Three-dimensional limit variation analysis on the ultimate pullout capacity of the anchor cables based on the Hoek-Brown failure criterion

Only simplified two-dimensional model and a single failure mode are adopted to calculate the ultimate pullout capacity(UPC)of anchor cables in most previous research.This study focuses on a more comprehensive combination failure mode that consists of bond failure of an anchorage body and failure of an anchored rock mass.The three-dimensional ultimate pullout capacity of the anchor cables is calculated based on the Hoek-Brown failure criterion and variation analysis method.The numerical solution for the curvilinear function in fracture plane is obtained based on the finite difference theory,which more accurately reflects the failure state of the anchor cable,as opposed to that being assumed in advance.The results reveal that relying solely on a single failure mode for UPC calculations has limitations,as changes in parameter values not only directly impact the UPC value but also can alter the failure model and thus the calculation method.

Unified strength model based on the Hoek-Brown failure criterion for fibre-reinforced polymer-confined pre-damaged concrete columns with circular and square cross sections

Fibre-reinforced polymer(FRP)has the advantages of high strength,light weight,corrosion resistance and convenient construction and is widely used in repairing and strengthening damaged concrete columns.Most of the existing strength models were built by regression analysis of experimental data;however,in this article,a new unified strength model is proposed using the Hoek-Brown failure criterion.To study the strength of FRP-confined damaged and undamaged concrete columns,900 test data were collected from the published literature and a large database that contains the cross-sectional shape of each specimen,the damage type,the damage level and the FRP-confined stiffness was established.A new strength model using the Hoek-Brown failure criterion was established and is suitable for both circular and square columns that are undamaged,load-damaged and fire-damaged.Based on the database,most of the existing strength models from the published literature and the model proposed in this paper were evaluated.The evaluation shows that the proposed model can predict the compressive strength for FRP-confined pre-damaged and undamaged concrete columns with good accuracy.

Elasto-plastic Analysis of Circular Tunnels in Jointed Rock Masses Satisfy the Hoek-Brown Failure Criterion

The relationship between the Hoek-Brown parameters and the mechanical response of circular tunnels is il-lustrated. Closed-form and approximate solutions are given for the extent of the plastic zone and the stress and dis-placement fields under axisymmetrical and asymmetric stress conditions. For the same rock masses and under axisym-metrical stress conditions,the radius of the plastic zone in terms of Hoek-Brown criterion is generally an approximation of the radius in terms of the Mohr-Coulomb criterion. The radius in terms of the Hoek-Brown criterion is larger under low stress conditions. For poor quality rock masses (GSI<25),measures (such as grouting,setting rock bolts,etc.) that improve the GSI of rock masses are effective in improving the stability of tunnels. It is not advisable to improve the sta-bility of the tunnels by providing a small support resistance p through shotcrete,except for very poor quality jointed rock masses. Without reference to the quality of the rock mass,the disturbance factor D should not less than 0.5. Meas-ures which disturb rock masses during tunnel construction should be taken carefully when the tunnel depth increases.

Enhanced Hoek-Brown(H-B)criterion for rocks exposed to chemical corrosion

Underground constructions often encounter water environments,where water–rock interaction can increase porosity,thereby weakening engineering rocks.Correspondingly,the failure criterion for chemically corroded rocks becomes essential in the stability analysis and design of such structures.This study enhances the applicability of the Hoek-Brown(H-B)criterion for engineering structures operating in chemically corrosive conditions by introducing a kinetic porosity-dependent instantaneous mi(KPIM).A multiscale experimental investigation,including nuclear magnetic resonance(NMR),X-ray diffraction(XRD),scanning electron microscopy(SEM),pH and ion chromatography analysis,and triaxial compression tests,is employed to quantify pore structural changes and their linkage with the strength responses of limestone under coupled chemical-mechanical(C-M)conditions.By employing ion chromatography and NMR analysis,along with incorporating the principles of free-face dissolution theory accounting for both congruent and incongruent dissolution,a kinetic chemical corrosion model is developed.This model aims to calculate the kinetic porosity alterations within rocks exposed to varying H+concentrations and durations.Subsequently,utilizing the generalized mixture rule(GMR),the kinetic porositydependent mi is formulated.Evaluation of the KPIM-enhanced H-B criterion using compression test data from 5 types of rocks demonstrated a high level of consistency between the criterion and the experimental results,with a coefficient of determination greater than 0.96,a mean absolute percentage error less than 4.84%,and a root-mean-square deviation less than 5.95 MPa.Finally,the physical significance of the porosity-dependent instantaneous mi is clarified:it serves as an indicator of a rock’s capacity to leverage the confining pressure effect.

