Parametric Monte Carlo studies of rock slopes based on the Hoek–Brown failure criterion

Probabilistic evaluation of slope failures is increasingly seen as the most appropriate framework for accounting for uncertainties in design. This paper performs reliability assessments for rock slopes based on the latest version of the Hoek–Brown failure criterion. The purpose of this study is to demonstrate the use of a new form of stability number for rock slope designs that has been recently developed from finite element upper and lower bound limit analysis methods, and to provide guidance for its use in probabilistic assessments. The analyses show that by using this newly proposed stability number, the probability of failure (P_f) obtained from case studies agrees well with the true state of the slope. In addition, this paper details a procedure to determine the magnitude of safety factor required for rock slope design.     

Parametric studies of disturbed rock slope stability based on finite element limit analysis methods

This study employs the finite element upper bound and lower bound limit analysis methods to investigate the stability of inhomogeneous rock slopes. The differences in the stability numbers of the upper and lower bound solutions are bracketed within ±10.5% or better, and the stability numbers obtained are presented in rock slope stability charts. These stability charts can provide a convenient tool for preliminary stability designs of inhomogeneous rock slopes. Various recommended blasting damage zones are considered, and disturbance factors are used to represent damage levels. Results showed that rock mass disturbance could significantly influence the evaluation of rock slope stability.     

Pseudo-dynamic stability of rock slope considering Hoek–Brown strength criterion

Rock slopes with planar joints or weak structural planes are vulnerable in nature, especially suffering from the natural hazards, instabilities of slopes are more prone to occur. Therefore, concerning to the influence of earthquakes, this paper performs a new procedure to evaluate slope stability in a geomaterial governed by Hoek–Brown strength criterion. A rotational failure mechanism determined by 21 dependent angle variables is introduced to respect the Hoek–Brown strength criterion. The earthquake load is characterized by a modified pseudo-dynamic method that does not violate the zero boundary condition and considers the damping properties of geomaterials. A slice approach is adopted to calculate the earthquake-induced inertial force work rate. The stability number of rock slope is considered to measure the safety. The stability number is formulated as a classical optimization problem controlled by 21 dependent angle variables and a time variable which need to be optimized by the genetic algorithm toolbox. Comparisons with the literature are made to prove rationality and accuracy of the proposed procedure. Parametric study is carried out to reveal the influence of dynamic properties. For engineering application, stability charts are provided for a quick assessment of slope safety.

     

Stability analysis of rock slopes with a modified Hoek–Brown failure criterion

The original Hoek–Brown (HB) failure criterion was used to analyse the stability of rock slopes. For highly fractured rock, the original HB failure criterion has been modified, but its effect on the stability of rock slopes has not been studied. Within the framework of the kinematical approach of limit analysis, this paper computes the rigorous upper bounds of stability factors of homogeneous rock slopes with the modified HB failure criterion under the plane strain condition, by employing a ‘generalized tangential’ technique. In such technique, instead of using the modified HB failure criterion, a series of linear failure surfaces tangent to the actual non-linear failure surface are utilized to derive the upper bound solutions, incorporating a new parameter n ranging from 0.5 to 0.65. The numerical results are compared with other published solutions for the case of n=0.5. The effects of the n on the stability factors of rock slopes are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of rock mass disturbance on the stability of rock slopes using the Hoek–Brown failure criterion

The process of creating man made or “cut” slopes in rock invariably leads to stress relief within the rock mass which in turn induces a certain degree of fracturing and disturbance. The level of disturbance can be particularly significant when the slope is formed using blasting techniques. However, the effects of this disturbance on the overall rock slope stability have not been investigated thoroughly in the current literature. In order to account for rock mass disturbance during construction, a disturbance factor has been included in the Hoek–Brown failure criterion [1]. This paper uses finite element upper and lower bound limit analyses to estimate rock slope stability based on the Hoek–Brown failure criterion whilst including the effect of rock mass disturbance. A rigorous set of analyses have been performed where the level of disturbance is considered as constant or linearly varying throughout the slope. The results are then compared to a number of reported case histories for verification purposes. From the results of this study, the disturbance factor was found to have significant influence on the rock slope stability assessment, especially for poorer quality rock masses. Hence, cautious engineering judgement must be exercised when estimating the level of disturbance. In addition, utilising stability charts to estimate the stability of cut rock slopes without considering the rock mass disturbance may lead to significant overestimations.     

