Mechanical characteristics of the exponential strength criterion under conventional triaxial stresses

The minor principal stress is the key factor influencing the mechanical property of rock, and the strength criterion of conventional triaxial stresses is the basis of any true triaxial criterion. This paper examines the Coulomb criterion, Hoek–Brown criterion, the generalized Hoek–Brown criterion, and the exponential criterion proposed by the author. The procedure to determine the parameters in criteria is studied. Parameters in the generalized Hoek–Brown criterion are sensitive to the power n, thus Coulomb criterion with n=1.0 and the original form of Hoek–Brown criterion with n=0.5, especially that determined by linear regression, are not the best choice for the fitting solution of test data. The exponential criterion determined on the least absolute deviation will approximate a great number of normal test data, and expose the oddity data. It can describe the relation between strength and confining pressure in the entire stress range, with low misfit. The uniaxial compressive strengths predicted by the exponential criterion are nearly the same as the real magnitudes for all rocks. The deviator of strengths under conventional triaxial extension and compression is larger than half of the maximum strength increase from the intermediate principal stress. Therefore, the parameters in the exponential criterion may be completely determined from the strengths under conventional triaxial compression and extension.     

Experimental verification of parameter m in Hoek–Brown failure criterion considering the effects of natural fractures

Hoek–Brown failure criterion is one of the widely used rock strength criteria in rock mechanics and mining engineering. Based on the theoretical expression of Hoek–Brown parameter m of an intact rock, the parameter m has been modified by crack parameters for fractured rocks. In this paper, the theoretical value range and theoretical expression form of the parameter m in Hoek–Brown failure criterion were discussed. A critical crack parameter B was defined to describe the influence of the critical crack when the stress was at the peak, while a parameter b was introduced to represent the distribution of the average initial fractures. The parameter m of a fractured rock contained the influences of critical crack (B), confining pressure (σ₃) and initial fractures (b). Then the triaxial test on naturally fractured limestones was conducted to verify the modification of the parameter m. From the ultrasonic test and loading test results of limestones, the parameter m can be obtained, which indicated that the confining pressure at a high level reduced the differences of m among all the specimens. The confining pressure σ₃ had an exponential impact on m, while the critical crack parameter B had a negative correlation with m. Then the expression of m for a naturally fractured limestone was also proposed.     

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.     

Numerical simulation of coal pillar strength

The present paper describes an approach for estimation of strain-softening constitutive behaviour of coal-mass through calibration of a numerical model with field cases. A three-dimensional finite element model was used, in which the coal-mass was considered to be a Hoek–Brown strain-softening material. Dilation behaviour was considered to be a function of confinement and plastic shear strain. The numerical model was calibrated by satisfying the (1) various failed/stable cases of coal pillar of Indian coal mines and (2) post-failure behaviour of large-scale coal pillars determined by various researchers through conducting in-situ tests. The calibration of the model was done by taking various permutation and combination of strain-softening parameters. It has been concluded that the yielding of the pillars start almost at 2/3 of the peak strength. The statistical expressions for estimation of pillar strength for Indian coal mines and post-failure modulus have been developed by analyzing the results of the simulations. For Indian coal pillars having width-to-height ratio (w/h) less than five, the pillar strength was almost linearly dependent on w/h, and non-linearly dependent on uniaxial compressive strength of the coal specimen. The post-failure modulus of coal pillars was non-linearly dependent on w/h, and not significantly dependent on the uniaxial compressive strength.     

