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.     

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.     

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.     

A new elasto-plastic solution for analysis of underwater tunnels considering strain-dependent permeability

An analytical–numerical solution is proposed in this paper for calculating the mechanical and hydraulic characteristics of underwater tunnels, excavated in an elasto-plastic strain-softening and Hoek–Brown rock material considering strain-dependent permeability. The problem is considered with axial-symmetry condition and thus, the initial stress state is assumed to be hydrostatic. Hydraulic and mechanical differential equations governing the plastic and the elastic rock mass around the tunnel are derived. As the derived equations do not have closed-form solutions, a computer program has been prepared, in order to solve the corresponding equations numerically and to examine the analysis. The results obtained are compared with the results obtained using a commercial finite difference code for investigating the effects of utilised simplifying assumptions. The proposed approach is also used to analyse the response of tunnels under different hydro-mechanical conditions. The results show that the proposed solution can be used for the actual design of underwater tunnels, hydraulically and mechanically.     

The strength of hard-rock pillars

Observations of pillar failures in Canadian hard-rock mines indicate that the dominant mode of failure is progressive slabbing and spalling. Empirical formulas developed for the stability of hard-rock pillars suggest that the pillar strength is directly related to the pillar width-to-height ratio and that failure is seldom observed in pillars where the width-to-height ratio is greater than 2. Two-dimensional finite element analyses using conventional Hoek–Brown parameters for typical hard-rock pillars (Geological Strength Index of 40, 60 and 80) predicted rib-pillar failure envelopes that did not agree with the empirical pillar-failure envelopes. It is suggested that the conventional Hoek–Brown failure envelopes over predict the strength of hard-rock pillars because the failure process is fundamentally controlled by a cohesion-loss process in which the frictional strength component is not mobilized. Two-dimensional elastic analyses were carried out using the Hoek-Brown brittle parameters which only relies on the cohesive strength of the rock mass. The predicted pillar strength curves were generally found to be in agreement with the observed empirical failure envelopes.     

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.     

Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone

The excavation impact (e.g. due to blasting, TBM drilling, etc.) induces an excavation damaged or disturbed zone around a tunnel. In this regard, in drill and blast method, the damage to the rock mass is more significant. In this zone, the stiffness and strength parameters of the surrounding rock mass are different. The real effect of a damage zone developed by an excavation impact around a tunnel, and its influence on the overall response of the tunnel is of interest to be quantified. In this paper, a fully analytical solution is proposed, for stresses and displacements around a tunnel, excavated in an elastic–brittle–plastic rock material compatible with linear Mohr–Coulomb criterion or a nonlinear Hoek–Brown failure criterion considering the effect of the damaged zone induced by the excavation impact. The initial stress state is assumed to be hydrostatic, and the damaged zone is assumed to have a cylindrical shape with varied parameters; thus, the problem is considered axial-symmetric. The proposed solution is used to explain the behavior of tunnels under different damage conditions. Illustrative examples are given to demonstrate the performance of the proposed method, and also to examine the effect of the damaged zone induced by the excavation impact. The results obtained by the proposed solution indicate that, the effects of the alteration of rock mass properties in the damaged zone may be considerable.     

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.     

岩石剪胀角模型与验证

Mohr-Coulomb模型和基于Mohr-Coulomb的应变软化模型均通常假设剪胀角为恒定值,然而这种假设不能正确表达岩石在破坏变形过程中的非线性体积变化行为。根据7种岩石类型在不同围压条件下的体积应变测量数据,结合塑性力学理论,采用非线性拟合方法建立能同时考虑围压和塑性剪切应变影响的剪胀角模型。分析模型的响应并结合岩石内部颗粒尺寸以及单轴抗压强度,将该模型划分为4种岩石类型:粗粒径硬岩、中粒径硬岩、中–细粒径软岩和细粒径软岩。根据FLAC应变软化模型中非关联塑性流动法则的计算原理,推导剪胀角模型中的塑性剪切应变与应变软化模型中塑性参数的关系,将剪胀角模型嵌入应变软化模型中,构建剪胀角模型模块。最后,采用建立的剪胀角模型预测Moura煤岩在三轴压缩条件下的体积应变–轴向应变关系曲线。研究结果表明,数值模拟与试验结果具有很好的一致性。     

A theoretical solution for analysis of tunnels below groundwater considering the hydraulic–mechanical coupling

An analytical solution for the analysis of tunnels below groundwater table in plane strain axisymmetric condition is presented. Seepage body force and secondary permeability of the rock mass due to the mechanical–hydraulic coupling are taken into account. The strain-softening behavior model and Hoek–Brown empirical strength criterion for the rock mass are used in the analysis. As the derived analytical equations do not have closed form solutions, a computer program has been prepared for solving the corresponding equations numerically and examining the analysis. It is shown that the tunnel stability depends on the seepage and the pore water pressure particularly in the case of high pore pressure gradient.     

