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.

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.

Energy analysis of face stability of deep rock tunnels using nonlinear Hoek-Brown failure criterion

The nonlinear Hoek-Brown failure criterion was introduced to limit analysis by applying the tangent method. Based on the failure mechanism of double-logarithmic spiral curves on the face of deep rock tunnels, the analytical solutions of collapse pressure were derived through utilizing the virtual power principle in the case of pore water, and the optimal solutions of collapse pressure were obtained by using the optimization programs of mathematical model with regard of a maximum problem. In comparison with existing research with the same parameters, the consistency of change rule shows the validity of the proposed method. Moreover, parametric study indicates that nonlinear Hoek-Brown failure criterion and pore water pressure have great influence on collapse pressure and failure shape of tunnel faces in deep rock masses, particularly when the surrounding rock is too weak or under the condition of great disturbance and abundant ground water, and in this case, supporting measures should be intensified so as to prevent the occurrence of collapse.

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.

Limit load and failure mechanisms of a vertical Hoek-Brown rock slope

The problem considered in this short note is the limit load determination of a vertical rock slope.The classical limit theorem is employed with the use of adaptive finite elements and nonlinear programming to determine upper and lower bound limit loads of a Hoek-Brown vertical rock slope.The objective function of the mathematical programming problem is such as to optimize a boundary load,which is known as the limit load,resembling the ultimate bearing capacity of a strip footing.While focusing on the vertical slope,parametric studies are carried out for several dimensionless ratios such as the dimensionless footing distance ratio,the dimensionless height ratio,and the dimensionless rock strength ratio.A comprehensive set of design charts is presented,and failure envelopes shown with the results explained in terms of three identified failure mechanisms,i.e.the face,the toe,and the Prandtl-type failures.These novel results can be used with great confidence in design practice,in particularly noting that the current industry-based design procedures for the presented problem are rarely found.

Stability analysis of intermittently jointed rock slopes based on the stepped failure mode

In practical engineering,due to the noncontinuity characteristics of joints in rock slopes,in addition to plane failure,stepped sliding failure may occur for intermittently jointed rock slopes.Especially for intermittently bedding jointed rock slopes,the correlation and difference in strength parameters between joints and rock bridges,along with the various failure modes and intermittency of rock bridges,contribute to the complexity of stepped failure modes and the unpredictability of failure regions.Based on the upper-bound limit analysis method and multi-sliders step-path failure mode,considering the shear and tensile failure of rock bridges and the weakened relationship between the strength parameters of rock bridges and jointed surfaces,by introducing the modified M-C failure criterion and the formula for calculating the energy consumption of tensile failure of rock bridges,two failure mechanisms are constructed to obtain the safety factor(F_(s))of intermittently jointed rock slopes.The sequential quadratic programming method is used to obtain the optimal upper-bound solution for F_(s).The influence of multiple key parameters(slope height H,horizontal distance L,Slope angleβ,shear strength parameters of the rock bridgeφr and cr,Dimensionless parameter u,weakening coefficients of the internal friction angle and cohesion between the rock bridges and joint surfaces Kφand Kc)on the stability analysis of intermittently jointed rock slopes under the shear failure mode of rock bridges as well as under the tensile failure mode is also explored.The reliability of the failure mechanisms is verified by comparative analysis with theoretical results,numerical results,and landslide cases,and the variation rules of F_(s)with each key parameter are obtained.The results show that F_(s) varies linearly withφr and cr of the rock bridge and with K_(φ)and K_(c),whereas F_(s)changes nonlinearly with H and L.In particular,with the increase in Kφand Kc,Fs increases by approximately 52.78%and 171.02%on average,respectively.For rock bridge tensile failure,F_(s) shows a nonlinearly positive correlation withφr,cr,Kφand Kc.In particular,with the increase in Kφand Kc,Fs increases by approximately 13%and 61.69%on average,respectively.Fs decreases rapidly with increasing slope gradientβand decreasing dimensionless parameterμ.When Kφand Kc are both less than 1.0,the stepped sliding surface occurs more easily than the plane failure surface,especially in the case of tensile failure of the rock bridge.In addition,rock slopes with higher strength parameters,taller heights,and greater weakening coefficients are prone to rock bridge tension failure with lower Fs,and more attention should be given to the occurrence of such accidents in actual engineering.

