Hoek-Brown Failure Criterion Research Articles

Data-Driven Rock Strength Parameter Identification Using Artificial Bee Colony Algorithm

Rock strength parameters are essential to understanding the rock failure mechanism and safely constructing rock excavation. It is a challenging problem for determining the rock failure criterion and its parameters due to the complexity of rock media. This study adopts an artificial bee colony (ABC) algorithm to determine the Hoek-Brown failure criterion, widely used in rock engineering practice, based on experimental data. The ABC-based approach is presented in detail and applied to a collection of experimental data collected from the literature. The ABC-based approach successfully determines the Hoek-Brown failure criterion, and the determined failure envelope is in excellent agreement with the measured curve. The maximum relative error obtained by ABC is only 2.15% and is far less than the 12.24% obtained by the traditional method. Then, the developed approach is applied to the Goupitan Hydropower Station, China, and determines the rheological parameters of soft rock based on the Burgers model. The deformation of an experiment located in the Goupitan Hydropower Station is evaluated based on obtained parameters by the developed approach. The predicted deformation matches the monitored displacement in the field. The obtained parameters of the failure criterion characterize the mechanical behavior of rock mass well. Thus, the method used provides a reliable and robust approach to determining the mechanical parameters of the failure criterion.

Prediction of the Stability of Various Tunnel Shapes Based on Hoek–Brown Failure Criterion Using Artificial Neural Network (ANN)

In this paper, artificial neural network (ANN) models are presented in order to enable a prompt assessment of the stability factor of tunnels in rock masses based on the Hoek–Brown (HB) failure criterion. Importantly, the safety assessment is one of the serious concerns for constructing tunnels and requires a reliable and accurate stability analysis. However, it is challenging for engineers to construct finite element limit analysis (FELA) algorithms with the HB failure criterion for tunnel stability solutions in rock masses. For the first time, a machine-learning-aided prediction of tunnel stability based on the HB failure criterion is proposed in this paper. Three different shapes of tunnels, i.e., heading tunnel, dual square tunnels, and dual circular tunnels, are considered. The inputs include four dimensionless parameters for the heading tunnel including the cover-depth ratio, the normalized uniaxial compressive strength, the geological strength index (GSI), and the mi parameter. Moreover, dual square and circular tunnels include one more additional parameter namely the distance ratio. The results present the best ANN models for each tunnel shape, providing very reliable solutions for predicting the tunnel stability based on the HB failure criterion.

Roof stability in flat-ceiling deep rock cavities and tunnels

Three-dimensional stability of roofs in deep flat-ceiling cavities is analyzed. The stability number, factor of safety, and required supporting stress are used as measures of roof stability. Despite the simplicity of the flat roof geometry, the three-dimensional stability analysis presents some complexities owed to the shape of the failure surface geometry in the collapse mechanism. The failure mode assumes a rock block moving downward into the cavity, and the study aims to recognize the most critical shape of the failing block. Three specific block shapes are described in some detail, but more have been analyzed. Blocks defined by a special case of a 4th order conical surface (quartic) on a rectangular base, and a 2nd order elliptic surface (quadric) are found to be the most critical in the stability analysis. The kinematic approach of limit analysis was used, with the rock strength governed by the Hoek-Brown failure criterion. The parametric form of the Hoek-Brown function was employed. Interestingly, an absence of diagonal symmetries in the most critical failure mechanisms was observed in roof collapse of square-ceiling cavities. Computational results in terms of dimensionless measures of stability are presented in charts and tables.

Effects of water saturation and loading rate on direct shear tests of andesite

For estimating the long-term stability of underground framework, it is vital to learn the mechanical and rheological characteristics of rock in multiple water saturation conditions. However, the majority of previous studies explored the rheological properties of rock in air-dried and water saturated conditions, as well as the water effects on compressive and tensile strengths. In this study, andesite was subjected to direct shear tests under five water saturation conditions, which were controlled by varying the wetting and drying time. The tests were conducted at alternating displacement rates under three vertical stresses. The results reveal that the shear strength decreases exponentially as water saturation increases, and that the increase in shear strength with a tenfold increase in displacement rate is nearly constant for each of the vertical stresses. Based on the findings of the shear tests in this study and the compression and tension tests in previous studies, the influences of both water saturation and loading rate on the Hoek-Brown failure criterion for the andesite was examined. These results indicate that the brittleness index of the andesite, which is defined as the ratio of uniaxial compressive strength to tensile strength, is independent of both water saturation and loading rate and that the influences of the water saturation dependence and the loading rate dependence of the failure criterion can be converted between each other.

