Hoek-Brown Strength Parameters Research Articles

Establishing Hoek–Brown Strength Parameters for Artificially Cemented Soils: Practical Methodology

Establishing Hoek–Brown Strength Parameters for Artificially Cemented Soils: Practical Methodology

Critical strain method incorporating calculation of equivalent H-B strength parameters for back analysis during tunnel excavation

A comprehensive method is proposed for the continuous update of the design rockmass parameters during tunnel excavation coupled to novel calculation of the Hoek-Brown strength parameters for given Mohr-Coulomb ones. The method starts with the strength parameters of Mohr-Coulomb or Hoek-Brown criterion used at the design stage of tunnels, and then updates them by using the displacements measured during the excavation of tunnels. The update is performed based on a well established back analysis procedure that determines both cohesion C and internal friction angle φ of the Mohr-Coulomb criterion. Then, these parameters either directly update or are converted to equivalent Hoek-Brown parameters that update the original design parameters of the tunnel. The need arising for conversion of parameters during the back analysis is thus addressed: A method is developed for the evaluation of the equivalent Hoek-Brown parameters for an axisymmetric structure subjected to major principal initial stress p0 and supported by internal pressure pi. Four characteristic cases are analysed theoretically and numerically with a finite differences code and found to be in excellent agreement. The equivalent parameters solution is developed in Matlab source code under a creative commons license and greatly enables the automation of the update procedure.Based on the updated parameters, it is possible to continuously modify the original design of underground support structures, according to the results of back analysis of measured displacements. The modified strength parameters of the H-B criterion can also be used for assessing the stability of the tunnels for the next excavation stage of the tunnels.

Interaction effects and an optimization study of the microparameters of the flat-joint model using the Plackett-Burman design and response surface methodology

In view of the large number of microparameters of the flat-joint model (FJM) and the inefficiency of the calibration process, this paper uses the Plackett-Burman (PB) design, response surface methodology (RSM), and optimization techniques to calibrate the FJM3D. The values of each lower and upper bound in the PB design are selected based on the relationship between 10 microparameters and the UCS, BTS, UCS/BTS, elastic modulus E, Poisson’s ratio υ, Hoek-Brown strength parameter mi, average coordination number (CN), and crack initiation stress σci. Then, the PB design is used in the sensitivity analysis to screen out the three most influential factors for each response for the central composite design (CCD). The three microparameters most influencing E are the effective modulus Empb, the ratio of the normal to shear stiffness KnKs, and the installation gap ratio Gapr; the three microparameters most influencing υ are KnKs, the mean bond cohesion strength Coh, and Empb; the three microparameters most influencing UCS are Coh, Gapr, and the mean bond tensile strength Mbts; and the three microparameters most influencing BTS are Mbts, KnKs, and the residual friction angle Rfa. CCD analysis reveals that there are interaction effects between the FJM3D microparameters and the real effect of each microparameter differs at different levels of other microparameters. The linear and nonlinear equations obtained by fitting the PB design and the CCD design are applied as constraints to the optimization problem, and the problem of obtaining the best set of FJM3D microparameters is solved. The optimal sets were successfully matched for a variety of rock types from both the FJM3D simulations and experiments, and Lac du Bonnet granite was used as an example to compare the strength, deformation characteristics, and rock failure modes. This method can be used as a new way to quickly calibrate the microparameters of the FJM3D.

Mobilized Mohr-Coulomb and Hoek-Brown Strength Parameters during Failure of Granite in Alxa Area in China for High-Level Radioactive Waste Disposal