A failure criterion for shale considering the anisotropy and hydration based on the shear slide failure model

A failure criterion fully considering the anisotropy and hydration of shale is essential for shale formation stability evaluation.Thus,a novel failure criterion for hydration shale is developed by using Jaeger’s shear failure criterion to describe the anisotropy and using the shear strength reduction caused by clay minerals hydration to evaluate the hydration.This failure criterion is defined with four parameters in Jaeger’s shear failure criterion(S_(1),S_(2),a andφ),three hydration parameters(k,ω_(sh)andσ_(s))and two material size parameters(d and l0).The physical meanings and determining procedures of these parameters are described.The accuracy and applicability of this failure criterion are examined using the published experimental data,showing a cohesive agreement between the predicted values and the testing results,R^(2)=0.916 and AAREP(average absolute relative error percentage)of 9.260%.The error(|D_(p)|)is then discussed considering the effects ofβ(angle between bedding plane versus axial loading),moisture content and confining pressure,presenting that|Dp|increases whenβis closer to 30°,and|D_(p)|decreases with decreasing moisture content and with increasing confining pressure.Moreover,|D_(p)|is demonstrated as being sensitive to S1and being steady with decrease in the data set whenβis 0°,30°,45°and 90°.

3D limit analysis of rock slopes based on equivalent linear failure criterion with tension cut-off

Hoek-Brown (HB) failure criterion is widely used to predict the strength of intact or heavily jointed rock mass. For stability analysis of rock slopes governed by the HB failure criterion, the equivalent linearity to Mohr-Coulomb (MC) criterion is often adopted, leading to the well-known equivalent Mohr-Coulomb method (EMCM). Existing studies on EMCM analysis mainly consider the shear strength of rock material, while consideration of the tensile strength is rare. This contradicts the fact that the underlying tensile strength of rock mass has considerable impact on the rock slope stability in real world. In this regard, this paper proposes a limit analysis-based approach that can account for tension in the three-dimensional (3D) stability analysis of HB rock slope. This approach is established on the equivalent linearity of the HB criterion with consideration of tensile strength, known as the equivalent tension cut-off MC method (ETMCM), and using a horn-like 3D mechanism of limit analysis. The safety factor solutions given by the proposed approach are validated by previous studies and numerical results. Parametric studies are conducted to investigate the effect of rock tensile strength on slope stability. Results show that the consideration of tension leads to a more conservative safety factor and a sharper curvature of the failure surface, and these impacts tend to be more obvious with the increases in slope inclination and slope width. Finally, the stability of the HB rock slope under seepage conditions is studied using the proposed approach. The results indicate that the effect of tensile strength is highly remarkable in seepage circumstances.

Prediction and Verifcation of Forming Limit Diagrams Based on a Modifed Shear Failure Criterion

The forming limit diagram plays an important role in predicting the forming limit of sheet metals.Previous studies have shown that,the method to construct the forming limit diagram based on instability theory of the original shear failure criterion is efective and simple.The original shear instability criterion can accurately predict the left area of the forming limit diagram but not the right area.In this study,in order to improve the accuracy of the original shear failure criterion,a modifed shear failure criterion was proposed based on in-depth analysis of the original shear failure criterion.The detailed improvement strategies of the shear failure criterion and the complete calculation process are given.Based on the modifed shear failure criterion and diferent constitutive equations,the theoretical forming limit of TRIP780 steel and 5754O aluminum alloy sheet metals are calculated.By comparing the theoretical and experimental results,it is shown that proposed modifed shear failure criterion can predict the right area of forming limit more reasonably than the original shear failure criterion.The efect of the pre-strain and constitutive equation on the forming limits are also analyzed in depth.The modifed shear failure criterion proposed in this study provides an alternative and reliable method to predict forming limit of sheet metals.

Experimental Study and Failure Criterion Analysis of Rubber Fibre Reinforced Concrete under Biaxial Compression-Compression