地震效应下非线性抗剪强度参数对裂缝边坡稳定性影响的上限解析

基于非线性Mohr-Coulomb破坏准则,结合极限分析上限法和拟静力分析法,建立功能方程,推导了地震效应下裂缝边坡的安全系数计算方程。采用数学规划方法,计算了不同参数组合条件下的边坡安全系数值,详细分析了非线性条件下一系列参数对边坡稳定性的影响。研究表明,边坡安全系数随非线性参数和地震效应的增大而减小。对比分析可知,在非线性破坏准则下,裂缝深度较大时,裂缝对边坡稳定性影响显著,且边坡越陡影响越大;当裂缝深度超过某个值后,临界破坏面起始端可能不穿过裂缝最底端,而是从裂缝中间某部位穿过。在地震效应作用下,非线性抗剪强度参数对安全系数影响显著。研究成果进一步完善了裂缝边坡稳定性分析内容,所列图表为边坡的设计与施工提供有益参考。     

An Empirical Failure Criterion for Intact Rocks

The parameter m _i is an important rock property parameter required for use of the Hoek–Brown failure criterion. The conventional method for determining m _i is to fit a series of triaxial compression test data. In the absence of laboratory test data, guideline charts have been provided by Hoek to estimate the m _i value. In the conventional Hoek–Brown failure criterion, the m _i value is a constant for a given rock. It is observed that using a constant m _i may not fit the triaxial compression test data well for some rocks. In this paper, a negative exponent empirical model is proposed to express m _i as a function of confinement, and this exercise leads us to a new empirical failure criterion for intact rocks. Triaxial compression test data of various rocks are used to fit parameters of this model. It is seen that the new empirical failure criterion fits the test data better than the conventional Hoek–Brown failure criterion for intact rocks. The conventional Hoek–Brown criterion fits the test data well in the high-confinement region but fails to match data well in the low-confinement and tension regions. In particular, it overestimates the uniaxial compressive strength (UCS) and the uniaxial tensile strength of rocks. On the other hand, curves fitted by the proposed empirical failure criterion match test data very well, and the estimated UCS and tensile strength agree well with test data.     

Seismic bearing capacity of shallow foundations near rock slopes using the generalized Hoek–Brown criterion

The seismic bearing capacity of shallow foundations resting on a modified Hoek–Brown rock mass is investigated within the framework of the kinematic approach of limit analysis theory. The analysis focuses on evaluating the reduction in bearing capacity induced by seismic loading and by the proximity of a rock slope. A pseudo‐static approach is adopted to account for the earthquake effects for the seismic bearing capacity evaluations. At the rock material level, the closed‐form expressions previously obtained for the support functions of the rock failure criterion allow the implementation of different failure mechanisms families, and thus to derive rigorous upper bounds estimates of the load‐bearing capacity in both static and seismic conditions. The effects of geometrical, strength and loading parameters are assessed through a large number of parametric computations. Finally, design tables are presented for practical use in rock engineering. Copyright © 2010 John Wiley & Sons, Ltd.     

基于模糊因素的岩质边坡地震稳定性多模型组合评价

岩质边坡地震稳定性评价是岩土边坡地震失稳防治的基础工作,针对评价过程中评价因素的多源模糊性,借鉴工程可变模糊集理论,将模糊可变评价模型应用于岩质边坡地震稳定性评价中。综合考虑岩质边坡内在结构和外部自然条件,选取岩土体特性、新构造运动特征、坡高、坡角、年均降雨量和场地地震烈度6个指标作为评价指标体系并建立等级标准;将改进熵权算法引入该模型中,利用指标实际数据离散性求权重;通过改变模糊可变评价模型参数对岩质边坡地震稳定性进行线性与非线性组合评价,并将均值作为最终评价结果。将该方法应用于天然边坡与路堑边坡实例中,结果表明,模糊可变评价模型评价结果合理、客观,具有更高的可靠性与稳定性,为岩质边坡地震稳定性评价工作提供了一种新的研究方法与思路     

Upper bound finite element analysis of slope stability using a nonlinear failure criterion

In this study, upper bound finite element (FE) limit analysis is applied to stability problems of slopes using a nonlinear criterion. After formulating the upper bound analysis as the dual form of a second-order cone programming (SOCP) problem, the stress field and corresponding shear strength parameters can be determined iteratively. Thus, the nonlinear failure criterion is represented by the shear strength parameters associated with stress so that the analysis of slope stability using a nonlinear failure criterion can be transformed into the traditional upper bound method with a linear Mohr–Coulomb failure criterion. Comparison with published solutions illustrates the accuracy and feasibility of the proposed method for a simple homogeneous slope stability problem. The proposed approach is also applied to a seismic stability problem for a rockfill dam to study the influence of different failure criterions on the upper bound solutions. The results show that the seismic stability coefficients obtained using two different nonlinear failure criteria are similar but that the convergence differs significantly.     