Strength parameters of the host rock is of paramount importance for modelling the behaviors of underground disposal repository of high-level radioactive waste (HLW). Mobilization of strength parameters should be studied for a better understanding and modelling on the mechanical behaviors of the surrounding rock, considering the effect of temperature induced by the nuclear waste. The granite samples cored from NRG01 borehole in Alxa candidate area in China for HLW disposal are treated by different temperatures (T = 20 °C, 100 °C and 200 °C), and then are used to carry out a series of uniaxial and tri-axial compression experiments under various confining pressures (σ3 = 0, 5, 10, 20, and 30 MPa) in this study. With the recorded axial stress—axial strain and axial stress—lateral strain curves, mobilization of both Mohr-Coulomb and Hoek-Brown strength parameters are analyzed with the increasing plastic shear strain. It has been found that NRG01 granite samples show generally similar cohesion weakening and friction strengthening behaviors, as well as the non-simultaneous mobilization of Hoek-Brown strength parameters (mb and s), under the effect of various treatment temperatures. Furthermore, the samples treated by higher temperatures show lower initial values of cohesion, but their initial friction angle and mb values are relatively higher. This should be mainly owing to the thermally induced cracks in the samples. This study should be helpful for a better modelling on the mechanical behaviors of NRG01 granite samples as the host rock of a possible HLW disposal repository.

The role of transgranular capability in grain-based modelling of crystalline rocks

Considering that many numerical methods have been developed to study the mechanical properties of rocks based on micro heterogeneities such as mineral size, morphology, composition and boundary defects, an overview of grain-based modelling is presented in this study to compare the advantages and limitations of different grain-mimicking methods. Intergranular and transgranular fracturing models and difficulties in parameter calibrations are discussed. Four typical grain models, UDEC-GBM, GB-FDEM, cluster and clump, are used to represent the two main families: block-based grains and particle-based grains with and without considering grain crushing. Subsequently, how grain crushing affects crack stresses, Hoek-Brown strength parameters mi, localized shearing and cracking transformation is simulated by these grain models. The simulated results indicate that the approaches capable of grain breakage lead to more consistent results in comparison with laboratory tests, for example, evident dilatancy in uniaxial compressive loading, high strength ratios, nonlinear failure strength envelopes and shear bands under high confinement. The role of transgranular capability is significantly important in the simulation of rock deformation using grain models.

A Multicomponent Particle Model and Linear Fitting Calibration Method for Heterogeneous Rocks

Deformation and failure of rock are very important to underground space engineering. In this paper, a new three‐dimensional multicomponent bonded‐particle model is developed for heterogeneous rocks. Granite samples from the Sanshandao Gold Mine are first analyzed using a microscope. Cylindrical and disc models consisting of four minerals (i.e., plagioclase, potash feldspar, quartz, and biotite) are built to simulate the behavior of the granite under compressive and tensile tests. To improve the calibration efficiency, a new method for determining the microparameters of minerals is proposed. Uniaxial compression, triaxial compression, and Brazilian tests are carried out in the lab. The failure pattern and stress‐strain curves are obtained from numerical simulations and verified with those observed in the experiments. Furthermore, the Mohr–Coulomb and Hoek–Brown strength parameters are obtained and compared with the experimental results. The multicomponent particle model is shown to well reproduce the behavior of granite under the compressive and tensile tests. The multicomponent bonded‐particle model significantly improves the accuracy of three parameters: the ratio of tensile strength to uniaxial compressive strength, the friction angle, and the parameter mi of the Hoek–Brown criterion. The linear fitting calibration method provides a fast and efficient way to determine the microparameters of particles in heterogeneous rocks.

Numerical Simulation of Bedrock Sagging Sinkholes in Strain‐Softening Rock Induced by Embankment Construction

A bedrock sagging sinkhole occurred in Jiangxi Province of China when constructing the Changli freeway above shallow karst caves. It was chosen as a case to investigate the failure mechanism and potential evolution. The in situ stress of the study area was measured and numerically reproduced. The Hoek–Brown strength parameters were obtained by laboratory tests. A strain‐softening constitutive model was established according to the strain‐softening behaviour exhibited by the specimens in the triaxial test. The stress‐strain curves of the specimens were reproduced by numerical methods. Then, bedrock sagging sinkholes in strain‐softening rock induced by embankment construction were simulated. The occurrence of the strain‐softening zone and its transition to the residual zone were observed and classified into four stages. The stress paths of the four stages were analysed. Interestingly, in this case, the supports at both ends of the bedrock began to yield from the top and extended downward, while the midspan position began to yield from the bottom and extended upward, and the reasons for yielding were related to tension. Further analysis found that the failure mode was basically consistent with the size and direction of the bending moment. In fact, this failure mode was quite similar to a fixed supported beam. Then, the feasibility of calculating the stability of karst caves based on beam assumptions was discussed. Finally, potential evolution of the bedrock sagging sinkhole was discussed.