In order to examine the biaxial compression-compression properties of rubber fibre reinforced concrete(RFRC),an experimental study on RFRC under different lateral compressive stresses was carried out by considering different rubber replacement rates and polypropylene fibre contents.The failure modes and mechanical property parameters of different RFRC working conditions were obtained from the experiment to explore the effects of rubber replacement rate and polypropylene fibre content on the biaxial compression-compression properties of RFRC.The following conclusions were drawn.Under the influence of lateral compressive stress,the biaxial compression-compression failure mode gradually developed from a columnar pattern to a flaky pattern,suggesting that the incorporation of rubber and polypropylene fibres into the concrete resulted in a significant change in the development of cracks.For different rubber replacement rates and polypropylene fibre contents,the vertical compressive stress exhibited the same developing trend under the influence of lateral compressive stress.Specifically,the lateral compressive stress imposed the minimum effect on the vertical compressive stress when the rubber replacement rate and polypropylene fibre content were 20%and 0.4%,respectively,and imposed the maximum effect when the rubber replacement rate and polypropylene fibre content were 20%and 0%,respectively.With the increase of rubber replacement rate,the vertical peak stress was significantly reduced,which implies that an appropriate amount of polypropylene fibres can increase the vertical peak stress to a certain extent.Then,the biaxial compression-compression mechanism of RFRC was analysed from the microscopic level by using scanning electron microscope(SEM).Meanwhile,based on Kupfer’s biaxial compression-compression failure criterion and the octahedral stress space,a biaxial compression-compression failure criterion for RFRC was proposed,which was proven to have good applicability.The research results of this study provide important theoretical basis for the engineering application and development of RFRC.

The Hoeke-Brown failure criterion and GSI-2018 edition

The HoekeBrown criterion was introduced in 1980 to provide input for the design of underground excavations in rock.The criterion now incorporates both intact rock and discontinuities,such as joints,characterized by the geological strength index(GSI),into a system designed to estimate the mechanical behaviour of typical rock masses encountered in tunnels,slopes and foundations.The strength and deformation properties of intact rock,derived from laboratory tests,are reduced based on the properties of discontinuities in the rock mass.The nonlinear HoekeBrown criterion for rock masses is widely accepted and has been applied in many projects around the world.While,in general,it has been found to provide satisfactory estimates,there are several questions on the limits of its applicability and on the inaccuracies related to the quality of the input data.This paper introduces relatively few fundamental changes,but it does discuss many of the issues of utilization and presents case histories to demonstrate practical applications of the criterion and the GSI system.

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 and deformational properties.The HoekeBrown(HeB) failure criterion is an empirical strength criterion widely applied to rock mechanics and engineering.A direct modification to HeB failure criterion to account for rock anisotropy is considered as the base of the research.Such modification introduced a new definition of the anisotropy as direct parameter named the anisotropic parameter(Kb).However,the computation 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 the minimum value of anisotropic parameter(Kmin),and to predict the Kmindirectly from the uniaxial compression tests instead of triaxial tests,and also to decrease the amount of experimental work.

Energy analysis of rock plug thickness in karst tunnels based on non-associated flow rule and nonlinear failure criterion

The geological hazards, such as water inrush and mud outburst, are easily induced by the high water pressure caverns ahead of a karst tunnel face. Therefore, it is a pivotal issue to determine the reserved thickness of rock plug during the construction of tunnels. The limit analysis principle is employed to analyze the safe thickness from the point of energy dissipation, and the nonlinear and non-associated characteristics of geotechnical materials are both considered. On the basis of a plane failure pattern of rock plug, the expressions of detaching curve and rock plug thickness are derived. The effect of each parameter on the safe thickness of rock plug is discussed in detail, which interprets the corresponding failure scope of rock plug. The obtained results indicate that the thickness of rock plug is highly influenced by the nonlinear dilatancy coefficient and the nonlinear coefficient. The proposed method is validated by a comparison of the calculated results with those of the engineering project of the "526 karst cavern" of Yunwushan tunnel. This proposed method can provide reference basis for the design and excavation of karst tunnels in the future.

Seismic bearing capacity of strip footings on rock masses using the Hoeke Brown failure criterion

In this paper,the bearing capacity of strip footings on rock masses has been studied in the seismic case.The stress characteristics or slip line method was used for analysis.The problem was analyzed in the plane strain condition using the Hoeke Brown failure criterion.First,the equilibrium equations along the stress characteristics were obtained and the rock failure criterion was applied.Then,the equations were solved using the finite difference method.A computer code has been provided for analysis.Given the footing and rock parameters,the code can calculate the stress characteristics network and obtain the stress distribution under the footing.The seismic effects have been applied as the horizontal and vertical pseudo-static coefficients.The results of this paper are very close to those of the other studies.The seismic bearing capacity of weightless rock masses can be obtained using the proposed equations and graphs without calculating the whole stress characteristics network.

