of dilatancy angle and softening parameter are considered to investigate the effects of elastic strain increments and variable dilatancy within the plastic region. The effects of variable dilatancy and peak dilatancy angle on the ground response curve are investigated for tunnels in poor-to-good-quality rock masses. The results show the importance of correctly estimating peak dilatancy angle in elastic-perfectly plastic and elastic-strain softening Hoek–Brown media.     

Applicability of Mohr–Coulomb and Hoek–Brown strength criteria to the dynamic strength of brittle rock

A series of dynamic uniaxial and triaxial compression, uniaxial tension and unconfined shear tests were conducted on the Bukit Timah granite of Singapore. The results are analyzed in this paper in order to examine the validity and applicability of the Mohr–Coulomb and the Hoek–Brown criteria to the rock material strength properties subjected to dynamic loads. The study indicates that rock material strength under dynamic loads can be approximately described by the Mohr–Coulomb criterion, at low confining pressure range. The change of strength is primarily due to the variation of cohesion with loading rate. The rock material strength under dynamic loads is better described by the Hoek–Brown criterion. Assessment of the Hoek–Brown criterion shows that the uniaxial compressive strength increases with increasing loading rate, and the parameter m appears unaffected by the loading rate.     

Influence of degree of interlock on confined strength of jointed hard rock masses

The strength of jointed rock mass is strongly controlled by the degree of interlock between its constituent rock blocks. The degree of interlock constrains the kinematic freedom of individual rock blocks to rotate and slide along the block forming joints. The Hoek–Brown (HB) failure criterion and the geological strength index (GSI) were developed based on experiences from mine slopes and tunneling projects in moderately to poorly interlocked jointed rock masses. It has since then been demonstrated that the approach to estimate the HB strength parameters based on the GSI strength scaling equations (called the ‘GSI strength equations’) tends to underestimate the confined peak strength of highly interlocked jointed rock masses (i.e. GSI > 65), where the rock mass is often non-persistently jointed, and the intact rock blocks are strong and brittle. The estimation of the confined strength of such rock masses is relevant when designing mine pillars and abutments at great depths, where the confining pressure is high enough to prevent block rotation and free sliding on block boundaries. In this article, a grain-based distinct element modeling approach is used to simulate jointed rock masses of various degrees of interlock and to investigate the influences of block shape, joint persistence and joint surface condition on the confined peak strengths. The focus is on non-persistently jointed and blocky (persistently jointed) rock masses, consisting of hard and homogeneous rock blocks devoid of any strength degrading defects such as veins. The results from this investigation confirm that the GSI strength equations underestimate the confined strength of highly interlocked and non-persistently jointed rock masses. Moreover, the GSI strength equations are found to be valid to estimate the confined strength of persistently jointed rock masses with smooth and non-dilatant joint surfaces.     

Tensile resistance of rock anchors

Using the Euler's variational method and assuming a rock mass failure criterion of Hoek and Brown type (Hoek E, Brown ET. Empirical strength criterion for rock masses. J Geotech Eng Division, American Society of Civil Engineers 1980;106(GT9)1013–35. Hoek E, Wood D, Shah S. A modified Hoek-Brown criterion for jointed rock masses. In: Hudson JA, editor. Proc., Rock Characterization Symp. of ISRM: Eurock 92. London: British Geotechnical Society, 1992. p. 209–14), the tensile resistance of rock anchors is obtained. The rupture surface shape through the rock mass is also obtained and checked against some published data. Depending on the slenderness ratio (L/D; L=anchor length, D=anchor diameter) two types of failure surfaces are obtained; short and long ones. This second type of surface is a complex one and it is composed by a cylinder and a surface of revolution with an hyperbolic type shape. The values of the ultimate pullout strength, depending on the rock type and its Bieniawski indexes, are obtained and compared with the values published in technical literature. A reasonable agreement is obtained.     

