Hoek Brown Research Articles

Numerical Study of P-Waves Propagating across Deep Rock Masses Based on the Hoek–Brown Model

Strong dynamic disturbances constitute an important problem threatening the safety of deep underground engineering projects. The mechanical mechanisms responsible for such disturbances in deep rock masses can be revealed by studying stress waves propagating across them. In this study, the propagation of P-waves across a deep rock mass was investigated both numerically and theoretically. The Hoek–Brown (HB) model, which is able to account for the stress state and nonlinear failure of the rock mass, was employed. In particular, the time-domain recursive method was extended to analyze the propagation of P-waves across the rock mass following the HB failure criterion. Good agreement is found between the theoretical results and simulations, which demonstrates that using the Universal Distinct Element Code (UDEC) with the HB model to study the transmission of P-waves across rock masses is eminently feasible. The effects of the disturbance factor, confining pressure, incident amplitude, and coupling of joints and rock mass quality on wave propagation were also studied using the UDEC in a systematic manner. The results show that for an incident P-wave of given amplitude, the greater the rock mass quality and smaller the disturbance factor, the greater the transmission coefficient. In addition, there is an upper limit to the effect the confining pressure has on the transmission coefficient. The amplitude of the incident wave also has an important influence on the transmission coefficient. Finally, if the quality of the rock mass is good and the disturbance slight, then the attenuation of the wave is primarily caused by the joints.

Numerical investigations of rock bridge effect on open pit slope stability

Abstract In this study, the effect of rock bridges on rock slope stability was investigated by incorporating non-persistent joint networks in numerical models, and the critical profiles of an open pit mine were analysed. Parallel deterministic networks of infinite and finite lengths, ubiquitous joint network model and Veneziano joint network model were used in order to simulate the rock fractures. Materials were modelled based on the generalised Hoek―Brown and equivalent Mohr–Coulomb failure criteria. The parallel deterministic infinite and the ubiquitous joint network models produced lower safety factors. The introduction of rock bridges along discontinuity planes in the parallel deterministic network and Veneziano joint network models significantly contributed to the stability and strain distribution, which should be considered in stability analysis of rock mass in open pit by rock slope practitioners. The results show the significance of joints in hard rock behaviour and the joints should be included in order to attain practical and realistic simulations.

A Mathematical Model Based on Artificial Neural Networks to Predict the Stability of Rock Slopes Using the Generalized Hoek–Brown Failure Criterion

Stability of rock slopes is a critical issue in many mining and civil engineering projects. The current state of practice for slope stability analysis is based on obtaining the factor of safety (FOS). Stability charts are widely used by engineers to obtain a FOS for a quick assessment of the initial stability of slopes. The stability of rock slopes with vertical walls in urban areas adjacent to existing structures is another important issue in this regard. However, the stability of earth or rock slopes are usually analyzed ignoring surcharge loads. The effect of adjacent structures (as surcharge load) on slope stability (considering these loads) can be very useful in slope stability analyses in urban or even non-urban areas. In the present study, it is tried to investigate the effect of surcharge load on the stability of rock slopes based on the generalized Hoek–Brown failure criterion using a finite element numerical software and the related charts are presented. Since there is a stability chart for each slope angle, a comprehensive mathematical model utilizing artificial neural networks is proposed to predict the stability factor of rock slopes. The independent variables in this study were slope angles, slope height, the intensity of surcharge load, Geological Strength Index (GSI), and unconfined compressive strength of the intact rock. Sensitivity analysis showed that the changes in GSI, effect of surcharge and unconfined compressive strength of rock have the highest effect on the slope stability assurance coefficient.

Numerical simulation of karst environments to study subsidence

Karstic zones, due to dissolution of limestone rocks, are a problem in several areas of the world, being associated with damage sustained by facilities, among other issues. A numerical study is presented using Plaxis 2D to analyse terrain subsidence on the surface, considering a transversal section to apply a road load, while cavities in the ground increase due to karsticity. Results allow perception of the surface subsidence, associated with the number of cavities, before and up to collapse of the first voids created. Stress paths developed in the cavities’ opening process are built from the interpretation of results provided by the numerical analyses. These stress paths make it possible to obtain the way in which stress values change between initial and final states, defining an in situ stress variation for the void's presence. Furthermore, it is concluded that there is a tendency towards stability for the superficial displacement and a maximal affected horizontal length associated with a specific number of cavities, despite the fact that more of these are developed in the zone. The rocks and the soil were simulated by Hoek–Brown and Mohr–Coulomb constitutive models, considering the specific characteristics of rocks and soil endemic to Yucatan, Mexico.