Comprehensive statistical analysis of intact rock strength for reliability-based design

Rock engineering design is currently evolving from the traditional use of safety factors and towards the more rational reliability-based design (RBD), as witnessed by the introduction of a number of geotechnical limit states design (LSD) standards worldwide. The probabilistic nature of RBD requires statistical characterization of design parameters, including the strength of intact rock. The triaxial compressive strength of intact rock is commonly characterized as a function of confining pressure, and values of the parameters that define this function are generally obtained by regression analysis of laboratory test data. However, the problem of fitting strength criteria to intact rock strength data has been historically tackled as a problem of obtaining best fit curves only, and has inadvertently omitted characterization of the variability inherent in the strength data. As a result, the uncertainty in parameter estimations resulting from this variability are often not quantified rigorously. Not only does this omission render such regression analyses incomplete, it also limits development of reliability-based design protocols for rock engineering which require statistical characterization of design parameters. To overcome both of these deficiencies, this paper presents frequentist (i.e. classical) and Bayesian regression models that rigorously incorporate variability and uncertainty associated with estimated Hoek-Brown strength parameters. In particular, it discusses the limitations of the frequentist model when dealing with limited data or a combination of tensile and compressive strength data. It discusses the potential of Bayesian data analysis techniques to overcome the issue of limited data in rock engineering design by using informative prior distributions. The paper also demonstrates the main issue in fitting strength criteria to tensile and compressive data simultaneously from a statistical perspective, and presents a Bayesian regression model that rigorously fits the strength criterion to a combination of tensile and compressive data. The paper concludes with suggested statistical approaches – Bayesian or frequentist – for conditions encountered in the analysis of intact rock strength data.

Effect of Small Numbers of Test Results on Accuracy of Hoek–Brown Strength Parameter Estimations: A Statistical Simulation Study

The Hoek–Brown empirical strength criterion for intact rock is widely used as the basis for estimating the strength of rock masses. Estimations of the intact rock H–B parameters, namely the empirical constant m and the uniaxial compressive strength $$\sigma_{\text{c}}$$ , are commonly obtained by fitting the criterion to triaxial strength data sets of small sample size. This paper investigates how such small sample sizes affect the uncertainty associated with the H–B parameter estimations. We use Monte Carlo (MC) simulation to generate data sets of different sizes and different combinations of H–B parameters, and then investigate the uncertainty in H–B parameters estimated from these limited data sets. We show that the uncertainties depend not only on the level of variability but also on the particular combination of parameters being investigated. As particular combinations of H–B parameters can informally be considered to represent specific rock types, we discuss that as the minimum number of required samples depends on rock type it should correspond to some acceptable level of uncertainty in the estimations. Also, a comparison of the results from our analysis with actual rock strength data shows that the probability of obtaining reliable strength parameter estimations using small samples may be very low. We further discuss the impact of this on ongoing implementation of reliability-based design protocols and conclude with suggestions for improvements in this respect.