Validation of Hoek-Brown failure criterion charts for rock slopes

Although stability charts suggested by Hoek and Bray on the basis of Mohr-Coulomb criterion are used for rock slopes,but complete and precise recognition is required for distinguishing cohesive strength and Mohr-Coulomb equivalent internal friction angle for rock mass.The paper by Lia et al.[6]is the only one that introduced rock slope charts according to Hoek-Brown failure criterion.In this paper,at first,this type of charts is introduced.Then,Mohr-Coulomb failure criterion charts[2]are compared and validated with Hoek-Brown failure criterion ones[6].Next,Bishop method utilizing Slide software is compared with Hoek-Brown failure criterion stability charts.Average standard deviation(ASD),root mean square error(RMSE)and variance account for(VAF)were used for the comparison.According to the results,because of high distribution and very low correlation among the comparisons,Hoek-Brown failure criterion charts are not efficient.

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

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

A generalized three-dimensional failure criterion for rock masses

The smooth convex generalized failure function, which represents 1/6 part of envelope in tile deviatoric plane, is proposed. The proposed function relies on four shape parameters （L, 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 tbr 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 （L~） that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters （c~. 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 parameterfis introduced. The modified extension ratio is related tofand extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range off 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 betweenfand 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.

A simplified approach to directly consider intact rock anisotropy in Hoek——Brown failure criterion

Many rock types have naturally occurring inherent anisotropic planes, such as bedding planes, foliation,or flow structures. Such characteristic induces directional features and anisotropy in rocks＇ strength anddeformational properties. The HoekeBrown （HeB） failure criterion is an empirical strength criterionwidely applied to rock mechanics and engineering. A direct modification to HeB failure criterion toaccount for rock anisotropy is considered as the base of the research. Such modification introduced a newdefinition of the anisotropy as direct parameter named the anisotropic parameter （Kb）. However, thecomputation of this parameter takes much experimental work and cannot be calculated in a simple way.The aim of this paper is to study the trend of the relation between the degree of anisotropy （Rc） and theminimum value of anisotropic parameter （Kmin）, and to predict the Kmin directly from the uniaxialcompression tests instead of triaxial tests, and also to decrease the amount of experimental work. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.

Failure probability analysis of coal crushing induced by uncertainty of influential parameters under condition of in-situ reservoir

The uncertainties of some key influence factors on coal crushing,such as rock strength,pore pressure and magnitude and orientation of three principal stresses,can lead to the uncertainty of coal crushing and make it very difficult to predict coal crushing under the condition of in-situ reservoir.To account for the uncertainty involved in coal crushing,a deterministic prediction model of coal crushing under the condition of in-situ reservoir was established based on Hoek-Brown criterion.Through this model,key influence factors on coal crushing were selected as random variables and the corresponding probability density functions were determined by combining experiment data and Latin Hypercube method.Then,to analyze the uncertainty of coal crushing,the firstorder second-moment method and the presented model were combined to address the failure probability involved in coal crushing analysis.Using the presented method,the failure probabilities of coal crushing were analyzed for WS5-5 well in Ningwu basin,China,and the relations between failure probability and the influence factors were furthermore discussed.The results show that the failure probabilities of WS5-5 CBM well vary from 0.6 to 1.0; moreover,for the coal seam section at depth of 784.3-785 m,the failure probabilities are equal to 1,which fit well with experiment results; the failure probability of coal crushing presents nonlinear growth relationships with the increase of principal stress difference and the decrease of uniaxial compressive strength.

Numerical modeling of permeability evolution based on degradation approach during progressive failure of brittle rocks

The permeability evolution of rock during the progressive failure process is described. In combination with the strength degradation index, the degradation formulas of s and a, which are dependent on the plastic confining strain component, the material constants of Hock-Brown failure criterion are presented, and a modified elemental scale elastic-brittle-plastic constitutive model of rock is established. The rela- tionship between volumetric strain and permeability through tri-axial compression is investigated. Based on the above, a permeability evolution model is established. The model incorporates confining pressure- dependent degradation of strength, dilatancy and corresponding permeability evolution. The model is implemented in FLAC by the FISH function method. The permeability evolution behavior of rock is inves-tigated during the progressive failure process in a numerical case. The results show that the model is cap- able of reproducing, and allowing visualization of a range of hydro-mechanical responses of rock. The effects of confining pressure on degradation of strength, dilatancy and permeability evolution are also reflected.

