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 new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass

A simple numerical procedure for calculating the distribution of stresses and radial displacements around a circular tunnel excavated in a strain-softening Mohr–Coulomb or generalized Hoek–Brown rock mass is proposed. The problem is considered as axisymmetric, i.e. the initial stress state is assumed to be hydrostatic and the rock mass is said to be isotropic. By invoking the finite difference approximation of the equilibrium and compatibility equations, the increments of stresses and strains for each ring, starting from the outmost one for which boundary conditions are known a priori, are calculated in a successive manner. In the proposed approach, the potential plastic zone is divided into a finite number of concentric rings whose thicknesses are determined internally to satisfy the equilibrium equation. For the strain-softening behavior, it is assumed that all the strength parameters are a linear function of deviatoric plastic strain. Several illustrative examples are given to demonstrate the performance of the proposed method. For the brittle–plastic case, the results show a very good agreement with the closed-form solution. For strain-softening cases, the predictions by the proposed method are also in good agreement with the known rigorous numerical solutions. It is shown that the approximate solution converges to the exact solution when the increment of stress for each ring becomes smaller. The influence of the strength parameter ‘a’, appearing in the generalized Hoek–Brown criterion, on the elasto-plastic solutions is examined through the establishment of ground reaction curves and the discussion for the locations of the plastic radii.     

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.     

基于统一强度理论的临坡地基极限承载力解答

基于统一强度理论,综合考虑中间主应力、基底至坡肩水平距离和坡角的影响,推导了条形基础下临坡地基的极限承载力解答,给出其适用条件和计算步骤,讨论与水平地基极限承载力公式的差异,分析所得解答的可比性,并对比文献模型试验结果和上限法解答进行验证,最后探讨各参数的影响特性。研究表明：本文极限承载力解答可退化为Mohr-Coulomb强度准则解答并得到一系列新强度准则解答,具有很好的可比性;与文献模型试验结果和上限法解答吻合良好,验证了所得解答的正确性;中间主应力对临坡地基极限承载力的增强作用显著,未考虑中间主应力效应的结果偏于保守;临坡地基极限承载力随基底至坡肩水平距离的增加逐渐增大,当达到临界距离时与水平地基的极限承载力一致;坡角的增加导致临坡地基极限承载力大幅减小,且中间主应力效应较大时承载力降低的更加明显。本文解答可为临坡地基优化设计提供一定的理论参考。     

Revisiting rock classification to estimate rock mass properties

This paper presents the results of ongoing research carried out by the author exploring methods to provide a more robust estimate of rock mass properties specifically for use in tunnel design. Data from various large-scale rock mass failures are introduced, including coal pillars. The damage-initiation,spalling-limit approach is compared to the coal pillar database. New comparisons of estimating the geological strength index(GSI) and relationships to estimate the Hoeke Brown failure criterion parameters, mb, s and a, are presented.     

Interaction of shotcrete with rock and rock bolts—A numerical study

The shotcrete–rock interaction is very complex and is influenced by a number of factors. The influence of the following factors was investigated by a series of numerical analyses: the surface roughness of the opening, the rock strength and Young's modulus, the discontinuities, the extent and properties of the excavated disturbed zone, the mechanical properties of the interface between shotcrete and rock, and the thickness of the shotcrete lining and the rock bolts. The study was carried out as a sensitivity analysis. The results showed that the rock strength and the surface roughness had significant impact on the number of failures at the rock–shotcrete interface and in the shotcrete lining. Furthermore, the behaviour of the lining is sensitive to small amplitudes of the surface roughness. In all the cases investigated, a high interface strength was favourable. The results indicate that if a thick shotcrete lining is dependent on the bond strength. The benefit of using a thicker lining can be doubtful. The analyses showed that for an uneven surface the extent of the EDZ had a minor effect on the behaviour of the shotcrete lining. Furthermore, if rock bolts were installed at the apex of the protrusion instead of at the depression, the number of failures decreased both at the interface and in the lining.     

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 quantitative comparison of strength criteria for hard rock masses

Knowledge of the rock mass strength is important for the design of all types of underground excavations. A frequently applied approach for estimation of the rock mass strength is through an empirical failure criterion, often in conjunction with rock mass classification/characterisation systems. This paper presents a review of existing methods to estimate the rock mass strength using empirical failure criteria and classification/characterisation systems—in this study, commonly denoted as estimation methods. A literature review of existing methods is presented, after which a set of methods were selected for further studies. The selected methods were used in three case studies, to investigate their robustness and quantitatively compare the advantages and disadvantages of each method. A Round Robin test was used in two of the cases. The case studies revealed that the N, Yudhbir-RMR₇₆, RMi, Q-, and Hoek–Brown-GSI methods, appeared to yield a reasonable agreement with the measured strengths. These methods are thus considered the best candidates for realistic strength estimation, provided that care is taken when choosing values for each of the included parameters in each method. This study has also clearly shown the limits of presently available strength estimation methods for rock masses and further work is required to develop more precise, practical, and easy-to-use methods for determining the rock mass strength. This should be based on the mechanical behaviour and characteristics of the rock mass, which implies that parameters that consider the strength of intact rock, block size and shape, joint strength, and physical scale, are required.     

