Estimating shear strength of high-level pillars supported with cemented backfilling using the HoekeBrown strength criterion

Deep metal mines are often mined using the high-level pillars with subsequent cementation backfilling(HLSCB)mining method.At the design stage,it is therefore important to have a reasonable method for determining the shear strength of the high-level pillars(i.e.cohesion and internal friction angle)when they are supported by cemented backfilling.In this study,a formula was derived for the upper limit of the confining pressure σ3max on a high-level pillar supported by cemented backfilling in a deep metal mine.A new method of estimating the shear strength of such pillars was then proposed based on the Hoek eBrown failure criterion.Our analysis indicates that the horizontal stress σhh acting on the cemented backfill pillar can be simplified by expressing it as a constant value.A reasonable and effective value for σ3max can then be determined.The value of s3max predicted using the proposed method is generally less than 3 MPa.Within this range,the shear strength of the high-level pillar is accurately calculated using the equivalent MohreCoulomb theory.The proposed method can effectively avoid the calculation of inaccurate shear strength values for the high-level pillars when the original HoekeBrown criterion is used in the presence of large confining pressures,i.e.the situation in which the cohesion value is too large and the friction angle is too small can effectively be avoided.The proposed method is applied to a deep metal mine in China that is being excavated using the HLSCB method.The shear strength parameters of the high-level pillars obtained using the proposed method were input in the numerical simulations.The numerical results show that the recommended level heights and sizes of the high-level pillars and rooms in the mine are rational.

A new three-dimensional Hoek-Brown strength criterion

The Hoek-Brown （HB） strength criterion has been widely applied to the estimation of strength of intact rock and rock mass, while evolving ever since. However, negligence of the effect of the intermediate principal stress still remains in the criterion＇s latest version. At the same time, several three-dimensional （3D） HB strength, which can takes into account the influence of the intermediate principal stress, have already been proposed, among which the 3D HB criterion proposed by Zhang and Zhu seems to be the most reasonable one. However, the Zhang 3D HB criterion may have problems with some stress path close to triaxial extension state because of the non-convexity characteristic of its failure surface. In this paper, a new 3D HB strength criterion is presented based on a generalized form of the HB criterion, which also considers the effect of the intermediate principal stress and inherits all the merits of the original version of the HB criterion. In addition, this new criterion can remedy to some extent the shortcomings observed in the Zhang 3D HB criterion. Polyaxial tests for five different rocks from pub- lished literatures are used for evaluating this new criterion and comparing it with the Zhang 3D HB criterion. The re- sults show that this new criterion may over-predict or under- predict the polyaxial strength of rocks but the errors are rela- tively small, and similar results are also found for the Zhang 3D HB criterion, which one is better depends on the type of the rock under estimation.

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.

Shear behavior of intact granite under thermo-mechanical coupling and three-dimensional morphology of shear-formed fractures

The shear failure of intact rock under thermo-mechanical(TM)coupling conditions is common,such as in enhanced geothermal mining and deep mine construction.Under the effect of a continuous engineering disturbance,shear-formed fractures are prone to secondary instability,posing a severe threat to deep engineering.Although numerous studies regarding three-dimensional(3D)morphologies of fracture surfaces have been conducted,the understanding of shear-formed fractures under TM coupling conditions is limited.In this study,direct shear tests of intact granite under various TM coupling conditions were conducted,followed by 3D laser scanning tests of shear-formed fractures.Test results demonstrated that the peak shear strength of intact granite is positively correlated with the normal stress,whereas it is negatively correlated with the temperature.The internal friction angle and cohesion of intact granite significantly decrease with an increase in the temperature.The anisotropy,roughness value,and height of the asperities on the fracture surfaces are reduced as the normal stress increases,whereas their variation trends are the opposite as the temperature increases.The macroscopic failure mode of intact granite under TM coupling conditions is dominated by mixed tensileeshear and shear failures.As the normal stress increases,intragranular fractures are developed ranging from a local to a global distribution,and the macroscopic failure mode of intact granite changes from mixed tensileeshear to shear failure.Finally,3D morphological characteristics of the asperities on the shear-formed fracture surfaces were analyzed,and a quadrangular pyramid conceptual model representing these asperities was proposed and sufficiently verified.