Lateral Loading of a Rock–Socketed Pile Using the Strain Wedge Model Based on Hoek–Brown Criterion

Rock–socketed pile under lateral loading is important in engineering practice. It is very significant to calculate the lateral bearing capacity of rock–socketed piles since few studies focus on this problem. The rock cohesion and instantaneous angle of friction, which have a high correlation with confining pressure, are obtained. Moreover, the strain wedge model is modified from three aspects: the assumption of nonlinear displacement; the stress level related to cohesion and friction angle; and the pile side resistance. Then, the modified strain wedge model is employed to deduce p–y criterion for rock–socketed pile considering Hoek–Brown failure criterion. The fourth-order partial differential equation constructed according to the p–y curve is solved by using the finite difference method. A numerical method with 2 m diameter rock-socketed pile is given to validate the rationality of the proposed method. It is shown that the proposed could predict the pile deformation well, and the responses are considered acceptable.

Numerical modelling for geotechnical assessment of rock mass behaviour and performance of support system for diversion tunnels using optimized Hoek-Brown parameters

Purpose. Empirical and numerical methods play a vital role in assessing rock mass behaviour quantitatively and qualitatively to design underground structures/caverns and support systems. This research aims to assess and evaluate the rock mass be-haviour for safe, stable, efficient, and economical design of support system for underground structures especially tunnels in diverse rock mass conditions. Methods. In this research, such empirical design methods as Rock Mass Rating (RMR), Q-system and GSI were used to characterize and classify the rock mass environment along the tunnel for the preliminary design of twin tunnels and support systems. The geomechanical parameters, Hoek-Brown failure criterion, and its variants for assessing rock mass behaviour were optimized using multiple regression of Stewart, generalized and globalized variant of nonlinear regression method. The rock mass was classified for the selected section A-A. The excavation method and support system for the said section were designed based on the results obtained from empirical modelling. 2D elasto-plastic finite element method (FEM) was used for numerical analysis of rock mass behaviour and performance of the designed supports in section A-A. Findings. The major rock type encountered in the diversion scheme comprises gabbronorite (GN) and Ultramafic Association (UMA). Based on the quantification of RMR, Q-system, and GSI, section A-A’s rock mass ranges from very poor to poor. From the numerical analysis for the said rock mass environment both RMR and Q system support recommendations are equally efficient to support the rock mass surrounding the tunnel. However, keeping in view the yield zone, especially in the crown, the rock bolt’s length should not be less than 5 meters. Based on the analysis of results, both the tunnels are at a safe distance from each other. Originality. In this research, the design input parameters for numerical modeling were optimized by using different techniques to eliminate the chances of error in evaluating rock mass behaviour and designing an optimum support system in the said rock mass environment. Practical implications. The assessment of rock mass behaviour and the design of optimum support systems in heterogenous conditions is quite challenging and requires thorough investigation through different design techniques. This research provides a refined meth-od to be used for the safe, stable, and economical design of tunnels.

Sandstone Slope Stability Analysis Under Wetting-Drying Cycles Based on Generalized Hoek-Brown Failure Criterion