Ultimate Bearing Capacity of Rock Mass Foundations Subjected to Seepage Forces Using Modified Hoek–Brown Criterion

The experimental data shows that most rocks behave nonlinearly in nature. The modified nonlinear Hoek–Brown failure criterion was considered to investigate the bearing capacity problem of shallow rigid foundations on rock masses subjected to horizontal seepage forces. Two multi-wedge translational failure mechanisms, including symmetrical and non-symmetrical mechanisms were used in the closed-form of the upper bound method of the limit analysis theory. The symmetrical failure mechanism was used in the case of no seepage, while the seepage effect was considered in the non-symmetrical mechanism. The variation of seepage forces was obtained as a function of gradient ratio i(γw/γsub) in the developed formulation. The bearing capacity coefficients Nγ, Nq and Nσ are introduced for the case of seepage flow condition. The results show that the magnitude of the bearing capacity coefficients reduces continuously with an increase in the value of gradient ratio i(γw/γsub). The obtained results were compared and offered for functional use in foundation engineering.

Practical Use of Modified Hoek–Brown Criterion for Soil Slope Stability Analysis

Many slopes are comprised of soils that exhibit a nonlinear shear strength or failure envelope, and multiple mathematical relationships have been developed to account for this nonlinearity. At the same time, the numerical shear strength reduction (SSR) method has become a common method for analyzing the stability of slopes. Despite these developments, a practical, commercially available method to include nonlinear shear strength in numerical analysis has not been established for soil. The Generalized Hoek–Brown (GHB) model provides a nonlinear failure criterion, but is formulated for use with rock. This paper proposes a Modified Hoek–Brown (MHB) criterion to make the model applicable to soil and leverage the GHB criterion present in many numerical analysis packages. Past applications of SSR to the GHB are discussed and a numerical method for reduction of the parameters in the context of soil slopes is proposed. A simple wedge analysis validates the MHB method for a linear envelope. Three examples of increasing complexity compare results of limit equilibrium with both finite element and finite difference SSR analyses. In general, the different numerical methods yield very similar results. The SSR method using MHB predicts critical strength reduction factors 2–5% lower than the limit equilibrium factors of safety. The approach presented in this paper allows practitioners to model nonlinear shear strength in finite element strength reduction analysis for cases where this nonlinearity is judged to be an important factor.

Estimation of the Mechanical Properties of Igneous Rocks in Consideration of Seismic Effects

It is not widely accepted that the dynamic strain rate experienced by a rock mass during an earthquake is lower than that during a high strain rate scenario such as an explosion or a rock burst. The present study investigated the problem of determining the mechanical properties of rock considering earthquake effects using experimental tests and empirical formulas. First, to obtain the most concentrated frequency band of earthquake energy, a statistical analysis was conducted to obtain the response spectra of ground motion records near the example projects. The mean response spectrum was then fitted with a design response spectrum formula. The most concentrated frequency band of earthquake energy was then determined with the flat segment of the design response spectrum. Based on this frequency range, cyclic loading-unloading testing that simulated the earthquake effect was conducted on rock samples. The strain rate range experienced by rocks during an earthquake was then obtained. This strain rate range is between the quasi-static and intermediate strain rate ranges and is significantly lower than the high strain rate range. Based on this strain rate range, dynamic triaxial compression tests were conducted on igneous rock specimens from the case study areas. It is confirmed that the fit of the dynamic compression test results with the Hoek-Brown (HB) criterion is more accurate than that with the Mohr-Coulomb criterion. The HB strength envelope under static conditions can be used to predict the seismic strength behaviors, requiring an update of only the intercept of the envelope. Thus, the HB strength criterion considering the seismic effect was obtained. Moreover, the seismic modulus of the intact rock can be estimated with the values of the uniaxial compressive strength (UCS) under different strain rates and the modulus ratio, which was found to be independent of the strain rate. Hence, it is practical to estimate the mechanical properties of rock masses in consideration of seismic effects with the seismic HB strength criterion and the Hoek and Diederichs equation based on the UCSs under various strain rates. Ultimately, the rock mechanical properties in consideration of seismic effects were estimated via the proposed approach for the Baihetan hydropower plant project as an example.