一个用于预测金属板材断裂的损伤破坏准则

基于连续介质损伤力学 ,提出了描述金属板材塑性损伤的演变方程 ,并依此得到了一个用于预测金属板材断裂破坏的简单准则 .其中特别考虑了剪切机制和剪切应力在金属薄板损伤发展和最终破坏过程中的作用 ,同时也引入了材料微观损伤对材料机械性能的影响 .提出的损伤演变规律 ,可近似表征金属薄板在成型过程中损伤的发展变化 ,计算得出的损伤曲线与有关实验曲线比较吻合 .给出的损伤破坏准则 ,可直接用于预测金属板材的断裂极限应变     

A simplified three-dimensional extension of Hoek-Brown strength criterion

The Hoek-Brown(HB) strength criterion has been applied widely in a large number of projects around the world.However,this criterion ignores the intermediate principal stress σ_2.Many evidences have demonstrated that the rock strength is dependent on σ_2. Thus it is necessary to extend the HB criterion into a three-dimensional(3D) form.In this study,the effect of σ_2 on the strength of rocks is identified by reviewing the true triaxial tests of various rock types reported in the literature.A simple 3D strength criterion is developed.The modified criterion is verified by the true triaxial tests of 13 rock types.The results indicate that the modified criterion can achieve a good fit to most of rock types.It can represent a series of criteria as b varies.For comparisons,several existing 3D versions of the HB criterion are selected to predict the strengths of these rock types.It is indicated that the proposed criterion works better than other criteria.A substantial relationship between parameter b and the unconfined compressive strength is established,which guarantees that the proposed criterion can still work well even in the absence of true triaxial test data.     

Shear strength criteria for rock,rock joints,rockfill and rock masses: Problems and some solutions

Although many intact rock types can be very strong, a critical confining pressure can eventually be reached in triaxial testing, such that the Mohr shear strength envelope becomes horizontal. This critical state has recently been better defined, and correct curvature or correct deviation from linear Mohr–Coulomb (M-C) has finally been found. Standard shear testing procedures for rock joints, using multiple testing of the same sample, in case of insufficient samples, can be shown to exaggerate apparent cohesion. Even rough joints do not have any cohesion, but instead have very high friction angles at low stress, due to strong dilation. Rock masses, implying problems of large-scale interaction with engineering structures, may have both cohesive and frictional strength components. However, it is not correct to add these, following linear M-C or nonlinear Hoek–Brown (H-B) standard routines. Cohesion is broken at small strain, while friction is mobilized at larger strain and remains to the end of the shear deformation. The criterion ‘c then σ_n tan φ’ should replace ‘c plus σ_ntan φ’ for improved fit to reality. Transformation of principal stresses to a shear plane seems to ignore mobilized dilation, and caused great experimental difficulties until understood. There seems to be plenty of room for continued research, so that errors of judgement of the last 50 years can be corrected.     

A generic method for rock mass classification

Rock mass classification (RMC) is of critical importance in support design and applications to mining, tunneling and other underground excavations. Although a number of techniques are available, there exists an uncertainty in application to complex underground works. In the present work, a generic rock mass rating (GRMR) system is developed. The proposed GRMR system refers to as most commonly used techniques, and two rock load equations are suggested in terms of GRMR, which are based on the fact that whether all the rock parameters considered by the system have an influence or only few of them are influencing. The GRMR method has been validated with the data obtained from three underground coal mines in India. Then, a semi-empirical model is developed for the GRMR method using artificial neural network (ANN), and it is validated by a comparative analysis of ANN model results with that by analytical GRMR method.     

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.     

Effects of confinement on rock mass modulus: A synthetic rock mass modelling (SRM) study

The main objective of this paper is to examine the influence of the applied confining stress on the rock mass modulus of moderately jointed rocks (well interlocked undisturbed rock mass with blocks formed by three or less intersecting joints). A synthetic rock mass modelling (SRM) approach is employed to determine the mechanical properties of the rock mass. In this approach, the intact body of rock is represented by the discrete element method (DEM)-Voronoi grains with the ability of simulating the initiation and propagation of microcracks within the intact part of the model. The geometry of the pre-existing joints is generated by employing discrete fracture network (DFN) modelling based on field joint data collected from the Brockville Tunnel using LiDAR scanning. The geometrical characteristics of the simulated joints at a representative sample size are first validated against the field data, and then used to measure the rock quality designation (RQD), joint spacing, areal fracture intensity (P₂₁), and block volumes. These geometrical quantities are used to quantitatively determine a representative range of the geological strength index (GSI). The results show that estimating the GSI using the RQD tends to make a closer estimate of the degree of blockiness that leads to GSI values corresponding to those obtained from direct visual observations of the rock mass conditions in the field. The use of joint spacing and block volume in order to quantify the GSI value range for the studied rock mass suggests a lower range compared to that evaluated in situ. Based on numerical modelling results and laboratory data of rock testing reported in the literature, a semi-empirical equation is proposed that relates the rock mass modulus to confinement as a function of the areal fracture intensity and joint stiffness.     