The deterioration of rock strength due to cyclic fluctuations in the groundwater level in drawdown regions influences bank slope stability. Based on wetting–drying cycling and uniaxial compression tests of sandstone specimens, the intact rock strength deterioration characteristics were analyzed. Considering different types of rock mass discontinuities, the rock mass strength deterioration features were shown. The results of rock mass strength deterioration characteristics were then applied in sandstone slope stability analysis. Using the finite element method, the safety factors of the slope under different wetting–drying cycles were computed based on the generalized Hoek–Brown failure criterion. Meanwhile, statistical distribution features of strength parameters were considered, and safety factor distribution of studied slopes was obtained with Monte Carlo simulation. The results show that rock mass deformation modulus decreased during wetting–drying cycling. The mean safety factors of the slope with any type of discontinuities were reduced by wetting–drying cycling in probabilistic analyses. Wetting–drying cycling plays an important role in damaging slope stability at initial stages since the degree of rock mass strength deterioration is remarkable during initial wetting–drying cycles. However, the rock mass strength and safety factor of the slope are reduced gently after 10 wetting–drying cycles. This research presents the phases of the effect of wetting–drying cycles on sandstone slope stability in drawdown regions.

Uncertainty in the Stability for Excavated Surface during the Cutter Replacement in Tunneling Construction

In the process of tunnel excavation, the cutter wear rate is accelerated when the tunnel boring machine (TBM) passes through the rock stratum and the cutter is commonly replaced frequently during the excavation. In the replacement stage, the stability of the excavation surface is crucial to the operation of the project. In this study, commercial software PLAXIS3D is adopted for the evaluation of the stability of the excavation surface by considering different scenarios. The Hoek–Brown failure criterion is adopted as the constitutive model for the rock in the numerical analysis. Various failure modes under different working conditions are obtained. The results from the numerical analyses indicate that the excavation surface during the replacement is stable for those different scenarios. The factor of safety (FoS) for the stability of the excavation surface varies from 2.88 to 14.04. The failure modes near the excavation surface can be divided into four types: (i) central inward invasion, (ii) bilateral inward invasion, (iii) sliding, and (iv) outward invasion. With the increase of support pressure, the safety factor increases first and then decreases.

Physical and Mechanical characteristic relationships of Late-Cretaceous to Eocene reservoir rocks in the Maui, Maari and Manaia Fields, New Zealand

Constraint on the rock strength parameters within the subsurface is a fundamental requirement for accurate geomechanical modelling of aspects of reservoir stability and regional scale basin interactions. Insufficient rock strength data for offshore lithologies within the Taranaki Basin leads to a dependence on uncalibrated, empirical relationships applied in conjunction with wireline measurements for rock strength predictions. Use of these uncalibrated empirical relationships can lead to unreliable strength estimates, which reduces the confidence in geomechanical modelling and the subsequent solutions for the region.We conducted uniaxial and triaxial experiments on cores from offshore Taranaki reservoir rocks from 2000 to 4000 m depth to develop the first, calibrated, empirical rock strength relationships for reservoir rocks of the Taranaki Basin, using both grain size and porosity as input parameters. We show that grain size and porosity can be used as predictive tools for determining Hoek-Brown and Mohr-Coulomb failure criterion parameters for petroleum geomechanics. As mean grain diameter and porosity are the dominant control on rock strength, we infer that rock strength parameters within the Taranaki Basin will follow a similar spatial distribution as the reservoir sandstone facies, which are dominantly defined by grain size.We also show that the empirical relationships we developed with this dataset can be locally calibrated for other parts of the Taranaki basin, and indeed for other sandstones, such as the Buntsandstein from the Rhine Graben, France. Finally, we provide an approach by which grain size (+/− porosity) can be used to approximate the input parameters for the Hoek-Brown failure criterion in the absence of laboratory experiments. We propose, therefore, that the empirical relationships presented herein can be used to link facies descriptions with first-order estimates of mechanical properties at the basin scale.

Application of Artificial Neural Networks for Predicting the Stability of Rectangular Tunnels in Hoek–Brown Rock Masses

An artificial neural network (ANN) model for predicting the stability of rectangular tunnels in rock masses based on the Hoek–Brown (HB) failure criterion is presented in this study. Since the safety assessment of the tunnel stability is one critical issue for civil engineers during the construction, it is very important to develop a reliable and accurate stability analysis of such problems. The finite element limit analysis (FELA) with the HB failure criterion is used to develop the numerical upper and lower bound solutions of the problem of rectangular tunnels in rock masses. A novel machine learning-aided prediction of this problem is then developed based on the datasets of the numerical bound solutions obtained from the FELA. The inputs consist of six dimensionless parameters including the cover-depth ratio of tunnels, the width ratio of tunnels, the normalized uniaxial compressive strength, the geological strength index, the mi parameter, and the degree of disturbance of rock masses. The results show that the optimal ANN models provide very great accuracy in predicting the stability of the rectangular tunnels based on the HB failure criterion. The solutions will provide a prompt assessment of tunnel stability in rock masses for geotechnical engineers during the construction of rock tunnels.