Limit analysis for the seismic stability of three-dimensional rock slopes using the generalized Hoek-Brown criterion

The parameters that influence slope stability and their criteria of failure are fairly understood but over-conservative design approaches are often preferred,which can result in excessive overburden removal that may jeopardize profitability in the context of open pit mining.Numerical methods such as finite element and discrete element modelling are instrumental to identify specific zones of stability,but they remain approximate and do not pinpoint the critical factors that influence stability without extensive parametric studies.A large number of degrees of freedom and input parameters may make the outcome of numerical modelling insufficient compared to analytical solutions.Existing analytical approaches have not tackled the stability of slopes using non-linear plasticity criteria and threedimensional failure mechanisms.This paper bridges this gap by using the yield design theory and the Hoek-Brown criterion.Moreover,the proposed model includes the effect of seismic forces,which are not always taken into account in slope stability analyses.The results are presented in the form of rigorous mathematical expressions and stability charts involving the loading conditions and the rock mass properties emanating from the plasticity criterion.

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.

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.

Correction of dynamic Brazilian disc tensile strength of rocks under preloading conditions considering the overload phenomenon and invoking the Griffith criterion

Dynamic tensile failure is a common phenomenon in deep rock practices,and thus accurately evaluating the dynamic tensile responses of rocks under triaxial pressures is of great significance.The Brazilian disc(BD)test is the suggested method by the International Society for Rock Mechanics and Rock Engineering(ISRM)for measuring both the static and dynamic tensile strengths of rock-like materials.However,due to the overload phenomenon and the complex preloading conditions,the dynamic tensile strengths of rocks measured by the BD tests tend to be overestimated.To address this issue,the dynamic BD tensile strength(BTS)of Fangshan marble(FM)under different preloading conditions were measured through a triaxial split Hopkinson pressure bar(SHPB).The fracture onset in BD specimen was captured through a strain gage around the disc center.The discrepancy between the traditional tensile strength(TTS,determined by the peak load P_(f) of the BD specimen)and the nominal tensile strength(NTS,obtained from the load P_(i) when the diametral fracture commences in the tested BD specimen)was applied to quantitatively evaluating the overload phenomenon.The Griffith criterion was used to rectify the calculation of the tensile stress at the disc center under triaxial stress states.The results demonstrate that the overload ratio(s)increases with the loading rate(σ)and decreases with the hydrostatic pressure(σ_(s)).The TTS corrected by the Griffith criterion is independent of theσ_(s)due to the overload phenomenon,while the NTS corrected by the Griffith criterion is sensitive to both the andσ.Therefore,it is essential to modify the tensile stress in dynamic confined BD tests using both the overload correction and the Griffith criterion rectification to obtain the accurate dynamic BTS of rocks.

Three-Dimensional Analysis of Rolling by Twin Shear Stress Yield Criterion

Using the twin shear stress yield criterion, the surface integral of the co-line vectors, and the integration depending on upper limit, Kobayashi＇s three-dimensional velocity field of rolling was analyzed and an analytical expression of rolling torque and single force was obtained. Through redoing the same experiment of rolling pure lead as Sims, the calculated results by the above expression were compared with those of Kobayashi and Sims formulae. The results show that the twin shear stress yield criterion is available for rolling analysis and the calculated results by the new formula are a little higher than those by Kobayashi and Sims ones if the reduction ratio is less than 30%.