Reliability analysis of jointed rock slope considering uncertainty in peak and residual strength parameters

Stability analysis of rock slopes is a complex problem because of uncertainties involved in the rock mass properties. The probabilistic approach is a rational way to deal with these uncertainties. This article investigates the stability of a rock slope by deterministic and probabilistic approaches by considering uncertainty in peak and residual strength parameters and strength-drop in stress-strain behavior of the rock mass. A Geological Strength Index (GSI) based on a quantitative approach was used to estimate the statistical parameters of peak and residual strength parameters. Reliability index of the slope was then estimated using Hong’s Point Estimate Method coupled with the finite element method. The approach is demonstrated using an important case study of a Himalayan rock slope supporting the piers of the world’s highest railway bridge. It was observed that the factor of safety and reliability index for the slope was highly sensitive to residual strength parameters, and; hence, ignoring strength-drop from the rock mass behavior and uncertainty in residual strength parameters can overestimate the factor of safety and the reliability index of rock slopes. A parametric study is carried out to evaluate the influence of the coefficient of variation of uniaxial compressive strength of intact rock, the Hoek-Brown strength parameter of intact rock, roughness and alteration parameters of the joints on the probability of failure and the reliability index. The approach is verified by comparing the estimated displacements along the slope with in-situ measured displacements observed during field monitoring over the years. The approach used can be extended to the rock slopes or rock slides where high displacements in the slopes are expected due to triggering forces like seismic forces, excavation or structural loads.

The effect of microstructure on mi strength parameter variation of common rock types

A series of laboratory tests was conducted on selected intact sedimentary (sandstones and limestones) and igneous (granites and andesites-dacites) rocks obtained from several sites of Greece. Physical and mechanical properties were determined, while the investigation was focused on strength parameters determination and especially on Hoek-Brown strength parameter mi through the triaxial compression. Mineral composition of grains, as well as microstructure (grain size, shape and interlocking) was estimated through microscopic examination using some well known petrographic indices. The alteration and weathering state for the studied igneous rocks were also microscopically quantified. Regression analysis of laboratory data have clearly shown a direct strong intrinsic influence of microstructure as quantitatively described by the petrographic indices and by textural interlocking of the grains, as well as of grain deterioration on measured mi value. The mi value generally increases with the increasing grain size of studied clastic and igneous rocks, but it decreases to a power expression with sparitic material of limestones. The increase of weathering and alteration state also decreases the mi parameter of tested igneous rocks.

Estimation of rock block size distribution for determination of Geological Strength Index (GSI) using discrete fracture networks (DFNs)

Rock mass classification provides fundamental data for a numerical stability analysis of rock structures. Among rock mass classification systems, the RMR and Q systems often are used for rock support system selection and the Geological Strength Index (GSI) system for estimating rock mass strength and deformation parameters. Moreover, the GSI system is the only rock mass classification that is directly linked to engineering design parameters such as the Mohr–Coulomb or Hoek–Brown strength parameters or the rock mass modulus. However, the original application of the GSI system requires long-term experience and a careful approach because of the fact that its use is a subjective decision. A quantitative approach to assist a less experienced engineer in assigning representative GSI values was presented. It employed the rock block volume and joint conditions as quantitative characterisation factors. Their approach is founded on the linkage between descriptive geological terms and measurable field parameters, such as joint spacing and joint roughness. In this study, a discrete fracture network (DFN) model incorporated with stochastic simulation is applied to characterise rock block size distribution for determination of the GSI. The fracture frequency obtained from the core logging data is analysed and provided to the DFN model as input data. Realisation of the DFN and its verification are conducted to establish the joint systems corresponding to the original fracture frequency. As a result, the stochastic simulation can successfully provide the information on the rock block size distribution to the procedure of the GSI determination.

Stability Analysis of an Open Cut Slope in Wardha Valley Coal Field

Abstract Open cast mines are the prime source of coal production in India. Due to an increase in coal demand, deeper surface mines are being planned to ensure better productivity with enhanced safety of these mines. The safe working environment and continuous production of the coal calls for the safe and stable design of cut slopes. The stability analysis of this slope requires in depth geotechnical investigation which are aided by result orientated stability assessment using empirical methods and numerical simulation. In the present study, a slope was examined to understand the mechanism and comparisons were made with field observations. The investigation has involved a 32m high cut slope from an open cast mine in Wardha valley coal field which has been analyzed using a two dimensional numerical simulation. The bench slope consisted of a low strength sandstones, shales and clay sequence. Hoek-Brown strength parameters established and used as input parameters in the model. The results indicate that the slope is critically stable and may lead to failure without warning and needs proper attention.

