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.

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.

Determination of the constant m_(i) in the Hoek-Brown criterion of rock based on drilling parameters

The constant m_(i) in the Hoek-Brown(H-B) criterion is a fundamental parameter required for determining the compressive strength of rock. In this paper, drilling parameters provide a new basis for determining the constant mi. An analytical relationship between the drilling parameters and constant miis established in consideration of the contact response between the drilling bit and the cut rock in the crushed zone.New models are developed to predict the triaxial compressive strength(TCS), internal friction angle φand cohesion c of rock. Drilling tests are carried out on 6 rock types to study the correlation between φ and m_(i). A comparison between the predicted values of rock mechanical properties and the measured values from the laboratory is performed to verify the accuracy of the proposed model(yielding an error less than 10%). The TCSs and constant m_(i) values of fifteen rocks are cited to validate the accuracy of the proposed model. The result shows that the proposed model predicts the TCS and constant m_(i) within a maximum error of 20%. The method can be conveniently applied to the rock mechanical properties.

砂化白云岩Hoek-Brown准则参数取值方法研究及其岩体力学参数估算

武汉市碳酸盐岩天兴洲条带,发育一套巨厚层砂化白云岩,其岩体力学参数取值,目前尚无相关经验或建议。基于Hoek-Brown(简称H-B)准则及其Mohr-Coulomb(简称M-C)线性转换理论,归纳总结前人研究成果,在试验数据的基础上,研究了砂化白云岩的H-B准则参数,提出一种基于静止侧压力系数k_(1)的最大围压上限σ_(3max)简化取值方法,估算其岩体力学参数并给出了取值建议。结果表明:基于k_(1)的σ_(3m)_(ax)简化取值方法,合理缩减了围压σ_(3)的取值区间,提高了计算精度;砂化白云岩的地质强度指数GSI与无侧限抗压强度σ_(ci)呈良好的线性关系,结合其砂化程度在岩体块度上的贴切表现,制作了针对砂化白云岩的改进GSI建议值表;岩体材料参数m_(i)的大小主要受控于岩石种类及细观颗粒的粗糙程度,其取值区间跨度小且对岩体力学参数影响较小,取用m_(i)=9;通过H-B准则与M-C准则的线性转换,建立了σ_(ci)与岩体抗剪强度之间的计算关系;各试验值、拟合值及理论计算值之间关系自洽、结果合理,有效解决了极破碎白云岩岩体力学参数的取值问题。上述基础性工作对H-B准则的应用起到了有益的补充,对武汉市类似岩层的岩体力学参数取值方法研究及运用具有重要借鉴意义。

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.

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.

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 modified Hoek-Brown failure criterion considering the damage to reservoir bank slope rocks under water saturation-dehydration circulation

After water is impounded in a reservoir, rock mass in the hydro-fluctuation belt of the reservoir bank slope is subject to water saturation- dehydration circulation （WSDC）. To quantify the rate of change of rock mechanical properties, samples from the Longtan dam area were measured with uniaxial compression tests after different numbers （1, 5, 10, 15, and 20） of simulated WSDC cycles. Based on the curves derived from these tests, a modified Hock- Brown failure criterion was proposed, in which a new parameter was introduced to model the cumulative damage to rocks after WSDC. A case of an engineering application was analyzed, and the results showed that the modified Hock-Brown failure criterion is useful. Under similar WSDC-influenced engineering and geological conditions, rock mass strength parameters required for analysis and evaluation of rock slope stability can be estimated according to this modified Hoek-Brown failure criterion.

Rockburst criterion and evaluation method for potential rockburst pit depth considering excavation damage effect

Excavation-induced disturbances in deep tunnels will lead to deterioration of rock properties and formation of excavation damaged zone(EDZ).This excavation damage effect may affect the potential rockburst pit depth.Taking two diversion tunnels of Jinping II hydropower station for example,the relationship between rockburst pit depth and excavation damage effect is first surveyed.The results indicate that the rockburst pit depth in tunnels with severe damage to rock masses is relatively large.Subsequently,the excavation-induced damage effect is characterized by disturbance factor D based on the Hoek-Brown criterion and wave velocity method.It is found that the EDZ could be further divided into a high-damage zone(HDZ)with D=1 and weak-damage zone(WDZ),and D decays from one to zero linearly.For this,a quantitative evaluation method for potential rockburst pit depth is established by presenting a three-element rockburst criterion considering rock strength,geostress and disturbance factor.The evaluation results obtained by this method match well with actual observations.In addition,the weakening of rock mass strength promotes the formation and expansion of potential rockburst pits.The potential rockburst pit depth is positively correlated with HDZ and WDZ depths,and the HDZ depth has a significant contribution to the potential rockburst pit depth.