Analytical Solution for Three-dimensional Forging Taking into Account Bulging of Sides by Mean Yield Criterion

Currently, for some complex plastic deformations, the analytical solution can not be obtained by using Mises yield criterion, because Mises yield criterion is nine dimensions, the velocity field is complex, and the solving methods are not innovative. Corresponding solutions of these problems are that yield criterion is linearized to reduce the variable numbers, and the velocity field and the solving methods are reasonably simplified, respectively. In this paper, a new linear yield criterion--mean yield（MY） criterion and inner-product of strain rate vector are used to analytically solve 3D forging taking into account bugling of sides. The velocity field is expressed as a vector in three dimensions, and rotation and divergence are applied to confirm that the velocity field is kinematically admissible. Then, the corresponding strain rate tensor of the velocity field is transformed into principal one by making the determinant of coefficients of the tensor cubic equation be zero. By using MY criterion, the plastic power is term by term integrated and summed according to inner-product of strain rate vector. An upper bound analytical solution is obtained for the forging, and verified by a pure lead press test. The test result turns out that the total pressure calculated by MY criterion is higher by 2.5%-15% than measuring value. In addition, a measuring formula of bulging parameter （a） is proposed, but the values of a measured by the formula are lower than those optimized by the golden section search. The total pressure calculated by MY criterion is compared with the ones by twin shear, Trasca yield, and Mises yield criterion. The comparing result shows that the total pressure calculated by MY criterion is slightly higher than the mean value of that by twin shear and Trasca yield criterion, and lower than that by Mises yield criterion, but more close to that by Mises yield criterion compared with that by other two. The proposed analytical solving methods can be effectively used to other complex plastic deformation, simplifying the solving process and obtaining the reasonable results.

Investigation of anisotropic strength criteria for layered rock mass

Layered rock mass is a type of engineering rock mass with sound mechanical anisotropy,which is generally unfavorable to the stability of underground works.To investigate the strength anisotropy of layered rock,the Mohr-Coulomb and Hoek-Brown criteria are introduced to establish the two transverse isotropic strength criteria based on Jaeger's single weak plane theory and maximum axial strain theory,and parameter determination methods.Furthermore,the sensitivity of strength parameters(K 1,K 2,and K 3)that are used to characterize the anisotropy strength of non-sliding failure involved in the strength criteria and confining pressure are investigated.The results demonstrate that strength parameters K 1 and K 2 affect the strength of layered rock samples at all bedding angles except for the bedding angle of 90°and the angle range that can cause the shear sliding failure along the bedding plane.The strength of samples at any bedding angle decreases with increasing K 1,whereas the opposite is for K 2.Except for bedding angles of 0°and 90°and the bedding angle range that can cause the shear sliding along the bedding plane,K 3 has an impact on the strength of rock samples with other bedding angles that the specimens'strength increases with increase of K 3.In addition,the strength of the rock sample increases as confining pressure rises.Furthermore,the uniaxial and triaxial tests of chlorite schist samples were carried out to verify and evaluate the strength criteria proposed in the paper.It shows that the predicted strength is in good agreement with the experimental results.To test the applicability of the strength criterion,the strength data of several types of rock in the literature are compared.Finally,a comparison is made between the fitting effects of the two strength criteria and other available criteria for layered rocks.

Analysis on the turning point of dynamic in-plane compressive strength for a plain weave composite

Experimental investigations on dynamic in-plane compressive behavior of a plain weave composite were performed using the split Hopkinson pressure bar. A quantitative criterion for calculating the constant strain rate of composites was established. Then the upper limit of strain rate, restricted by stress equilibrium and constant loading rate, was rationally estimated and confirmed by tests. Within the achievable range of 0.001/s-895/s, it was found that the strength increased first and subsequently decreased as the strain rate increased. This feature was also reflected by the turning point(579/s) of the bilinear model for strength prediction. The transition in failure mechanism, from local opening damage to completely splitting destruction, was mainly responsible for such strain rate effects. And three major failure modes were summarized under microscopic observations: fiber fracture, inter-fiber fracture, and interface delamination. Finally, by introducing a nonlinear damage variable, a simplified ZWT model was developed to characterize the dynamic mechanical response. Excellent agreement was shown between the experimental and simulated results.