Evaluation of rock strength criteria for wellbore stability analysis

Polyaxial strength test data of five rocks are used to examine the Mohr–Coulomb, Drucker–Prager, modified Lade, Mogi–Coulomb and three-dimensional (3D) Hoek–Brown criteria regarding their ability, with parameters determined based on the triaxial compression test data, to represent the rock behavior under polyaxial stress states. Then the five strength criteria, with parameters determined based on the triaxial compression test data, are used to analyze wellbore stability of both vertical and inclined boreholes. The results show that the Mohr–Coulomb criterion under-predicts the polyaxial strength and estimates the highest minimum mud pressure required for wellbore stability while the Drucker–Prager criterion over-predicts the polyaxial strength and estimates the lowest minimum mud pressure. The modified Lade, Mogi–Coulomb and 3D Hoek–Brown criteria, with parameters based on triaxial test data, can either over-predict or under-predict the polyaxial strength. The over-prediction of the modified Lade criterion can be very large, and thus it may be unsafe to use it to estimate the minimum mud pressure. Both the over-prediction and under-prediction are relatively small for the 3D Hoek–Brown and Mogi–Coulomb criteria. Therefore, the 3D Hoek–Brown and Mogi–Coulomb criteria are recommended for wellbore stability analysis.     

Rock mechanics for design of Brisbane tunnels and implications of recent thinking in relation to rock mass strength

This paper explores the potential implications of recent thinking in relation to rock mass strength for future tunnelling projects in Brisbane, Australia, particularly as they are constructed within deep horizons where the in situ stress magnitudes is larger. Rock mass failure mechanisms for the current tunnels in Brisbane are generally discontinuity controlled and the potential for stress-induced failure is relatively rare. For the road tunnels which have been constructed in Brisbane over the last 12 years, the strength of the more massive rock masses for continuum analysis has been estimated by the application of the Hoek-Brown (H-B) failure criterion using the geological strength index (GSI) to determine the H-B parameters m_b, s and a. Over the last few years, alternative approaches to estimating rock mass strength for ‘massive to moderately jointed hard rock masses’ have been proposed by others, which are built on the work completed by E. Hoek and E.T. Brown in this area over their joint careers. This paper explores one of these alternative approaches to estimating rock mass strength for one of the geological units (the Brisbane Tuff), which is often encountered in tunnelling projects in Brisbane. The potential implications of these strength forecasts for future tunnelling projects are discussed along with the additional work which will need to be undertaken to confirm the applicability of such alternative strength criteria for this rock mass.     

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.     

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.     

Modification of rock mass strength assessment methods and their application in geotechnical engineering

Due to complicated structures and discontinuities in surrounding rock mass, existing empirical failure criteria cannot meet the requirements of engineering practice such as tunnels. To improve estimation accuracy on the strength of rock mass with joints, a modified chart of the Geological Strength Index using Hoek–Brown criteria was further tested to estimate rock mass strength [Lin et al. (2014) Bull Eng Geol Environ 4(73):1245–1258], and, in this paper, new strength estimation equations for jointed rock mass were then modified based on a large dataset obtained from Chinese projects. Here, standard drilling time is first introduced and described in this study, and then used as a parameter to estimate rock strength. Different empirical formulas based on joint density, rock mass classification, Hoek–Brown criteria, and elastic wave velocity are thus used to estimate rock mass strength by using data from the Jiubao tunnel. The results estimated based on different empirical formulas were similar, indicating that the modified assessment method presented in this paper can be used to estimate rock mass strength under certain circumstances. Cross-correlation of different empirical methods provides significant confidence in predicted rock mass strength calculations.     

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.     

Slope reliability and back analysis of failure with geotechnical parameters estimated using Bayesian inference