An analytical solution to wellbore stability using Mogi-Coulomb failure criterion

Deep wellbores/boreholes are generally drilled into rocks for oil and gas exploration, monitoring of tectonic stresses purposes. Wellbore and tunnel in depth are generally in true triaxial stress state, even if the ground is under axisymmetric loading condition. Stability of such wellbores is very critical and collapse of wellbore must be avoided. Mogi-Coulomb failure criterion is a better representation of rock strength under true triaxial condition. In this paper, an analytical solution is proposed using Mogi-Coulomb failure criterion. The solution is obtained for rock mass exhibiting elastic-perfectly plastic or elastic-brittle-plastic behaviour considering in-plane isotropic stresses. The proposed solution is then compared with exact analytical solution for incompressible material and experimental results of thick-wall cylinder. It is shown that the results obtained by the proposed analytical solution are in good agreement with the experimental results and exact analytical solution. A reduction of about 13%–20% in plastic zone from the proposed closed-form solution is observed, as compared to the results from the finite element method (FEM) based Mohr-Coulomb criterion. Next, the influences of various parameters such as Poisson's ratio, internal pressure (mud weight), dilation angle, and out-of-plane stress are studied in terms of stress and deformation responses of wellbore. The results of the parametric study reveal that variation in the out-of-plane stress has an inverse relation with the radius of plastic zone. Poisson's ratio does not have an appreciable influence on the tangential stress, radial stress and radial deformation. Dilation angle has a direct relation with the deformation. Internal pressure is found to have an inverse relation with the radial deformation and the radius of plastic zone.     

A closed-form solution for a spherical cavity in the elastic-brittle-plastic medium

A theoretically consistent closed-form solution for a spherical cavity in a brittle-plastic infinite medium subject to a hydrostatic stress field was derived. Both linear Mohr-Coulomb (M-C) and nonlinear Hoek-Brown (H-B) yield criteria are considered, and a non-associated flow rule is employed in the solution. Plastic radius, stresses and displacements are explicitly expressed as the functions of radial coordinates, internal pressure and strength parameters. In elastic-perfectly plastic analysis, results for displacements and stresses are in good agreement with those published. Comparison of brittle-plastic solutions with the elasto-plastic ones indicates that brittle-plastic behavior of rock mass increase the extent of plastic zone and the displacements around the cavity significantly. Ground response curves (GRC) are constructed for both M-C and H-B criteria and the potential applications are discussed.     

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 Hoek–Brown criterion with intrinsic material strength factorization

Probably the most common failure criterion for rock masses is the Hoek–Brown (HB) failure criterion. The HB criterion is an empirical relation that extrapolates the strength of intact rock to that of rock masses. For design purposes, the HB criterion is often fitted using equivalent Coulomb failure lines. However, equivalent Mohr–Coulomb (MC) shear strength parameters cannot yield the same failure characteristics as the HB criterion. The curvilinear HB criterion automatically accommodates changing stress fields; the MC criterion does not. The extended HB criterion proposed in this paper provides a solution to this problem by incorporating an intrinsic material strength factorization scheme. The original HB criterion is additionally enhanced by adopting the spatial mobilized plane (SMP) concept, first introduced by Matsuoka and Nakai (MN). The SMP concept accounts for the experimentally proven, influence of intermediate principal stresses on failure, which is disregarded in the original HB criterion. A small set of examples provided at the end of the article gives a good indication of the merits of using the extended HB criterion in practical applications.     

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%).     

Hoek-Brown准则奇异屈服面的圆化方法及其强度折减技术与应用

 为克服基于Hoek-Brown强度准则的屈服面数值奇异问题,提出新的棱角处屈服面圆化方法,在此基础上,借助ABAQUS二次开发平台,采用FORTRAN语言编制理想弹塑性模型子程序,应力更新算法为完全隐式的向后欧拉积分算法; 提出基于Hoek-Brown准则的有限元强度折减技术,并将其嵌入到圆化后的屈服函数中,探讨了强度折减技术应用于Mohr-Coulomb强度准则和Hoek-Brown非线性强度准则的异同点。对某圆形隧洞的模拟分析表明,围岩应力和塑性区的数值计算结果和解析结果吻合很好;对某岩质边坡的模拟分析表明,与等效Mohr-Coulomb模型的线性强度折减技术相比,基于Hoek-Brown模型的非线性强度折减技术获得的潜在滑动面位置较浅、形状较陡,更符合岩质边坡滑动面的特征,两个算例验证了该圆化方法和非线性强度折减技术的正确性和适用性。     

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.     

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.     

圆形巷道围岩应变软化弹塑性分析

 岩土类材料广义上都呈应变软化力学行为,考虑峰后岩石应变软化过程中引起后继强度面收缩的特性,认为岩体在足够小的局部范围内材料特性保持不变,将后继破坏岩体分为多个塑性区,给出基于平面应变和Mohr- Coulomb准则的围岩准应变软化弹塑性应力解析式,并给出符合围岩后继强度和变形演变规律的各软化区半径求解方程组。洞室围岩变形特征曲线是地下工程支护设计的重要依据,考虑岩体破裂过程中变形参数不断劣化的特性,分析剪切模量劣化对围岩变形、软化半径及破裂半径的影响。最后通过算例对模型进行验证,计算结果可以对地下工程的优化支护设计及安全性评价提供理论依据。