Discussion on “A generalized three-dimensional failure criterion for rock masses”

There are many methods to construct true triaxial strength criteria for rocks.Jaiswal and Shrivastva(2012)proposed a strength criterion,named J–S criterion,in the deviatoric plane,which provides nearly the same misfts for true triaxial test data as the exponential criterion.It is diffcult to calculate the strength at given2and3using the J–S criterion,and the multiple solutions to the nonlinear equation may induce confusion and mistake.Strength envelopes in deviatoric planes are not geometric similar;therefore,true triaxial test data cannot be grouped in the mean stress to check strength criteria in the deviatoric plane.

Reply to Discussion on “A generalized three-dimensional failure criterion for rock masses”

General:Jaiswal and Shrivastva(2012)proposed the mathematical formulations,i.e.the J–S criterion for converting generalized H-B failure criterion into 3D smooth convex failure criterion at deviatoric plane.The J–S strength criterion is in two versions:uniform and variable extension ratio.It has been observed from the analysis that at uniform extension ratio,the required strength parameters are only UCS and m(other parameters such as Ls,a,b and c are related with m).In the case of variable extension ratio,extra parameter f is required along with UCS and m.Thus,it has minimal strength parameters compared to You strength criterion.Furthermore,You strength criterion does not obey the smooth convex condition at deviatoric plane.

Strength criterion of composite solid propellants under dynamic loading

Based on the dynamic loading(1-100 s^(-1)) experiments under different temperatures(223-298 K) and stress states, uniaxial and biaxial strength criterion of a Hydroxyl-terminated polybutadiene(HTPB)based composite solid propellant were further investigated. These experiments were conducted through the use of a new uniaxial INSTRON testing machine, different new designed gripping apparatus and samples with different configurations. According to the test results, dynamic uniaxial tensile strength criterion of the propellant was directly constructed with the master curve of the uniaxial maximum tensile stress. Whereas, a new method was proposed to determine the dynamic uniaxial compressive strength of the propellant in this study. Then uniaxial compressive strength criterion of the propellant was constructed based on the related master curve. Moreover, it found that the uniaxial tensilecompressive strength ratio of the propellant is more sensitive to loading temperature under the test conditions. The value of this parameter is about 0.4 at room temperature, and it reduces to 0.2-0.3 at low temperatures. Finally, the theoretical biaxial strength criterion of HTPB propellant under dynamic loading was constructed with the unified strength theory, the uniaxial strength and the typical biaxial tensile strength. In addition, the theoretical limit lines of the principal stress plane for the propellant under dynamic loading at different temperatures were further plotted, and the scope of the limit line increases with decreasing temperature.

STRENGTH CRITERION FOR PLAIN CONCRETE UNDER MULTIAXIAL STRESS BASED ON DAMAGE POISSON’S RATIO

A new unified strength criterion in the principal stress space has been proposed for use with normal strength concrete （NC） and high strength concrete （HSC） in compressioncompression-tension, compression-tension-tension, triaxial tension, and biaxial stress states. The study covers concrete with strengths ranging from 20 to 130 MPa. The conception of damage Poisson＇s ratio is defined and the expression for damage Poisson＇s ratio is determined basically. The failure mechanism of concrete is illustrated, which points out that damage Poisson＇s ratio is the key to determining the failure of concrete. Furthermore, for the concrete under biaxial stress conditions, the unified strength criterion is simplified and a simplified strength criterion in the form of curves is also proposed. The strength criterion is physically meaningful and easy to calculate, which can be applied to analytic solution and numerical solution of concrete structures.