A generalized three-dimensional failure criterion for rock masses

The smooth convex generalized failure function, which represents 1/6 part of envelope in the deviatoric plane, is proposed. The proposed function relies on four shape parameters (Ls, a, b and c), in which two parameters (a and b) are dependent on the others. The parameter Ls is called extension ratio. The proposed failure function could be incorporated with any two-dimensional (2D) failure criteria to make it a three-dimensional (3D) version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion for geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering (1) constant extension ratio and (2) variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio (Ls) that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters (σc and m) are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameter f is introduced. The modified extension ratio is related to f and extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range of f varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression between f and Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures.

Estimating rock mass properties using Monte Carlo simulation: Ankara andesites

In the paper, a previously introduced method ( Sari, 2009) is applied to the problem of estimating the rock mass properties of Ankara andesites. For this purpose, appropriate closed form (parametric) distributions are described for intact rock and discontinuity parameters of the Ankara andesites at three distinct weathering grades. Then, these distributions are included as inputs in the Rock Mass Rating ( RMR) classification system prepared in a spreadsheet model. A stochastic analysis is carried out to evaluate the influence of correlations between relevant distributions on the simulated RMR values. The model is also used in Monte Carlo simulations to estimate the possible ranges of the Hoek–Brown strength parameters of the rock under investigation. The proposed approach provides a straightforward and effective assessment of the variability of the rock mass properties. Hence, a wide array of mechanical characteristics can be adequately represented in any preliminary design consideration for a given rock mass.

Elasto‐plastic analytical model for the design of grouted bolts in a Hoek–Brown medium

AbstractThis paper presents an elasto‐plastic analytical solution of an axi‐symmetrical problem for a circular tunnel reinforced by grouted bolts. Considered as the improved model of Indraratna and Kaiser (Int. J. Rock. Mech. Min. Sci. Geomech. Abstr. 1990; 27:269–281; Int. J. Numer. Anal. Meth. Geomech. 1990; 14:227–251), in proposed solution the rock mass obeys the non‐linear Hoek–Brown yield criterion (version 2002) in terms of its peak and residual strength parameters (the most spread strength criterion for the rock masses). The proposed approach considers a⩾0.5 for the rock mass and is based on the assumption that after the peak strength of the rock is reached, the material loses its strength, as dictated by a strength loss parameter. The strength loss parameter makes it possible to model either elastic–perfectly plastic or elastic–brittle–plastic behaviour. Because of the mathematical complexity, numerical treatments have been used to assist the solution in order to evaluate the equilibrium and compatibility equations. The concept of equivalent material for Hoek–Brown strength parameters is introduced to describe the rock mass improvement due to bolting effect. The results of the numerical analyses reveal a linear relation between the improved Hoek–Brown strength parameters and residual ones, taking into consideration the bolt density parameter (β). The proposed solution is able to analyse the stress and displacement state in the presence of a bolting intervention with the objective of improving the degree of stability of the rock around the tunnel. Descriptive applications of the derived elasto‐plastic solutions are also presented to explain the effectiveness of the grouted bolts in convergence reduction. Evidences obtained by numerical analysis verify the analytical solution. Copyright © 2009 John Wiley & Sons, Ltd.

Practical Estimates of Tensile Strength and Hoek–Brown Strength Parameter m i of Brittle Rocks