A Bayesian approach is proposed for the inference of the geotechnical parameters used in slope design. The methodology involves the construction of posterior probability distributions that combine prior information on the parameter values with typical data from laboratory tests and site investigations used in design. The posterior distributions are often complex, multidimensional functions whose analysis requires the use of Markov chain Monte Carlo (MCMC) methods. These procedures are used to draw representative samples of the parameters investigated, providing information on their best estimate values, variability and correlations. The paper describes the methodology to define the posterior distributions of the input parameters for slope design and the use of these results for evaluation of the reliability of a slope with the first order reliability method (FORM). The reliability analysis corresponds to a forward stability analysis of the slope where the factor of safety (FS) is calculated with a surrogate model from the more likely values of the input parameters. The Bayesian model is also used to update the estimation of the input parameters based on the back analysis of slope failure. In this case, the condition FS = 1 is treated as a data point that is compared with the model prediction of FS. The analysis requires a sufficient number of observations of failure to outbalance the effect of the initial input parameters. The parameters are updated according to their uncertainty, which is determined by the amount of data supporting them. The methodology is illustrated with an example of a rock slope characterised with a Hoek-Brown rock mass strength. The example is used to highlight the advantages of using Bayesian methods for the slope reliability analysis and to show the effects of data support on the results of the updating process from back analysis of failure.     

Advanced Markov Chain Monte Carlo Approach for Finite Element Calibration under Uncertainty

Uncertainty involved in the experiment data prohibits the wide applications of the finite element (FE) model updating technique into engineering practices. In this article, the Markov Chain Monte Carlo approach with a Delayed Rejection Adaptive Metropolis algorithm is investigated to perform the Bayesian framework for FE updating under uncertainty. A major advantage of this algorithm is that it adopts global and local adaptive strategies, which makes the FE model updating be robust to uncertainty. Another merit of the studied method is that it not only quantitatively predicts structural responses, but also calculates their statistical parameters such as the confidence interval. Impact test data of a grid structure are investigated to demonstrate the effectiveness of the presented FE model updating technique, in which the uncertainty parameters include the vertical and longitudinal spring stiffness that simulate the boundary
conditions, the end‐fixity factor for modeling semi‐rigid connections, and the elastic modulus for simulating the uncertainty associated with material property.     

Reliability-based design in rock engineering: Application of Bayesian regression methods to rock strength data

Reliability-based design(RBD) is being adopted by geotechnical design codes worldwide, and it is therefore necessary that rock engineering practice evolves to embrace RBD. This paper examines the Hoek-Brown(He B) strength criterion within the RBD framework, and presents three distinct analyses using a Bayesian approach. Firstly, a compilation of intact compressive strength test data for six rock types is used to examine uncertainty and variability in the estimated He B parameters m and sc, and corresponding predicted axial strength. The results suggest that within-and between-rock type variabilities are so large that these parameters need to be determined from rock testing campaigns, rather than reference values being used. The second analysis uses an extensive set of compressive and tensile(both direct and indirect) strength data for a granodiorite, together with a new Bayesian regression model, to develop joint probability distributions of m and scsuitable for use in RBD. This analysis also shows how compressive and indirect tensile strength data may be robustly used to fit an He B criterion.The third analysis uses the granodiorite data to investigate the important matter of developing characteristic strength criteria. Using definitions from Eurocode 7, a formal Bayesian interpretation of characteristic strength is proposed and used to analyse strength data to generate a characteristic criterion.These criteria are presented in terms of characteristic parameters mkand s ck, the values of which are shown to depend on the testing regime used to obtain the strength data. The paper confirms that careful use of appropriate Bayesian statistical analysis allows the He B criterion to be brought within the framework of RBD. It also reveals that testing guidelines such as the International Society for Rock Mechanics and Rock Engineering(ISRM) suggested methods will require modification in order to support RBD. Importantly, the need to fully understand the implications of uncertainty in nonlinear strength criteria is identified.     

Ultimate bearing capacity of an anisotropic discontinuous rock massPart II: Determination procedure

The existence of a transverse strength isotropy concerning the strength in rock media due to the presence of a family of planes of weakness superimposed within an isotropic rock mass is considered. Six different mechanisms of failure are possible under a foundation depending on the boundary conditions and the orientation of the transverse isotropy. The theoretical bases for the determination of the ultimate load of a strip foundation on a rock mass are obtained, respectively, for the six different mechanisms. The following assumptions are adopted: (a) a non-linear behaviour through the rock mass, defined by the Hoek and Brown model (parameters m, s and the unconfined compressive strength, respectively) and (b) a linear strength behaviour along the planes of weakness, defined by their cohesion and angle of internal friction.     