Comparative analysis of non-recursive three-dimensional (3D) modifications of Hoek-Brown failure criterion

A comparative analysis of three-dimensional (3D) modifications of Hoek & Brown (1988) rock failure criterion was carried out. The correlations between failure stress and the other principal stresses were first determined using polyaxial test data for five geomaterials including KTB amphibolites, Westerly granite, Dunham dolomite, Shirahama sandstone and Yuubari shale. The prediction accuracies of five non-recursive, three-dimensional modifications to Hoek-Brown failure criteria and the original two-dimensional Hoek-Brown criterion were later investigated using root mean square error and coefficient of determination as measures of misfit. The results reveal that the intermediate principal stress significantly affects strength in geomaterials like the Dunham dolomite. It also moderately affects the strength of geomaterials like the KTB amphibolites, the Westerly granite and the Yuubari shale. Moreover, the intermediate principal stress has mixed effects on strength in the Shirahama sandstone. In addition, the original Hoek-Brown failure criterion could still be used with reasonable accuracy for geomaterials whose strength shows low dependence on the intermediate principal stress. While a three-dimensional Hoek-Brown criterion must be used for geomaterials like the Dunham dolomite, whose strength shows a high dependence on the intermediate principal stress. The original Hoek-Brown failure criteria should be used with caution for geomaterials like the Shirahama sandstone, the KTB amphibolites, the Westerly granite, and the Yuubari shale, whose strength shows either mixed or intermediate dependence on the intermediate principal stress. Average prediction accuracies followed the order: simplified Priest (2012), Ma et al. (2020), and Jiang & Zhao (2015). Both original Hoek & Brown (1988) and Li et al. (2021) criteria were tied, while Liu et al. (2019) was the least.

Thickness Calculation of Accumulative Damaged Zone by Rock Mass Blasting Based on Hoek–Brown Failure Criterion

Abstract Tunnel excavation by cyclic blasting caused accumulative damages to the surrounding rocks, reducing the intactness of the rock mass. The blasting impact contributes to the formation of an ...

Modelling of Critical Acceleration for Regional Seismic Landslide Hazard Assessments by Finite Element Limit Analysis

The critical acceleration model plays an important role in seismic slope stability and determines the predictive accuracy of regional seismic landslide hazard assessments. Recently, the critical acceleration model based on the limit equilibrium method has been used to evaluate the seismic stability of regional slopes. However, when the Hoek-Brown failure criterion is used to evaluate the seismic stability of slopes with angles greater than 60°, the results obtained is unconservative by limit equilibrium method. Therefore, based on the simulation of a typical slope model with finite element limit analysis, prediction equations of the critical acceleration are established. The corresponding results are compared with the prediction results from the limit equilibrium method. This comparison shows that the proposed critical acceleration model has higher predictive accuracy than the limit equilibrium method, especially when the Hoek-Brown failure criterion is used to evaluate the slopes with angles greater than 60°. The proposed model is applicable to the global scope and can be effectively applied to regional seismic landslide hazard assessments.