Wellbore-Stability Analysis by Integrating a Modified Hoek-Brown Failure Criterion With Dual-Porochemoelectroelastic Theory (includes associated erratum)

Summary When drilling through naturally fractured formations, the existence of natural fractures affects the fluid diffusion and stress distribution around the wellbore and induces degradation of rock strength. For chemically active formations, such as shale, the chemical–potential difference between the drilling mud and the shale–clay matrix further complicates the nonmonotonic coupled pore–pressure processes in and around the wellbore. In this work, we apply a recently formulated theory of dual–porosity/permeability porochemoelectroelasticity to predict the time evolution of mud–weight windows, while calculating stresses and pore pressure around an inclined wellbore drilled in a fractured shale formation. The effects of natural–fracture geometric and spatial distributions coupled with the chemical activity are considered in the wellbore–stability analysis. To account for the degrading effect of the fractured shale matrix on the bulk rock strength, a modified Hoek–Brown (MHB) criterion is developed to more closely describe the in–situ state of stress effects on the compressive shearing strength at great depth. Compared with the original Hoek–Brown (HB) failure criterion, the MHB criterion considers the influence of the intermediate principal stress and thus shows better agreement with true–triaxial data for various rocks at varying stress levels. The MHB criterion converges to the original HB criterion when the confining in–situ stresses are equal. Two field case studies indicate that this novel integrative methodology is capable of predicting the operational drilling–mud–weight windows used in these two cases. Another advantage of this newly developed technique is that it can be used as a back–analysis tool to estimate the fracture–matrix permeability from field operational data.

A cohesion loss model for determining residual strength of intact rocks

Residual strength of rock is required in stability evaluation of underground excavations such as tunnels and caverns. Accurate determination of residual strength of rock can facilitate cost-effective design of these underground structures. This study investigates the methods for determining residual strength of intact rocks. Firstly, previous methods for determining residual strength of intact rocks are examined from a literature review, and a new method, which uses the concept of cohesion loss, is proposed. The proposed cohesion loss model is based on the framework of the widely-used Hoek–Brown (H–B) strength model. It is simple and contains only one parameter that controls the nonlinearity of the residual strength envelope. The cohesion loss model is validated using two sets of test data and the results show that the model can represent residual strength envelopes of intact rocks satisfactorily. Finally, the model parameter is studied by examining a large amount of test data collected from previous publications. It is found that the model parameter varies for different rock types and shows a decreasing trend as the rock strength increases. Overall, the results in this study show that the proposed model can capture the residual strength of intact rocks.

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 (HB) 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 HB parameters m and σc, 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 σc suitable for use in RBD. This analysis also shows how compressive and indirect tensile strength data may be robustly used to fit an HB 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 mk and σck, 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 HB 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.

Study on the damage-softening constitutive model of rock and experimental verification

A damage-softening model is presented to describe the stress–strain curve of rock. By comparing the Hoek–Brown (H–B) and Mohr–Coulomb (M–C) yield criterion, the equivalent M–C yield criterion is selected as the strength criterion in this model. To better characterize the rock damage and failure processes with considering the relationship between damage and deformation, the concept of yield stress ratio is introduced to describe the yield-strengthening deformation before rock peak stress. Damage events are described by two cumulative damage evolution laws. The evolution equations of tensile and shear damage are presented based on the equivalent plastic strains, and the maximum value between tensile and shear damage represents the total damage for rock. Considering that rock cannot bear tensile load after tensile failure but still has a certain shear strength, its tensile and shear strengths are small after shear failure. The elastic modulus is affected by tensile damage, whereas the angle of internal friction, the cohesion, and dilation angles are influenced by shear damage. The proposed damage-softening model describes the strain softening, brittle stress drop, and residual strength of rock after peak stress, and finally the model is implemented in FLAC3D. Comparing the test and the numerical calculation results, the damage-softening model better describes the total stress–strain curve of rock.

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 mb, 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.

Upper Bound Solution of the Safety Factor for a Shield Tunnel Face Subjected to the Hoek–Brown Failure Criterion

Assessment of the stability of a tunnel face under a setting support pressure is a key challenge in shield tunneling engineering, especially when a tunnel is excavated in a heavily fractured rock mass. In this paper, a factor of safety is introduced for investigating the face stability of a tunnel using the upper bound theorem in combination with the Hoek–Brown nonlinear failure criterion. The factor of safety for the tunnel face is defined as the ratio of the rate of the energy dissipation and the external rate of work in the kinematically admissible velocity field. The upper bound solution of the factor of safety is obtained from the optimal computation. The solutions provided by the presented approach are compared with those derived from a numerical simulation. The values of the factor of safety derived from these two methods are found to be in agreement, indicating that the proposed method is valid. Furthermore, based on the measured data of the chamber pressure and the parameters provided by the geological survey report, the proposed method is used to evaluate the stability of the tunnel face for an actual project. The effects of different parameters on the factor of safety are also discussed.