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.     

Physical and mechanical properties of rock masses at Stromboli: a dataset for volcano instability evaluation

Stromboli island has a complex geological history with repeated changes in the volcanic activity alternating with destructive events, caldera collapses and flank landslides. The last activity resulted in the creation of the Sciara del Fuoco depression which was modified by the recent 2002–2003 landslide. The variation in lithology, degree of tectonization and disturbance has resulted in the presence of a wide spectrum of geotechnical materials. This paper summarises the physical and mechanical properties of Stromboli’s intact rocks, rock masses and loose deposits, based on field surveys and laboratory tests. A new classification of the rock succession is introduced and four lithotechnical units defined: Lava, Lava-Breccia, Breccia and Pyroclastic deposit. The range of variability in bulk volume, porosity, intact rock compressive strength and geological strength index is presented. The Hoek and Brown’s failure criterion was applied for each lithotechnical unit and the rock mass friction angle, apparent cohesion, tensile and compressive strength, global strength and modulus of deformation calculated in a specified stress range.      

Strength behaviour of a model rock intersected by non-persistent joint

Several constructions in the field of civil engineering quite often need to deal with rocks.Strength behaviour of rock intersected by a discontinuity or a set of discontinuities has been a topic of keen interest for engineering community.The popular attributes of discontinuities that have been given due importance are their frequency,orientation and surface characteristics.Non-persistency,however,has been given little attention.This article presents an experimental study wherein focus has been made on the effect of non-persistency of the joint on the uniaxial compressive strength(UCS) of a model rock for various geometries such as orientation,discontinuity length ratio and number of joint segments.The applicability of single plane of weakness theory(SPWT) to assess the strength of jointed specimens has also been evaluated.It has been noticed that SPWT captures the strength behaviour only for a narrow range of discontinuity orientations.As an improvement,an approach is suggested by extending concepts of degree of persistence and joint factor to have a better understanding towards strength behaviour of rocks intersected by non-persistent joints.     

用于纤维编织网增强混凝土的早强基体材料

为实现纤维编织网增强混凝土(textile-reinforced concrete,TRC)快速加固盾构隧道的目的,开发了一种用于TRC的磷酸无机复合类早强型基体材料.通过单轴压缩、坍落流动度、界面拉伸和剪切试验,对比分析了这种早强型基体材料与TRC传统基体材料——精细混凝土的工作性能;采用扫描电镜分析了其作用机理,并提出了力学作用模型.结果表明:与精细混凝土相比,早强型基体材料具有更优异的自密实、黏结以及渗透纤维编织网的能力;采用早强型基体材料的TRC加固方法可实现快速加固,在2h内即可发挥加固作用,具有广泛的应用前景.     

基于围岩本体Mogi-Coulomb强度准则的层理性岩层斜井井壁稳定模型

 层理性地层中进行大斜度井施工的井壁失稳问题较为突出,在传统井壁稳定模型基础上,以弱面对岩石强度的弱化作用实验为依据,引入欧拉变换充分考虑三维地应力方向的任意性,利用Mogi-Coulomb准则强化中主应力对围岩本体强度的影响,结合单弱面强度准则建立分析层理性地层斜井井壁稳定问题的模型,并进一步给出计算斜井坍塌压力与破裂压力的方法与公式。实验与计算结果表明：当加载方向与弱面夹角为30°时岩石强度最低;Mogi-Coulomb准则因考虑了中主应力的影响而对围岩本体强度的估计更为有效;考虑层理弱面影响的井壁坍塌破坏区域明显增大,破坏位置也发生改变;取得最小坍塌压力值的钻井方向在空间中与层理面并非简单的垂直关系;空间中关于地应力主平面对称的井孔破裂压力相同。利用所建模型进行安全泥浆压力窗口的计算可为安全钻井以及斜井轨迹设计提供理论依据。     

A waveform modification method for testing dynamic properties of rock under high temperature

In this study,a waveform modification method was proposed using a self-designed heating device combined with the split Hopkinson pressure bar(SHPB) technique for determination of dynamic behaviors of rock at high temperature.Firstly,the temperature gradient distribution on the incident bar was measured according to the variation of elastic modulus of the bar with temperature,and the relationship between the longitudinal wave velocity and temperature of the bar was obtained based on onedimensional stress wave theory.The incident bar with a temperature gradient was divided into a series of microelements,and then the transmission coefficient of the whole incident bar was obtained.Finally,the stress wave was modified by the transmission coefficient from 25℃ to 600℃.This method was used to study the dynamic properties of rock at high temperature,which not only preserves a classical SHPB device,but also effectively ensures the accuracy of the experimental results.A dynamic Brazilian disc experiment was carried out to explore the influences of loading rate and temperature on dynamic tensile strength of sandstone at high temperature using the proposed waveform modification method.