Study of the anisotropic tensile and compressive strength of a foliated phyllite

Anisotropy is an important characteristic of rocks, especially distinguishable in metamorphic rocks. Transverse isotropy is a particular case of anisotropy where foliation planes are distributed in the rock mass. Anisotropy can also originate from mineral foliation where minerals are oriented in a preferential direction. This inherent anisotropy can affect the rock strength significantly. Here the effect of foliation on the anisotropic strength of a phyllite is experimentally investigated. Phyllite specimens with defined foliation orientations are prepared and a series of laboratory tests carried out. Results for tensile strength and unconfined compressive strength are analysed. Maximum values of tensile strength and unconfined compressive strength are found for β = 0 and 90° and a significant reduction of strength is found for β = 45°. Compressive triaxial tests are conducted under confining pressures up to 20 MPa. The Hoek–Brown failure criterion is found to capture the experimental results in a better form than the Mohr–Coulomb criterion. Anisotropy indexes are adopted to evaluate the anisotropy effect on strength. Confinement is found to reduce the effect of anisotropy on phyllite strength. Cohesion and angle of shearing resistance were also found to be affected by the stress level, and further influenced by β.

Three-dimensional collapse mechanism and failure probability analysis of deep-buried tunnel based on the upper bound theorem of limit analysis and Hoek-Brown failure criterion

Three-dimensional collapse mechanism and failure probability analysis of deep-buried tunnel based on the upper bound theorem of limit analysis and Hoek-Brown failure criterion

Estimating the Geological Strength Index (GSI) in Regional Seismic‐Landslide Zonation Using the Empirical Regression Model

The assessment of the strength parameters of geological formations in regional scale which encounters thousands of slopes is a complicated approach and time consuming and needs huge field work. This issue is an important research topic concerning the regional seismic‐landslide susceptibility analysis or hazard zonation. An empirical regression model was presented to estimate the Geological Strength Index (GSI) with an implication on geological quadrangle of Gorgan region at Alborz mountains range (north of Iran). Two main sets of data were applied in this study: (1) geomorphological data including the slope height, aspect, and distance from faults and distance from thrusts and (2) the physical and mechanical properties of rocks including the unit weight, uniaxial compressive strength (σci), and the petrographic constant (mί) of intact rock. The first group was extracted from a 1 : 100,000 digital geologic map and 10 m digital elevation model (DEM) and the second group was obtained from the Hoek–Brown failure criterion recommended tables. Linear regression equations were generated applying data collected from 294 studied stations using SPSS software. The regression equation predicted GSI in terms of (1) the distance from faults, (2) the distance from thrusts, and (3) the uniaxial compressive strength (σci). The equation had an R2 value of 0.739 and thus fit well to the data. The new method in its present state was recommended for the estimation of the GSI values in regional scale conditions for the assessment of landslide susceptibility and hazard mapping or post events landslide occurrence prediction in the case of probable big earthquakes in Alborz area that is required for emergency responses. The results indicated that the estimation error was about ±30 while the average error was within +5 and −5 and average error percentage was about 3%.

Stability Assessment of Rock Mass System under Multiple Adjacent Structures

Numerical modeling is important for exploring the fundamental processes occurring in rock and for evaluating the real performance of structures built on and in rock mass system, and thus for supporting the design of rock engineering problems. Estimating the stability of rock mass foundation systems entirely based on a theoretical approach is a complicated task if there exists overlapping of their potential collapse modes. This paper applies finite element limit analysis to evaluate the bearing capacity of equally spaced multiple strip footings resting on rock mass obeying the modified non-linear Hoek–Brown failure criterion. Numerical solutions are expressed in terms of the efficiency factor that is dependent on the spacing between footings, as well as the rock mass properties. In addition, the effects of surface surcharge and footing roughness are quantified. The maximum spacing at which the interfering effect of adjacent footings becomes disappeared is evaluated and an algebraic expression for approximating the maximum spacing is proposed. Failure mechanisms for a few cases of rock mass under multiple strip footings are examined.