Two-dimensional face stability analysis in rock masses governed by the Hoek-Brown strength criterion with a new multi-horn mechanism

The face stability problem is a major concern for tunnels excavated in rock masses governed by the Hoek-Brown strength criterion.To provide an accurate prediction for the theoretical solution of the critical face pressure,this study adopts the piecewise linear method(PLM)to account for the nonlinearity of the strength envelope and proposes a new multi-horn rotational mechanism based on the Hoek-Brown strength criterion and the associative flow rule.The analytical solution of critical support pressure is derived from the energy-work balance equation in the framework of the plastic limit theorem;it is formulated as a multivariable nonlinear optimization problem relying on 2m dependent variables(m is the number of segments).Meanwhile,two classic linearized measures,the generalized tangential technique(GTT)and equivalent Mohr-Coulomb parameters method(EMM),are incorporated into the analysis for comparison.Surprisingly,the parametric study indicates a significant improvement in support pressure by up to 13%compared with the GTT,and as expected,the stability of the tunnel face is greatly influenced by the rock strength parameters.The stress distribution on the rupture surface is calculated to gain an intuitive understanding of the failure at the limit state.Although the limit analysis is incapable of calculating the true stress distribution in rock masses,a rough approximation of the stress vector on the rupture surface is permitted.In the end,sets of normalized face pressure are provided in the form of charts for a quick assessment of face stability in rock masses.

Failure criterion for soft rocks considering intermediate principal stress

The significant differences between hard rocks(more brittle)and soft rocks(more ductile)may suggest the use of different failure criteria.A strength criterion for soft rocks that includes intermediate principal stress was proposed.The new criterion includes two independent parameters:the uniaxial compressive strength(σ_(ci)),which can be obtained from common laboratory tests or indirectly estimated by alternative index tests in the laboratory or field;and f(joint),which is used to characterize the rock mass quality and can be easily estimated.The authors compared the predictive capabilities of the new criterion with other criteria using the database of soft rocks under two conditions:with and without triaxial data.For the estimation of triaxial and true-triaxial strengths,the new criterion generally produced a better fit.The proposed criterion is practical for an approximate first estimation of rock mass strength,even without triaxial data,as it balances accuracy(lower prediction error)and simplicity(fewer independent parameters).

A modified failure criterion for transversely isotropic rocks

A modified failure criterion is proposed to determine the strength of transversely isotropic rocks. Mechanical properties of some metamorphic and sedimentary rocks including gneiss, slate, marble, schist,shale, sandstone and limestone, which show transversely isotropic behavior, were taken into consideration. Afterward, introduced triaxial rock strength criterion was modified for transversely isotropic rocks.Through modification process an index was obtained that can be considered as a strength reduction parameter due to rock strength anisotropy. Comparison of the parameter with previous anisotropy indexes in literature showed reasonable results for the studied rock samples. The modified criterion was compared to modified Hoek-Brown and Ramamurthy criteria for different transversely isotropic rocks. It can be concluded that the modified failure criterion proposed in this study can be used for predicting the strength of transversely isotropic rocks.

Upper bound solutions for supporting pressures of shallow tunnels with nonlinear failure criterion

线性失败标准广泛地在对浅隧道起作用的地球压力的计算被使用。然而，试验性的证据证明那个非线性的失败标准能表示相当，很好，几乎所有的失败岩石打字。一个非线性的 Hoek 布郎失败标准被采用在限制分析框架估计浅隧道的支持的压力。二失败机制为计算外部力量和内部精力驱散的工作率被建议。到非线性的失败标准的一根正切的线被用来作为一个非线性的编程问题提出支持的压力问题。这样提出的客观功能关于失败机制和接触点的地点被最小化。以便估计有效性，为建议失败机制的支持的压力被计算，当非线性的标准被归结为线性标准时，与对方，和礼品相比，结果与以前出版的答案相比。协议支持建议失败机制的有效性。一个实验被进行在倒塌形状和浅隧道的支持的压力上调查非线性的标准的影响。

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 L s 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 fis 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.

Kinematical analysis of highway tunnel collapse using nonlinear failure criterion

For different kinds of rocks,the collapse range of tunnel was studied in the previously published literature.However,some tunnels were buried in soils,and test data showed that the strength envelopes of the soils followed power-law failure criterion.In this work,deep buried highway tunnel with large section was taken as objective,and the basic expressions of collapse shape and region were deduced for the highway tunnels in soils,based on kinematical approach and power-law failure criterion.In order to see the effectiveness of the proposed expressions,the solutions presented in this work agree well with previous results if the nonlinear failure criterion is reduced to a linear Mohr-Coulomb failure criterion.The present results are compared with practical projects and tunnel design code.The numerical results show that the height and width of tunnel collapse are greatly affected by the nonlinear criterion for the tunnel in soil.