Limit variation analysis of shallow rectangular tunnels collapsing with double-layer rock mass based on a three-dimensional failure mechanism

In the framework of upper bound theorem of limit analysis, the progressive collapse of shallow rectangular tunnels with double-layer rock mass has been theoretically analyzed based on the three-dimensional (3D) velocity discontinuity surfaces. According to the virtual work principle, the difference theorem and the variation method, the collapse surface of double-layer rock mass is determined based on the Hoek-Brown failure criterion. The formula can be degenerated to a single-layer rock collapsing problem when the rock mass is homogeneous. To estimate the validity of the result, the numerical simulation software PLAXIS 3D is used to simulate the collapse of shallow tunnels with double-layer rock mass, and the comparative analysis shows that numerical results are in good agreement with upper-bound solutions. According to the results of parametric analysis, the potential range of collapse of a double-layer rock mass above a shallow cavity decreases with a decrease in A1/A2,σci1/σci2 and σtm1/σtm2 and an increase in B1/B2,γ1/γ2. The range will decrease with a decrease in support pressure q and increase with a decrease in surface overload σs. Therefore, reinforced supporting is beneficial to improve the stability of the cavity during actual construction.

Calibration of Four Nonlinear Failure Envelopes from Triaxial Test Data and Influence of Nonlinearity on Geotechnical Computations

It is now recognized that many geomaterials have nonlinear failure envelopes. This non-linearity is most marked at lower stress levels, the failure envelope being of quasi-parabolic shape. It is not easy to calibrate these nonlinear failure envelopes from triaxial test data. Currently only the power-type failure envelope has been studied with an established formal procedure for its determination from triaxial test data. In this paper, a simplified procedure is evolved for the development of four different types of nonlinear envelopes. These are of invaluable assistance in the evaluation of true factors of safety in problems of slope stability and correct computation of lateral earth pressure and bearing capacity. The use of the Mohr-Coulomb failure envelopes leads to an overestimation of the factors of safety and other geotechnical quantities.

Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek-Brown criterion

The stability analysis of passive bolt-reinforced rock slopes under seismic loads is investigated within the framework of the kinematic approach of limit analysis theory.A pseudo-static method is adopted to account for the inertial forces induced in the rock mass by seismic events.The strength properties of the rock material are described by a modified Hoek-Brown strength criterion,whereas the passive bolts are modeled as bar-like inclusions that exhibit only resistance to tensile-compressive forces.Taking advantage of the ability to compute closed-form expressions for the support functions associated with the modified Hoek-Brown strength criterion,a rotational failure mechanism is implemented to derive rigorous lower bound estimates for the amount of reinforcement strength to prevent slope failure.The approach is then applied to investigating the effects of relevant geometry,strength and loading parameters in light of a preliminary parametric study.The accuracy of the approach is assessed by comparison of the lower bound estimates with finite element limit analysis solutions,thus emphasizing the ability of the approach to properly predict the stability conditions and to capture the essential features of deformation localization pattern.Finally,the extension of the approach to account for slipping at the interface between reinforcements and surrounding rock mass is outlined.

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

The Hoek-Brown(HB)strength criterion has been applied widely in a large number of projects around the world.However,this criterion ignores the intermediate principal stress s2.Many evidences have demonstrated that the rock strength is dependent on s2.Thus it is necessary to extend the HB criterion into a three-dimensional(3D)form.In this study,the effect of s2 on the strength of rocks is identified by reviewing the true triaxial tests of various rock types reported in the literature.A simple 3D strength criterion is developed.The modified criterion is verified by the true triaxial tests of 13 rock types.The results indicate that the modified criterion can achieve a good fit to most of rock types.It can represent a series of criteria as b varies.For comparisons,several existing 3D versions of the HB criterion are selected to predict the strengths of these rock types.It is indicated that the proposed criterion works better than other criteria.A substantial relationship between parameter b and the unconfined compressive strength is established,which guarantees that the proposed criterion can still work well even in the absence of true triaxial test data.