A nonlinear strength criterion for rock-like materials based on fracture mechanics

A nonlinear strength criterion for rock-like materials is developed in this paper. Taking α as an angle of micro-failure orientation in rock-like materials, a formulation between α and load is derived from a mixed-mode fracture criterion based on linear elastic fracture mechanics. According to micro-failure experimental phenomena of rock-like materials, a failure characteristic parameter under triaxial compression condition is chosen, which is relevant to confining pressure and is an invariant. A theoretical nonlinear strength criterion is also derived, which is exactly in the same mathematical form as the original Hoek–Brown empirical strength criterion. In addition, it is also found that the coefficient m in the Hoek–Brown criterion has physical meaning which is related to the ratio between the uniaxial compressive strength and the uniaxial tensile strength.     

Analytical solution for a circular opening in an elastic–brittle–plastic rock

This paper deals with the analytical solutions for the prediction of displacements around a circular opening in an elastic–brittle–plastic rock mass compatible with a linear Mohr–Coulomb or a nonlinear Hoek–Brown yield criterion. Three different cases of definitions for elastic strains in the plastic region, used in the existing solutions, are mentioned. The closed-form analytical solutions for the displacement in the plastic region are derived on a theoretically consistent way for all the cases by employing a non-associated flow rule. The results of the dimensionless displacements are compared using the data of the soft and hard rocks to investigate the effect of different definitions for elastic strains with the dilation angle.     

Measurements of induced stress and strength in the near-field around a tunnel and associated estimation of the Mohr–Coulomb parameters for rock mass strength

The state of induced stress measured by the compact conical-ended borehole overcoring technique in the immediate roof of an approach tunnel excavated under high rock stress is described. During the measurements, core disking was observed. An X-ray Computed Tomography (CT) scanner was used to select strain data uninfluenced by the core disking; then the induced rock stress was estimated from selected strain data. From these results, it is shown that the non-destructive investigation using X-ray CT is effective for visualization of the fracturing within cores and the selection of strains measured during overcoring. Furthermore, the Mohr–Coulomb failure criterion parameters for the rock mass were estimated by comparing the measured stresses with the shear strength of in situ rock and the uniaxial compressive strength determined in laboratory tests.     

A Unified Strength criterion for rock material

A non-linear Unified Strength criterion for rock material is presented. It is the development of the Unified Strength Theory (in: M. Jono, T. Inoue (Eds.), Mechanical Behaviour of Materials-VI (ICM-6), Pergamon, Oxford, 1991, pp. 841–846) and the modification of the Hoek–Brown strength criterion (Underground Excavations in Rock, The Institution of Mining and Metallurgy, London, 1980). The effect of intermediate principal stress on rock strength is considered in the non-linear Unified Strength criterion. The Hoek–Brown criterion is a single-shear strength criterion that forms the lower bound, and the non-linear twin-shear strength criterion forms the upper bound in the deviatoric plane. All the failure criteria ranging from the Hoek–Brown criterion (lower bound) and the non-linear twin-shear criterion (upper bound) and a series of criteria ranging between these two bounds may be introduced by the non-linear Unified Strength criterion. The theory can also be generalized to rock mass strength.     

A practical procedure for the back analysis of slope failures in closely jointed rock masses

Where closely jointed rock masses are encountered in slopes, failure can occur both through the rock mass, as a result of combination of macro and micro jointing, and through the rock substance. Determination of the strength of this category of rock mass is extraordinarily difficult since the size of representative specimens is too large for laboratory testing. This difficulty can be overcome by using a non-linear rock mass failure criterion or by back analysis of such slopes to estimate the rock mass strength. In this paper, a practical procedure and a computer program are presented for the back determination of shear strength parameters mobilized in slopes cut in closely jointed rock masses which obey a non-linear failure criterion rather than a linear one. The procedure shows that the constants to derive normal stress dependent shear strength parameters of the failed rock masses can be determined by utilizing a main cross-section and without a pre-determined value of rock mass rating (RMR). Trials are made for different RMR_m and RMR_s values corresponding to various possible combinations of the constant m and s, which are used in the Hoek–Brown failure criterion, satisfying the limit equilibrium condition. It is also noted that the procedure provides a quick check for the rock mass rating obtained from the site investigations. The method is used in conjunction with the Bishop's method of analysis based on circular slip surfaces. The procedure outlined in this paper has also been satisfactorily applied to documented slope failure case histories in three open pit mines in Turkey.     

