A generalized nonlinear three-dimensional Hoek‒Brown failure criterion

To understand the strengths of rocks under complex stress states,a generalized nonlinear threedimensional(3D)Hoek‒Brown failure(NGHB)criterion was proposed in this study.This criterion shares the same parameters with the generalized HB(GHB)criterion and inherits the parameter advantages of GHB.Two new parameters,b,and n,were introduced into the NGHB criterion that primarily controls the deviatoric plane shape of the NGHB criterion under triaxial tension and compression,respectively.The NGHB criterion can consider the influence of intermediate principal stress(IPS),where the deviatoric plane shape satisfies the smoothness requirements,while the HB criterion not.This criterion can degenerate into the two modified 3D HB criteria,the Priest criterion under triaxial compression condition and the HB criterion under triaxial compression and tension condition.This criterion was verified using true triaxial test data for different parameters,six types of rocks,and two kinds of in situ rock masses.For comparison,three existing 3D HB criteria were selected for performance comparison research.The result showed that the NGHB criterion gave better prediction performance than other criteria.The prediction errors of the strength of six types of rocks and two kinds of in situ rock masses were in the range of 2.0724%-3.5091%and 1.0144%-3.2321%,respectively.The proposed criterion lays a preliminary theoretical foundation for prediction of engineering rock mass strength under complex in situ stress conditions.

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.

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°.

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.

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.

Strength regularity and failure criterion of plain HSHPC under biaxial compression after exposure to high temperatures

Biaxial compression tests are performed on 100 mm × 100 mm × 100 mm cubic specimens of plain high-strength highperformance concrete （HSHPC） at seven kinds of stress ratios, σ2：σ3 =0 ： - 1, -0.20 ： - 1, -0.30 ： - 1, -0.40 ： - 1, -0.50 ： -1, -0. 75 ： - 1, and - 1.00 ： - 1 after exposure to normal and high temperatures of 20, 200, 300, 400, 500 and 600 ℃, using a large static-dynamic true triaxial machine. Frictionreducing pads are three layers of plastic membranes with glycerine in-between for the compressive loading plane. Failure modes of the specimens are described. The two principally static compressive strengths are measured. The influences of the temperatures and stress ratios on the biaxial strengths of HSHPC after exposure to high temperatures are also analyzed. The experimental results show that the uniaxial compressive strength of plain HSHPC after exposure to high temperatures does not decrease completely with the increase in temperature; the ratios of the biaxial to its uniaxial compressive strengths depend on the stress ratios and brittleness-stiffness of HSHPC after exposure to different high temperatures. The formula of the Kupfer-Gerstle failure criterion modified with the temperature and stress ratio parameters for plain HSHPC is proposed.

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

Linear failure criterion is widely used in calculation of earth pressure acting on shallow tunnels. However, experimental evidence shows that nonlinear failure criterion is able to represent fairly well the failure of almost all types of rocks, A nonlinear Hock-Brown failure criterion is employed to estimate the supporting pressures of shallow tunnels in limit analysis framework. Two failure mechanisms are proposed for calculating the work rate of extemal force and the internal energy dissipation. A tangential line to the nonlinear failure criterion is used to formulate the supporting pressure problem as a nonlinear programming problem. The objective function formulated in this way is minimized with respect to the failure mechanism and the location of tangency point. In order to assess the validity, the supporting pressures for the proposed failure mechanisms are calculated and compared with each other, and the present results are compared with previously published solutions when the nonlinear criterion is reduced to linear criterion. The agreement supports the validity of the proposed failure mechanisms. An experiment is conducted to investigate the influences of the nonlinear criterion on collapse shape and supporting pressures of shallow tunnels.

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.

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.