Strength criterion effect of the translator and destabilization model of gas-bearing coal seam

Coal seam destabilization inflicts damage to equipment, causes property loss and personnel casualties,and severely threatens mining safety and efficient production. To further understand this destabilization based on the basic theory of Lippmann seam destabilization, a mathematical model was introduced for gas pressure distribution by considering intermediate principal stress and support resistance.Subsequently, we established a translation model suitable for the entire roadway coal seam with rocky roof and floor by applying the unified form of yield criterion in the state of plane strain. We also obtained the analytic expressions of coal seam stress distribution on both sides of the roadway and the widths of plastic and disturbance zones. Afterward, we analyzed several typical cases with different material yield criteria, obtained the plastic zone widths of the coal seam under different gas pressures, and assessed the effects of support resistance, roadway size, and coal strength on coal seam destabilization. Results showed that: the results obtained on the basis of Wilson and Mohr–Coulomb criteria are considerably conservative, and the use of Druker–Prager criteria to evaluate the rockburst-induced coal seam destabilization is safer than the use of the two other criteria; coal seam stability is correlated with gas pressure;and high-pressure gas accelerates the coal seam destabilization.

EFFECT OF DENTIN TUBULES ON THE MECHANICAL PROPERTIES OF DENTIN. PART Ⅰ: STRESS-STRAIN RELATIONS AND STRENGTH CRITERION

As known, there is a large number of dentin tubules in dentin. These tubules have varying radii and are shaped into radially parallel pattern. The anisotropy of microstructure of dentin shows that dentin should be treated as a ma- terial of varying transverse isotropy. In this Part, the elastic stress-strain relations and the quadratic strength criterion are established in the form of having varying transverse isotropy, in the framework of micromechanics to take into account of the effect of the microstructures-dentin tubules. Simplified forms for isotropic and ho- mogeneous cases, as well as the corresponding plane stress form of the stress-strain relations are also given. These theoretical models are very well supported by the experiments shown later in the continued paper (Part Ⅱ).

An approximate nonlinear modified Mohr-Coulomb shear strength criterion with critical state for intact rocks

In this paper, the Mohr-Coulomb shear strength criterion is modified by mobilising the cohesion and internal friction angle with normal stress, in order to capture the nonlinearity and critical state concept for intact rocks reported in the literature. The mathematical expression for the strength is the same as the classical form, but the terms of cohesion and internal friction angle depend on the normal stress now,leading to a nonlinear relationship between the strength and normal stress. It covers both the tension and compression regions with different expressions for cohesion and internal friction angle. The strengths from the two regions join continuously at the transition of zero normal stress. The part in the compression region approximately satisfies the conditions of critical state, where the maximum shear strength is reached. Due to the nonlinearity, the classical simple relationship between the parameters of cohesion, internal friction angle and uniaxial compressive strength from the linear Mohr-Coulomb criterion does not hold anymore. The equation for determining one of the three parameters in terms of the other two is supplied. This equation is nonlinear and thus a nonlinear equation solver is needed. For simplicity, the classical linear relationship is used as a local approximation. The approximate modified Mohr-Coulomb criterion has been implemented in a fracture mechanics based numerical code FRACOD,and an example case of deep tunnel failure is presented to demonstrate the difference between the original and modified Mohr-Coulomb criteria. It is shown that the nonlinear modified Mohr-Coulomb criterion predicts somewhat deeper and more intensive fracturing regions in the surrounding rock mass than the original linear Mohr-Coulomb criterion. A more comprehensive piecewise nonlinear shear strength criterion is also included in Appendix B for those readers who are interested. It covers the tensile, compressive, brittle-ductile behaviour transition and the critical state, and gives smooth transitions.

Strength differential effect and influence of strength criterion on burst pressure of thin-walled pipelines

In the framework of the finite deformation theory, the plastic collapse analysis of thin-walled pipes subjected to the internal pressure is conducted on the basis of the unified strength criterion （USC）. An analytical solution of the burst pressure for pipes with capped ends is derived, which includes the strength differential effect and takes the influence of strength criterion on the burst pressure into account. In addition, a USC- based analytical solution of the burst pressure for end-opened pipes under the internal pressure is obtained. By discussion, it is found that for the end-capped pipes, the influence of different yield criteria and the strength differential effect on the burst pressure are significant, while for the end-opened pipes, the burst pressure is independent of the specific form of the strength criterion and strength difference in tension and compression.