Effect of intermediate principal stress on strength of anisotropic rock mass

The Mohr-Coulomb criterion needs to be modified for highly anisotropic rock material and jointed rock masses. Taking σ₂ into account, a new strength criterion is suggested because both σ₂ and σ₃ would contribute to the normal stress on the existing plane of weakness. This criterion explains the enhancement of strength (σ₂ – σ3) in the underground openings because σ₂ along the tunnel axis is not relaxed significantly. Another cause of strength enhancement is less reduction in the mass modulus in tunnels due to constrained dilatancy. Empirical correlations obtained from data from block shear tests and uniaxial jacking tests have been suggested to estimate new strength parameters. A correlation for the tensile strength of the rock mass is presented. Finally, Hoek and Brown theory is extended to account for σ₂. A common strength criterion for both supported underground openings and rock slopes is suggested.     

Rock failure with weak planes by self-locking concept

In order to investigate failure of a rock mass having planes of weaknesses, a three-dimensional model is proposed based on the self-locking concept in friction analysis. In the case of two-dimensional problems, the model gives the same results as that of the Mohr–Coulomb criterion. The three-dimensional model can be reduced to the two-dimensional model, if the weak plane is parallel to the intermediate principal stress and/or the intermediate stress is equal to the minimum principal stress. The results indicate that the influences of three principal stresses and mechanical parameters of the weak plane on spatial failure region are remarkable, in terms of shape and range, that the spatial failure region becomes smaller as the mechanical parameters increases, and the weak plane will never fail when some threshold of mechanical parameters is reached, no matter what values of weak plane strike and dip will be. The spatial failure region becomes smaller with increased values of the intermediate and minimum principal stresses, conversely it becomes larger with the maximum principal stress increased. Additionally, the influence of bottom hole pressure on the failure range of weak plane is analyzed, for bore holes in naturally fractured formations, with the help of a local coordinate system.     

Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek–Brown criterion

This paper applies numerical limit analyses to evaluate the ultimate bearing capacity of a surface footing resting on a rock mass whose strength can be described by the generalised Hoek–Brown failure criterion [Hoek E, Carranza-Torres C, Corkum B. Hoek–Brown failure criterion—2002 edition. In: Proceedings of the North American rock mechanics society meeting in Toronto, 2002]. This criterion is applicable to intact rock or heavily jointed rock masses that can be considered homogeneous and isotropic. Rigorous bounds on the ultimate bearing capacity are obtained by employing finite elements in conjunction with the upper and lower bound limit theorems of classical plasticity. Results from the limit theorems are found to bracket the true collapse load to within approximately 2%, and have been presented in the form of bearing capacity factors for a range of material properties. Where possible, a comparison is made between existing numerical analyses, empirical and semi-empirical solutions.     

Geomechanical characterisation of cataclastic rocks: experience from the Cleuson–Dixence project

The mechanical behaviour of cataclastic rocks is hard to characterise because of many difficulties arising in getting undisturbed samples during field investigations, in specimen preparation, and in performing appropriate laboratory testing. To acquire a better understanding of such kind of very poor rocks, the EPFL's Rock Mechanics Laboratory has carried out an extensive laboratory testing programme on cataclastic rocks from the Cleuson–Dixence project (Switzerland). After some reflections on the quantification of the intensity of tectonic degradation process by means of the Geological Strength Index, the paper describes briefly the sampling and testing techniques. Then it focuses on the failure criterion and the stress–strain relationship of two types of cataclastic rocks: quartzitic sandstones and phyllitic schists. A new failure criterion is suggested: it consists in an extension of the Hoek and Brown failure criterion that takes into account the progressive damage undergone by cataclastic rocks during the tectonic degradation process. Its parameters are bounded by an upper limit corresponding to the original rock and a lower limit corresponding to a soil-like extremely tectonised material. Finally, the stress–strain behaviour of the tested cataclastic rocks is found to depend on the confining stress and the tectonisation degree and is adequately approached by the hyperbolic law of Duncan and Chang.