An Empirical UCS Model for Anisotropic Blocky Rock Masses

The Hoek–Brown (HB) failure criterion is one of the most widely used failure criteria in rock engineering. Based on the Geological Strength Index (GSI) system, a number of empirical models have been proposed in parallel with this criterion to estimate the strength and deformation properties of rock masses such as uniaxial compressive strength (UCS) and deformation modulus. However, the GSI system does not incorporate the effects of joint orientation β on the quality of a rock mass. This means that these empirical models cannot capture anisotropic rock mass strength caused by joint orientations. In this research, UDEC rock mass models, which are calibrated by laboratory data, are used to investigate the effects of joint orientation on rock mass strength in an unconfined state. The values of UCS obtained from the numerical simulation are then compared with those calculated from traditional empirical UCS models based on the GSI system. The comparison study shows that the value of UCS is significantly overestimated by the traditional empirical model when 10° < β < 45°, which will have serious safety implications for engineering designs. To rectify the problem, based on the analysis of numerical simulation results, an anisotropic weighting factor fβ is proposed to be used to refine the empirical UCS model. The modified UCS model is demonstrated to be capable of giving conservative but more accurate prediction of the rock mass strength for various joint orientations, which will result in more optimal and safer engineering designs.

Limit Analysis of Tunnel Collapse According to the Hoek–Brown Criterion and Bolt Parameter Research

Based on the plasticity theory of upper bound limit analysis, the expression of a potential collapse profile of a tunnel can be obtained from the $$\sigma $$ $$_{n}$$ $$-$$ $$\tau $$ $$_{n}$$ form of the Hoek–Brown criterion. However, because the parameters Aand B in the Hoek–Brown expression are difficult to determine, the bolts parameters cannot be determined easily. Therefore, it is necessary to adopt a parameter transformation method to establish the relation between the parameters in the general form ( $$\sigma $$ $$_{1}$$ $$-$$ $$\sigma $$ $$_{3}$$ form) and the $$\sigma $$ $$_{n}$$ $$-$$ $$\tau $$ $$_{n}$$ form of the Hoek–Brown criterion. The field rock data can be used to determine the critical span of tunnel collapse under different parameters. Then, the impact of the parameters of the rock mass, such as GSI, $$m_{i}$$ , D, $$\sigma $$ $$_{ci}$$ , and $$\gamma $$ are studied. And based on the L and h of the profile (L is the half span of the profile and h is the height), we studied the reasonable bolt parameters. The result shows that if the rock is more intact (GSI is larger), $$m_{i}$$ is smaller, the blasting disturbance is smaller (D is smaller), the compressive strength is larger ( $$\sigma $$ $$_{ci}$$ is larger), and the density of the rock is smaller, the range of the potential collapse profile is smaller. Generally, the support effect is more efficient when the length of the bolts is 2h and the spacing between the bolts is 15–20% of L. This research can provide a reliable reference for the accurate judgment of tunnel stability and effective control of roof settlement.

The Equivalence of Three Shear–Normal Stress Forms of the Hoek–Brown Criterion

The Equivalence of Three Shear–Normal Stress Forms of the Hoek–Brown Criterion

Theoretical Analysis of Damaged Width &amp; Instability Mechanism of Rib Pillar in Open‐Pit Highwall Mining

To improve the resources’ recovery ratio and the economic benefits of open‐pit mines, the development of highwall mining is used to exploit residual coal. The design of rib pillar, formed by excavation and mining activities in intact coal seams, is crucial to the overall stability and safety of the highwall mining operations. This paper focuses on the damage caused to rib pillars by the large deformation, occurring with the application of highwall mining in an open‐pit in China. A mechanical model was established to investigate the damaged width of colinear rib pillars based on Hoek–Brown and Mohr–Coulomb failure criterion. The equations for calculating the damaged width of the rib pillar were obtained, respectively, by combining the Hoek–Brown failure criterion with the Mohr–Coulomb failure criterion. The failure mechanism for the width of the rib pillar and the factors affecting the colinear rib pillar were analyzed in detail. The results show that the application of the Hoek–Brown criterion has a unique advantage in analyzing the damaged width of the colinear rib pillars, in open‐pit highwall mining. The instability mechanism and failure process of the rib pillars are described in combination with the limiting equilibrium method and the ratio of the elastic zone’s width to the width of the entire rib pillar.