基于Hoek-Brown准则等效强度参数的工程应用

为研究重庆至长沙省际公路共和隧道层状围岩稳定性,减小模拟值与实测值误差,并充分考虑倾角效应、静水压力效应和罗德角效应,将适用于层状岩体的修正Hoek-Brown(H-B)准则在I_(1)-√J_(2)平面与Mohr-Coulomb(M-C)准则进行瞬时等效,推导出用于层状岩体的等效强度参数公式,计算得到不同倾角下砂质页岩等效c、φ值,然后输入FLAC^(3D)遍布节理模型中对原岩体参数进行修正。结果表明:1)修正岩体参数法所得关键点位移变化规律与修正前一致,且位移量大于修正前,但当倾角大于60°后,该法适用性有所降低;2)修正后围岩各关键点偏应力(σ_(1)-σ_(3))始终大于修正前,且二者塑性区随倾角变化趋势一致;3)当岩体内摩擦角小于节理内摩擦角时,围岩塑性区面积较大;反之,当岩体内摩擦角大于节理内摩擦角时,塑性区面积较小。

Retrospective and prospective review of the generalized nonlinear strength theory for geomaterials

Strength theory is the basic theory for calculating and designing the strength of engineering materials in civil,hydraulic,mechanical,aerospace,military,and other engineering disciplines.Therefore,the comprehensive study of the generalized nonlinear strength theory(GNST)of geomaterials has significance for the construction of engineering rock strength.This paper reviews the GNST of geomaterials to demonstrate the research status of nonlinear strength characteristics of geomaterials under complex stress paths.First,it systematically summarizes the research progress of GNST(classical and empirical criteria).Then,the latest research the authors conducted over the past five years on the GNST is introduced,and a generalized three-dimensional(3D)nonlinear Hoek‒Brown(HB)criterion(NGHB criterion)is proposed for practical applications.This criterion can be degenerated into the existing three modified HB criteria and has a better prediction performance.The strength prediction errors for six rocks and two in-situ rock masses are 2.0724%-3.5091%and 1.0144%-3.2321%,respectively.Finally,the development and outlook of the GNST are expounded,and a new topic about the building strength index of rock mass and determining the strength of in-situ engineering rock mass is proposed.The summarization of the GNST provides theoretical traceability and optimization for constructing in-situ engineering rock mass strength.

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.

Fracture initiation and propagation in intact rock—A review

The initiation and propagation of failure in intact rock are a matter of fundamental importance in rock engineering. At low confining pressures, tensile fracturing initiates in samples at 40%-60% of the uniaxial compressive strength and as loading continues, and these tensile fractures increase in density, ultimately coalescing and leading to strain localization and macro-scale shear failure of the samples. The Griffith theory of brittle failure provides a simplified model and a useful basis for discussion of this process. The Hoek-Brown failure criterion provides an acceptable estimate of the peak strength for shear failure but a cutoff has been added for tensile conditions. However, neither of these criteria adequately explains the progressive coalition of tensile cracks and the final shearing of the specimens at higher confining stresses. Grain-based numerical models, in which the grain size distributions as well as the physical properties of the component grains of the rock are incorporated, have proved to be very useful in studying these more complex fracture processes.

Numerical simulation of triaxial compression test for brittle rock sample using a modified constitutive law considering degradation and dilation behavior

The understanding of the rock deformation and failure process and the development of appropriate constitutive models are the basis for solving problems in rock engineering. In order to investigate progressive failure behavior in brittle rocks, a modified constitutive model was developed which follows the principles of the continuum damage mechanics method. It incorporates non-linear Hoek-Brown failure criterion, confining pressure-dependent strength degradation and volume dilation laws, and is able to represent the nonlinear degradation and dilation behaviors of brittle rocks in the post-failure region. A series of triaxial compression tests were carried out on Eibenstock(Germany) granite samples. Based on a lab data fitting procedure, a consistent parameter set for the modified constitutive model was deduced and implemented into the numerical code FLAC3 D. The good agreement between numerical and laboratory results indicates that the modified constitutive law is well suited to represent the nonlinear mechanical behavior of brittle rock especially in the post-failure region.

Pile end bearing capacity in rock mass using cavity expansion theory

Much empiricism is involved in design of rock-socketed piles in rock masses.In light of this,an analytical solution based on the cavity expansion theory is proposed for calculating the ultimate bearing capacity at the tip of a pile embedded in rock masses obeying the Hoek-Brown failure criterion.The ultimate end bearing capacity is evaluated by assuming that the pressure exerted at the boundaries of a pressure bulb immediately beneath the pile tip is equal to the limit pressure required to expand a spherical cavity.In addition,a relationship is derived to predict the pile load-settlement response.To demonstrate the applicability of the presented solution,the results of this study were compared to those of 91 field tests from technical literature.Despite the limitations,it is found that the end bearing resistance computed by the present work is in good agreement with the field test results.