Numerical Analysis of Slope Stability Due to Excavation of Diversion Tunnel at Pamukkulu Dam Site, Indonesia

Located in the Takalar Regency of South Sulawesi Province, the Pamukkulu Dam is planned to use a tunnel type as its diversion structure. One of the critical parts in the tunnel construction is the stability of portal slopes. This research aimed to estimate the effect of tunnel excavation on the stability of the portal inlet and outlet slopes under static and earthquake loads by using the finite element method. The slope stability analyses were carried out under conditions of prior to and after tunnel excavation. The input parameters used were laboratory test results in the forms of index properties and mechanical properties taken from rock core drilling samples, completed with the rock mass quality parameters based on the Geological Strength Index (GSI) classification. The Mohr-Coulomb failure criterion was used to model strength of the soil, while the Generalized Hoek-Brown failure criterion was used to model strength of the rocks. The results of rock cores analysis using the GSI method showed that the inlet tunnel slope consisted of four types of materials, namely residual soil, fair quality of basalt lava, good quality of basalt lava, and very good quality of basalt lava. Meanwhile, the outlet portal slope consisted of three types of materials, namely residual soil, good quality basalt lava, and very good quality basalt lava. The calculated horizontal seismic coefficient for the pseudo-static slope stability analysis was 0.0375. The analysis results of slope stability in the Y1 inlet section had a critical Strength Reduction Factor (SRF) value of 2.35 in a condition prior to the tunnel excavation and a critical SRF value of 2.34 after the tunnel excavation. The Y2 outlet section had a critical SRF value of 13.27 in a condition before tunnel excavation and a critical SRF value of 5.55 after the tunnel excavation. The earthquake load addition at the Y1 inlet section showed a critical SRF value of 2.05, both before and after the tunnel excavation. The Y2 outlet section showed a critical SRF value of 11.49 before the tunnel excavation and a critical SRF value of 5.54 after the tunnel excavation. The numerical analysis results showed that earthquake load reduced critical SRF values of the slopes. At the Y1 inlet section, the tunnel excavation did not have a significant effect on slope stability. It was demonstrated by an extremely small decrease in a critical SRF value of 0.43% for a condition without an earthquake load and an unchanged critical SRF in a condition with an earthquake load. At the Y2 outlet section, the tunnel excavation had a more significant effect on the slope stability. It was exhibited by the decrease in the critical SRF value of 58.18% in a condition without an earthquake load and a decrease in the critical SRF value of 51.78% in a condition with an addition of an earthquake load. However, the analysis of slope stability for both sections showed that all design slopes were above the required allowable safety factor value.

Assessment of the Bearing Capacity of Bridge Foundation on Rock Masses

This paper aims to study the bearing capacity of a shallow foundation on rock mass, considering the most usual bridge footing width and adopting a Hoek–Brown material. The dimension of the foundation has been shown to be very significant in soils with linear failure criteria (Mohr–Coulomb envelope), and its study is necessary in the case of non-linear failure criteria, typical of rock masses. Analytical solutions do not allow incorporating this effect. A parametric study by a finite difference method was carried out, studying a wide variety of rock mass through sensitivity analysis of three geotechnical parameters: geological origin of the rock mass (mi), uniaxial compressive strength, and geological strength index. The results obtained by the numerical solution for the Hoek–Brown failure criterion were compared with the analytical results by adopting the classical hypotheses of plane strain conditions, associated flow rule, and weightless rock mass. The variation of the numerical bearing capacity due to the consideration of the self-weight of the rock mass was also analyzed since its influence is conditioned by the volume of ground mobilized and therefore by the width of the foundation. Considering the similarities observed between the numerical and analytical results, a correlation factor function of the self-weight is proposed. It can be used in conjunction with the analytical method, to estimate in a semi-analytical way the bearing capacity of a bridge foundation.

Estimation of the Critical Seismic Acceleration for Three-Dimensional Rock Slopes

An analytical approach for the estimating of critical seismic acceleration of rock slopes was proposed in this study. Based on the 3D horn failure model, the critical seismic acceleration coefficient of rock slopes was conducted with the modified Hoek–Brown (MHB) failure criterion in the framework of upper-bound theory for the first time. The nonlinear Hoek–Brown failure criterion is incorporated into the three-dimensional rotational failure mechanism, and a generalized tangent technique is introduced and employed to convert the nonlinear Hoek–Brown failure criterion into a linear criterion. The critical seismic acceleration coefficients obtained from this study were validated by the numerical simulation results based on finite element limit analysis. The agreement showed that the proposed method is effective. Finally, design charts were provided for exceptional cases for practical use in rock engineering.