By applying the Griffith stress criterion of brittle failure, one can find that the uniaxial compressive strength (σc) of rocks is eight times the value of the uniaxial tensile strength (σt). The Griffith strength ratio is smaller than what is normally measured for rocks, even with the consideration of crack closure. The reason is that Griffith’s theories address only the initiation of failure. Under tensile conditions, the crack propagation is unstable so that the tensile crack propagation stress (σcd)t and the peak tensile strength σt are almost identical to the tensile crack initiation stress (σci)t. On the other hand, the crack growth after crack initiation is stable under a predominantly compressive condition. Additional loading is required in compression to bring the stress from the crack initiation stress σci to the peak strength σc. It is proposed to estimate the tensile strength of strong brittle rocks from the strength ratio of \( R = {\frac{{\sigma_{\text{c}} }}{{\left| {\sigma_{\text{t}} } \right|}}} = 8{\frac{{\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}}. \) The term \( {\frac{{\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}} \) accounts for the difference of crack growth or propagation in tension and compression in uniaxial compression tests. \( {\frac{{\sigma_{c} }}{{\sigma_{ci} }}} \) depends on rock heterogeneity and is larger for coarse grained rocks than for fine grained rocks. σci can be obtained from volumetric strain measurement or acoustic emission (AE) monitoring. With the strength ratio R determined, the tensile strength can be indirectly obtained from \( \left| {\sigma_{\text{t}} } \right| = {\frac{{\sigma_{\text{c}} }}{R}} = {\frac{{\sigma_{\text{ci}} }}{8}}. \) It is found that the predicted tensile strengths using this method are in good agreement with test data. Finally, a practical estimate of the Hoek–Brown strength parameter mi is presented and a bi-segmental or multi-segmental representation of the Hoek–Brown strength envelope is suggested for some brittle rocks. In this fashion, the rock strength parameters like σt and mi, which require specialty tests such as direct tensile (or Brazilian) and triaxial compression tests for their determination, can be reasonably estimated from uniaxial compression tests.

Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system

Abstract Rock mass characterization is required for many applications in rock engineering practice including excavation design, support design, stope design, amongst others. For these purposes, it is necessary to obtain design input parameters such as deformation moduli and strength parameters for numerical modeling. Although such parameters can ultimately be determined from in situ tests, at the preliminary design stage, where access to underground is limited, the practical way to obtain these parameters is to apply a rock mass classification system to characterize the rock mass and estimate the rock mass properties. Over the years, many classification systems, such as RQD, Rock Mass Rating, Q and Geological Strength Index (GSI) systems, have been developed. Amongst them, the Q system is widely used for rock support system design and the GSI system is used for estimating design parameters. The GSI system is the only rock mass classification system that is directly linked to engineering parameters such as Mohr–Coulomb, Hoek–Brown strength parameters or rock mass modulus. However, the application of the existing GSI system is hindered by the facts that the use of the system is to some extent subjective and requires long-term experience. In the present study, a quantitative approach to assist in the use of the GSI system is presented. It employs the block volume and a joint condition factor as quantitative characterization factors. The approach is built on the linkage between descriptive geological terms and measurable field parameters such as joint spacing and joint roughness. The newly developed approach adds quantitative means to facilitate use of the system, especially by inexperienced engineers. The GSI system is applied to characterize the jointed rock masses at two underground powerhouse cavern sites in Japan. GSI values are obtained from block volume and joint condition factor, which in turn are determined from site construction documents and field mapping data. Based on GSI values and intact rock strength properties, equivalent Mohr–Coulomb strength parameters and elastic modulus of the jointed rock mass are calculated and compared to in situ test results. The point estimate method is implemented to approximate the mean and variance of the mechanical properties of the jointed rock masses. It is seen that both the means and variances of strength and deformation parameters predicted from the GSI system are in good agreement with field test data.

Stability Analysis of Jointed/Weathered Rock Slopes Using the Hoek-Brown Failure Criterion

ABSTRACT The failure of rock slopes usually occurs along the discontinuities. But for weathered or highly jointed rock mass, failure surface is often curved as in soil slopes. To assess the stability of jointed/weathered rock slopes, shear strength reduction technique is adapted for use with the non-linear Hoek-Brown failure criterion, an empirical approach to estimating rock mass strength. Hoek-Brown strength parameters can be estimated using rock mass classification schemes such as RMR or GSI. Numerical results obtained by strength reduction are compared with limit analysis upper bound solutions derived using a series of linear failure surfaces tangent to the actual non-linear failure surface. Results are presented in graphs of normalized slope height versus RMR that could be used as stability charts.