Displacement Solution of Salt Cavern with Shear Dilatation Behavior Based on Hoek‐Brown Strength Criterion

In this study, an analytical solution of stress, strain, and displacement, in the elastic and plastic zone is proposed. The solution is derived on the basis of ideal elastoplastic mechanical model of spherical salt cavern with shear dilatation behavior, by adopting Hoek‐Brown (H‐B) criterion. The solution obtains not only in small and large strain stage but also in creep stage. The proposed solution is validated, by comparison of the obtained results with numerical results in FLAC3D. The results indicate that the result obtained adopting the H‐B criterion is closer to that one obtained adopting the Mohr‐Coulomb (M‐C). The H‐B criterion is more applicable for the salt cavern construction as it considers the structural characteristics of the rock salt formation. The displacement difference obtained by two different methods decreases with the increase of GSI or running pressure, but it increases with the enlarged angle of dilation. The influence of different assumptions of elastic strain of plastic zone on displacements is more significant under large strain conditions. The influence of the angle of dilation on displacements is more obvious when the elastic strain of plastic zone is given to stationary values, and the influence degree increases with the enlarged angle of dilation. Under the same conditions, the creep displacement decreases with the increase of GSI, and both the creep displacement and the effect degree enhance with the enlarged dilation angle. The proposed solutions can be used in the stability analysis of surrounding rock in the construction and operation of salt cavern storage.

Strength of massive to moderately jointed hard rock masses

The Hoek-Brown (HB) failure criterion and the geological strength index (GSI) were developed for the estimation of rock mass strength in jointed and blocky ground where rock mass failure is dominated by sliding along open joints and rotation of rock blocks. In massive, veined and moderately jointed rock in which rock blocks cannot form without failure of intact rock, the approach to obtain HB parameters must be modified. Typical situations when these modifications are required include the design of pillars, excavation and cavern stability, strainburst potential assessment, and tunnel support in deep underground conditions (around σ1/σci > 0.15, where σ1 is the major principal compressive stress and σci is the unconfined compressive strength of the homogeneous rock) in hard brittle rocks with GSI ≥ 65. In this article, the strength of massive to moderately jointed hard rock masses is investigated, and an approach is presented to estimate the rock mass strength envelope using laboratory data from uniaxial and triaxial compressive strength tests without reliance on the HB-GSI equations. The data from tests on specimens obtained from massive to moderately jointed heterogeneous (veined) rock masses are used to obtain the rock and rock mass strengths at confining stress ranges that are relevant for deep tunnelling and mining; and a methodology is presented for this purpose from laboratory data alone. By directly obtaining the equivalent HB rock mass strength envelope for massive to moderately jointed rock from laboratory tests, the HB-GSI rock mass strength estimation approach is complemented for conditions where the GSI-equations are not applicable. Guidance is also provided on how to apply the proposed approach when laboratory test data are not or not yet available.

Stope stability analysis of limestone ex-mined area at Ponjong sub-district Special Region of Yogyakarta using finite element method

Limestone mining activity in Gunungsewu karst terrain of Gunung Kidul, Yogyakarta has been long time to be the livelihood for local communities. Evidently, by the number of ex-mining that have not been maintained and one of them looks like a stope hole in underground mining and failures could be occur at the stope. In this case, the sustainability of geoheritage area required geothecnical analysis to anticipate possibility of failure. This research conducted on traditional mining area in Sidorejo Village, Ponjong sub-district, Gunungkidul by comparing between the Hoek Brown and Mohr Coulomb failure criterion. Hoek-Brown criterion considering the geologic structures of rock mass and classification of rock mass using Geological Strength Index (GSI) whereas Mohr-Coulomb criterion is based on laboratory testing of intact rocks that will be affect to the value estimation of sigma 1, sigma 3, strength factor, and total displacement at the stope. Stability of stope will be analysed using finite element method that assisted with software Phase 2. The result of this research can be used as a reference to determine an appropriate acts to preserve the Gunungsewu geoheritage area.