Upper bound analysis of slope stability with nonlinear failure criterion based on strength reduction technique

Based on the upper bound limit analysis theorem and the shear strength reduction technique, the equation for expressing critical limit-equilibrium state was employed to define the safety factor of a given slope and its corresponding critical failure mechanism by means of the kinematical approach of limit analysis theory. The nonlinear shear strength parameters were treated as variable parameters and a kinematically admissible failure mechanism was considered for calculation schemes. The iterative optimization method was adopted to obtain the safety factors. Case study and comparative analysis show that solutions presented here agree with available predictions when nonlinear criterion reduces to linear criterion, and the validity of present method could be illuminated. From the numerical results, it can also be seen that nonlinear parameter rn, slope foot gradient ,β, height of slope H, slope top gradient a and soil bulk density γ have significant effects on the safety factor of the slope.

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.

Safe retaining pressures for pressurized tunnel face using nonlinear failure criterion and reliability theory

Based on the active failure mechanism and passive failure mechanism for a pressurized tunnel face, the analytical solutions of the minimum collapse pressure and maximum blowout pressure that could maintain the stability of pressurized tunnel faces were deduced using limit analysis in conjunction with nonlinear failure criterion under the condition of pore water pressure. Due to the objective existence of the parameter randomness of soil, the statistical properties of random variables were determined by the maximum entropy principle, and the Monte Carlo method was employed to calculate the failure probability of a pressurized tunnel. The results show that the randomness of soil parameters exerts great influence on the stability of a pressurized tunnel, which indicates that the research should be done on the topic of determination of statistical distribution for geotechnical parameters and the level of variability. For the failure probability of a pressurized tunnel under multiple failure modes, the corresponding safe retaining pressures and optimal range of safe retaining pressures are calculated by introducing allowable failure probability and minimum allowable failure probability. The results can provide practical use in the pressurized tunnel engineering.

A modified failure criterion for transversely isotropic rocks

A modified failure criterion is proposed to determine the strength of transversely isotropic rocks. Me-chanical properties of some metamorphic and sedimentary rocks including gneiss, slate, marble, schist, shale, sandstone and limestone, which show transversely isotropic behavior, were taken into consider-ation. 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 in-dexes 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.

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.

Upper bound solution of surrounding rock pressure of shallow tunnel under nonlinear failure criterion

The nonlinear Baker failure criterion is introduced into the upper-bound limit analysis to examine the face stability of a shallow tunnel. The tunnel face under the ultimate limit state is analyzed from the perspective of energy balance. The work rates of external forces and internal energy dissipation are calculated. An analytical solution of necessary face pressures is derived. The optimal upper-bound solution of the face pressures is obtained by optimization. The results show that the three dimensionless parameters A, T, n of nonlinear Baker failure criterion have different effects on the necessary face pressures and the pattern failure mechanisms ahead of tunnel face. A is the most important one;n takes the second place, and T is the least one. The computed necessary face pressures are nonlinearly increasing when A is reduced. Combined with the actual monitoring data of Taxia tunnel, the calculation results in this paper is verified. It is suggested that the tunnel face supports should be strengthened timely in soft rocks to prevent the occurrence of face collapse.

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.

Three-dimensional upper bound limit analysis of underground cavities using nonlinear Baker failure criterion

A generalized nonlinear Baker failure criterion is employed with the upper bound limit analysis to study the surrounding rock stability of underground cavities. A three-dimensional(3D) failure mode is established by extending the two-dimensional(2D) failure mode, which offers an upper bound expression of the surrounding rock pressure. This method is validated with a series of examples before the influence of four parameters of scale parameter, curvature parameter, shift parameter and lateral pressure coefficient, on the surrounding rock pressure is analyzed. According to these results, failure ranges of the underground cavities are determined. The following conclusions are reached:(1) the proposed approach is more accurate to predict surrounding rock pressure than the Mohr-Coulomb failure criterion;(2) the surrounding rock with large scale parameter, curvature parameter, shift parameter, and lateral pressure coefficient can lead to a more stable underground cavity;(3) the failure range in 3D mode can be predicted according to the upper bound solutions.

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.

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.