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.     

Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media

A closed-form solution is presented in this paper for the prediction of displacements around circular openings in a brittle rock mass subject to a hydrostatic stress field. The rock mass is assumed to be governed by Hoek–Brown yield criterion and a non-associated flow rule is used. For the elastic–brittle–plastic analysis of circular openings in an infinite Hoek–Brown medium, the existing analytical solutions were found to be incorrect. The present closed-form solution is based on a theoretically consistent method and the solution does not require the use of any numerical method.The present closed-form solution was validated by using the finite element method. In the finite element analysis, the infinite boundary was simulated “exactly” by using the newly developed elastic support method. Several cases were analyzed and the present closed-form solutions for stresses and displacements were found to be in an excellent agreement with those obtained by using the finite element method.     

An exact implementation of the Hoek–Brown criterion for elasto-plastic finite element calculations

A simple stress update algorithm for generalised Hoek–Brown plasticity is presented. It is intended for use in elasto-plastic finite element computations and utilises the return mapping concept for computing the stress increment belonging to a given increment in strain at a material point. In the algorithm all manipulations are carried out in principal stress space, where the Hoek–Brown failure criterion has a very simple form compared to its formulation in general stress space. In principal stress space it is also simple to determine whether the stress should be returned to one of the edges or to the apex of the yield surface and to form the constitutive matrices. As opposed to earlier finite element implementations of Hoek–Brown plasticity the exact criterion is used, i.e. no rounding of the yield surface corners or edges is attempted. Numerical examples and a comparison with an often used method for dealing with the corner singularities indicates the efficiency of the presented.     

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.     

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.     

A modified Hoek–Brown failure criterion for anisotropic intact rock

The Hoek–Brown criterion parameters (σ_ci, m_i and s) are significantly influenced by the strength anisotropy of intact rock. In the present study, the criterion was modified by incorporating a new parameter (k_β) to account for the effect of strength anisotropy, thus being able to determine the strength of intact anisotropic rock under loading in different orientations of the plane of anisotropy. The range of the parameter (k_β) for the rocks tested has been analytically investigated by carrying out triaxial tests, in different orientations of the foliation plane. The proposed modification was studied for metamorphic rocks (gneiss, schist, marble), but could also be applied to other rock types exhibiting “inherent” anisotropy, e.g. sedimentary as well as igneous rocks. The proposed modified criterion is intended for use for prediction of strength of intact rock, but can also be extended to rock masses.     

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.     

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.     

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.     

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.     

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%).     

A simplified approach to directly consider intact rock anisotropy inHoekeBrown 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 anddeformational properties. The HoekeBrown （HeB） failure criterion is an empirical strength criterionwidely applied to rock mechanics and engineering. A direct modification to HeB failure criterion toaccount for rock anisotropy is considered as the base of the research. Such modification introduced a newdefinition of the anisotropy as direct parameter named the anisotropic parameter （Kb）. However, thecomputation 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 theminimum value of anisotropic parameter （Kmin）, and to predict the Kmin directly from the uniaxialcompression tests instead of triaxial tests, and also to decrease the amount of experimental work.
2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.     

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.     

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.     

Determination of stress–strain relationship of tubular material with hydraulic bulge test

Based on the plastic membrane theory and force equilibrium equations, etc., a unique approach is proposed with the curve fitting of experimental data to determine the stress–strain relationship of a thin-walled tube in hydroforming process (THF). A simple and practical hydraulic bulge test tooling was developed and free-bulged tests were performed on stainless steel and low carbon steel tubes to obtain required experimental deformation data. Finite element (FE) simulations of the free bulges were carried out to verify the approach indirectly. The results indicate that the present approach is accurate and acceptable to define the stress–strain behavior of tubular material, and furthermore, an extended flow stress curve with large strain can be obtained by the approach.     