Hoek-Brown强度准则研究进展与应用综述

1980年E. Hoek和E. T. Brown提出了Hoek-Brown(H-B)强度准则,已充分得到岩石力学与工程研究者的认同,并进行研究和应用。首先系统地阐述H-B强度准则研究进展：E. Hoek和E. T. Brown对H-B强度准则的研究成果、三维H-B强度准则、H-B强度准则岩石和岩体参数研究、考虑层状节理的H-B强度准则及其参数的各向异性研究。再对过去30 a国内外基于H-B强度准则工程应用的成果进行总结。最后对笔者所开展的H-B强度准则最新研究工作进行介绍：提出一种真正意义上的广义三维H-B强度准则,并采用3种Lode角函数进行屈服面修正,使其可以直接应用于后续本构模型建立和数值软件嵌入;采用三维颗粒流模型进行微观数值建模,对H-B强度准则岩石和岩体参数开展微观研究并建立多尺度联系,为参数的确定提供更加可靠的依据。     

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.     

Bayesian assessment of the characteristic concrete compressive strength using combined vague-informative priors

Bayesian statistics can be used in order to determine the characteristic value of an unknown distribution based on a limited number of test samples. In cases where no previous test results are available, most often a Bayesian method based on vague prior information is used. The assumption of a vague or uniform prior results in a conservative approach in cases where only a limited number of test results are available. However, in case of concrete, prior information on concrete strength can be found in literature or country-specific prior information can be determined. Therefore, the use of a combined vague-informative prior is of particular interest, more specifically in the form of scaled inverse-χ² distributions that can be used for updating the standard deviation of the strength distribution of concrete. The differences between the use of a vague and a combined vague-informative prior are investigated through Monte Carlo simulations. Because prior information is taken into account, the uncertainty regarding the standard deviation of the predictive strength distribution is significantly reduced.     

Correlation of the Hoek–Brown failure criterion for a sparsely jointed rock mass with an extended plane of weakness theory

An appropriate estimate of rock mass strength is necessary for the design of civil and mining structures built in or on rock. Rock mass is an inhomogeneous and anisotropic material with complex behaviour, which contains random planes of discontinuities that tend to reduce its strength. The direct estimation of this strength is practically unfeasible, due to difficulties in sampling and testing. This has led to the development of empirical failure criteria. These, express the strength of the rock mass in terms of properties of the intact rock and the discontinuities. The Hoek–Brown criterion is the most widely accepted one. However, albeit its use for many years, no experimental in situ validation with the actual rock mass strength has been demonstrated. Therefore, the Hoek–Brown criterion is investigated analytically through an extended plane of weakness theory, already validated with experimental evidence on physical specimens. Various intact rock qualities with blocky and very blocky structure are examined. The results indicate deviations in the rock mass strength predicted by the two approaches, especially when the intact rock strength is low.     

Polyaxial strength criterion and closed-form solution for squeezing rock conditions

In this paper,a nonlinear strength criterion is proposed using the average of intermediate(σ_2) and minor(σ_3) principal stresses in place of σ_3 in Ramamurthy(1994)'s strength criterion.The proposed criterion has the main advantages of negligible variation of strength parameters with confining stress and ability to link with conventional strength parameters.Additionally,a new closed-form solution based on the proposed criterion is derived and validated for Chhibro Khodri tunnel.Further,analytical solutions including Singh's elastoplastic theory,Scussel's approach,and closed-form solutions based on conventional and modified Ramamurthy(2007) criteria are compared with the results of proposed approach.It is shown that the in situ squeezing pressure predictions made by the proposed approach are more accurate.Also,a parametric study of the present analytical solution is carried out,which displays explicit dependency of tunnel stability on internal support pressure and tunnel depth.The influence of tunnel geometry is observed to be dependent on the applied support pressure.