基于Hoek-Brown准则的岩体高温-荷载耦合损伤模型及其验证

针对隧道岩体在高温和荷载共同作用下产生的耦合损伤问题,基于连续损伤力学理论,引入高温-荷载耦合损伤因子,建立了考虑高温作用的广义Hoek-Brown(H-B)准则的弹塑性损伤本构模型,解决了模型在数值积分过程中的奇异点问题,并利用Fortran语言编写了模型的有限元程序。结果表明:模型计算的应力-应变曲线与试验所得到的曲线趋势一致且计算结果误差在5%以内,温度作用下隧洞围岩损伤逐渐加剧,验证了模型的准确性和程序的稳健性,可为火灾作用下隧道围岩安全稳定性评价提供一种有效计算方法。

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.

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准则的底板破坏深度分析

为了研究煤层采动底板最大破坏深度,以1509工作面回风巷道为工程背景。利用半无限体理论建立力学模型,结合材料力学和Hoek-Brown准则,推导出最大破坏深度计算公式,并经过试验和现场监测进行验证。结果显示:底板最大破坏深度为20.3 m,且呈“水滴形”塑性区边界。底板破坏深度与岩体软硬程度呈负相关,与破碎程度呈正相关。现场监测结果显示,随底板埋深增加,采动应力影响减小;波速-垂深曲线表示底板最大破坏深度约为20 m,与理论计算基本吻合。研究成果对类似工作面有借鉴意义,有助于预测底板破坏深度,确保安全生产,并为相关领域研究提供参考。

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.

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.

考虑中间主应力效应的修正Hoek-Brown真三轴强度准则

为改善Hoek-Brown强度准则未考虑中间主应力对岩石强度的影响,通过岩石真三轴试验结果分析了岩石强度的演化规律,引入中间主应力系数,量化分析主应力系数对岩石真三轴强度的影响,考虑Hoek-Brown强度准则中的参数和岩石应力水平间的关联性,提出了基于拉格朗日插值方法的修正岩石真三轴Hoek-Brown强度准则,分析了其空间包络特征。最后,利用7种岩石的真三轴试验数据与其它3种真三轴强度准则进行最优拟合误差分析,探讨了修正强度准则的合理性。研究结果表明:岩石强度随着最小主应力的增加逐渐增大,随着中间主应力的增加则呈现先增大后减小的变化规律,表现出显著的区间效应;修正强度准则不仅能够继承Hoek-Brown强度准则在子午面上的非线性优点,且能够表征岩石强度在应力空间中的基本特性;线性和非线性插值形式的修正强度准则空间包络面分别为非等边的六棱锥面以及能够满足拉压子午面区间连续光滑要求的圆锥面;修正强度准则能够较好地预测岩石真三轴试验强度,较为合理地反映中间主应力对岩石强度的影响;相比于线性插值形式强度准则,二次插值形式强度准则能够提高岩石强度的预测精度1.2~2.0倍;相较于其他真三轴强度准则,修正强度准则对不同硬脆性岩石的真三轴试验结果均具有良好的强度预测精度,体现了修正强度准则的适用性和可靠性。

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.

基于Hoek-Brown强度准则的地下洞室开挖可靠度分析

地下洞室可靠度分析在地下空间开发工程中具有重要意义。采用拉丁超立方抽样法对工程实测获得的结构面特征指标进行抽样,将样本数据代入地质强度指标GSI评价体系中获得GSI概率分布函数,然后基于Hoek-Brown经验强度准则计算获得围岩力学参数的概率分布函数,进而利用数值模拟获得地下洞室变形量的概率分布函数,并结合洞室开挖的最大允许位移将求解可靠度指标转化为求解超过最大允许位移的概率。以五岳抽水蓄能电站为例,计算了主厂房不同部位的位移超过最大允许位移的概率,对主厂房的稳定性进行了评价。评价结果与工程实际情况基本相符,说明本文提出的方法可有效用于地下工程的开挖可靠度分析.

基于Hoek-Brown强度准则的溶蚀作用下灰岩力学参数计算方法

Hoek-Brown强度准则广泛使用于岩石及岩体工程中,针对地下工程中灰岩围岩洞室受溶蚀作用其稳定性会降低的问题,提出一种相应的参数计算方法.基于Hoek-Brown强度准则中可描述溶蚀作用下灰岩的力学特性的地质强度指标G,构建其与卸载点值和溶蚀时间的函数方程.通过灰岩单轴压缩试验结果,并结合Flac3D软件数值模拟得出函数方程的参数.为验证计算方法的合理性,将试验结果和数值模拟结果进行对比,结果表明计算结果和试验结果吻合度高.最后归纳出G的变化率随卸载点、溶蚀时间的演化规律.上述研究成果可对溶蚀作用下的洞室围岩稳定性动态演化规律的相关计算与数值模拟提供理论支撑.