Effective width of steel–concrete composite beam at ultimate strength state

In a steel–concrete composite beam section, part of the concrete slab acts as the flange of the girder in resisting the longitudinal compression. The well-known shear-lag effect causes a non-uniform stress distribution across the width of the slab and the concept of effective width is usually introduced in the practical design to avoid a direct analytical evaluation of this phenomenon. In the existing studies most researchers have adopted the same definition of effective width which might induce inaccurate bending resistance of composite beam to sagging moments. In this paper, a new definition of effective width is presented for ultimate analysis of composite beam under sagging moments. Through an experimental study and finite element modeling, the distribution of longitudinal strain and stress across the concrete slab are examined and are expressed with some simplified formulae. Based on these simplified formulae and some assumptions commonly used, the effective width of the concrete slab and the depth of the compressive stress block of composite beams with varying parameters under sagging moments are analytically derived at the ultimate strength limit. It is found that the effective width at the ultimate strength is larger than that at the serviceability stage and simplified design formulae are correspondingly suggested for the ultimate strength design.     

Role of overconsolidation on sand–geomembrane interface response and material damage evolution

The behavior of high-density polyethylene (HDPE) reinforcement elements is critical to the overall performance of many geotechnical structures including landfills, retaining walls, embankments, and shallow foundations. The reinforcement elements, which may consist of manufactured sheets or strips, must be properly installed in order to transfer stresses from areas of concentrated loading to reinforcing zones. While attention is given during construction to avoid contact damage of reinforcement elements, the construction process exerts stresses on the reinforcement element that may substantially exceed the stresses acting throughout the structure's service life. This paper examines the influence of overconsolidation of HDPE reinforcement elements on interface performance through examination of its effect on sand–smooth geomembrane interfaces. A range of overconsolidation and operational normal stress values are selected based on conditions typically present in practice. Results show the peak friction coefficient to increase substantially with overconsolidation magnitude while the effect on the residual friction coefficient is minimal. The magnitude of the increase in peak friction coefficient is shown to be dependent on both the damage induced by the smooth geomembrane and the degree to which sand particles adjacent to the interface undergo damage in the form of breakage during shear. The damage induced by the smooth geomembrane and the sand particles are presented in terms of the level of overconsonsolidation, confinement stress during shear, and particle characteristics such as angularity and hardness.     

A new testing method for investigating the loading rate dependency of peak and residual rock strength

One of the important methods for investigating viscoelasticity is to measure the loading-rate dependency of peak strength; however, no experimental method has been established for accurately measuring the loading-rate dependency of peak strength from a small number of samples. In this study we propose such a method. A single sample is loaded at alternating strain rates to obtain stress–strain curves for both strain rates. The loading-rate dependency of peak strength obtained via this method was compared with the findings of conventional methods. The loading-rate dependency indicated for Tage tuff, Sanjome andesite and Akiyoshi marble was nearly identical to previous results obtained using conventional methods, including results obtained under confining pressure. The loading-rate dependency of peak strength in these experiments shows a close relation with the creep stress-dependency of creep life. We also investigated the loading-rate dependency of the stress–strain curve for the post-failure region for which few results have been published. Under confining pressure, the corrected stress–strain curve, obtained by multiplying the stress of the complete stress–strain curve obtained at the fast strain rate by a constant determined by the ratio between the fast strain rate and slow strain rate, is nearly coincident with the stress–strain curve for the slow strain rate. This is an interesting result and represents new knowledge that may help elucidate failure mechanisms in the post-failure region. The loading-rate dependency of stress in the alternating strain rate experiment proposed here was most clearly observed when the stress–strain curve becomes flat, parallel to the strain axis. Some improvements to the proposed method are required to enable accurate investigations of loading-rate dependency during low stresses immediately after initiation of loading or during the abrupt decreases in stress that occur following peak strength.     

A beam model for the flexural–torsional buckling of thin-walled members with some applications

A beam model aimed at describing the flexural–torsional buckling of thin-walled members with non-symmetric cross-sections is presented. Two beam axes are introduced, and strain is defined with respect to both. The shearing strain between the cross-section and one of the two axes is assumed to vanish; the warping is supposed to be linear in the twist. Non-linear hyperelastic constitutive relations are introduced; by means of standard localization and static perturbation techniques, the field equations describing the flexural–torsional buckling are obtained. One benchmark example is given and some numerical values of the critical load for various warping constraints at the beam ends are provided.