Slope Stability Problems and Back Analysis in Heavily Jointed Rock Mass: A Case Study from Manisa, Turkey

This paper presents a case study regarding slope stability problems and the remedial slope stabilization work executed during the construction of two reinforced concrete water storage tanks on a steep hill in Manisa, Turkey. Water storage tanks of different capacities were planned to be constructed, one under the other, on closely jointed and deformed shale and sandstone units. The tank on the upper elevation was constructed first and an approximately 20-m cut slope with two benches was excavated in front of this upper tank before the construction of the lower tank. The cut slope failed after a week and the failure threatened the stability of the upper water tank. In addition to re-sloping, a 15.6-m deep contiguous retaining pile wall without anchoring was built to support both the cut slope and the upper tank. Despite the construction of a retaining pile wall, a maximum of 10 mm of displacement was observed by inclinometer measurements due to the re-failure of the slope on the existing slip surface. Permanent stability was achieved after the placement of a granular fill buttress on the slope. Back analysis based on the non-linear (Hoek–Brown) failure criterion indicated that the geological strength index (GSI) value of the slope-forming material is around 21 and is compatible with the in situ-determined GSI value (24). The calculated normal–shear stress plots are also consistent with the Hoek–Brown failure envelope of the rock mass, indicating that the location of the sliding surface, GSI value estimated by back analysis, and the rock mass parameters are well defined. The long-term stability analysis illustrates a safe slope design after the placement of a permanent toe buttress.     

Stability analysis of seismic slopes with cracks

Earthquakes can trigger slope instability, especially in the case of slopes with cracks. Studies of slope stability rarely account for the presence of cracks. In this study, the upper bound limit analysis technique and the pseudo-static method were used to examine the stability of homogeneous slopes with cracks subjected to seismic loading. A series of stability charts for slope inclinations of 2:1 (β = 63.4°), 1:1 (β = 45°), 2:3 (β = 33.7°), and 1:2 (β = 26.6°) (vertical to horizontal) and internal friction angles, φ, of 10°, 20°, 30°, and 40° are presented. These charts should be useful for readily determining the stability number (critical slope height), the critical crack depth, and the region affected by cracks for cracks of known depth but unknown location, cracks of known location but unspecified depth, and cracks of unspecified depth and location.     

Instantaneous Friction Angle and Cohesion of 2-D and 3-D Hoek–Brown Rock Failure Criteria in Terms of Stress Invariants

The Mohr–Coulomb (M–C) failure criterion is one of the most widely used failure criteria in rock mechanics, although it has a number of shortcomings such as neglecting the nonlinear strength observed in rock or the effect of the intermediate principal stress σ ₂. Other failure criteria have been proposed to effectively include in the predictions of failure the non-linear response of rock to confinement or the effects of the intermediate principal stress. The M–C criterion is still widely used, and it is arguably the criterion most used in practice. For example, stability evaluations of shallow rock structures such as slopes and foundations are routinely carried out by estimating a friction angle and a cohesion of the rock mass. To include the dependency of cohesion and friction angle on stresses, efforts are being made to estimate equivalent values of the M–C parameters for the range of stresses applicable to a particular design. The paper suggests a new and convenient approach to find the equivalent friction angle and cohesion from any failure criterion that can be expressed in terms of the Nayak and Zienkiewicz’s stress invariants. To demonstrate the capabilities and application of the methodology, the new approach is applied to two failure criteria: the Hoek–Brown (H–B) criterion and the Hoek–Brown and Willam–Warnke (HB–WW) criterion, 2-D and 3-D failure criteria, respectively. Results from the new method, in terms of equivalent friction and cohesion for the H–B criterion, are exactly the same as the results obtained from Balmer’s theory, which confirms the validity of the new method. The predicted equivalent friction and cohesion for the HB–WW criterion show a dependency on σ ₂, which does not occur for a 2-D failure criterion.     

Limit analysis solutions for three dimensional undrained slopes

This paper uses numerical finite element upper and lower bound limit analysis to produce stability charts for three dimensional (3D) homogeneous and inhomogeneous undrained slopes. Although the conventional limit equilibrium method (LEM) is used more often in practice for evaluating slope stability, the accuracy of the method is often questioned due to the underlying assumptions that it makes. Using the limit theorems can not only provide a simple and useful way of analysing the stability of slopes, but also avoid the shortcomings and arbitrary assumptions under pinning the LEM. The rigorous limit analysis results in this paper were found to bracket the slope stability number to within ±9% or better and therefore can be used to benchmark for solutions from other methods. In addition, it was found that using a two dimensional (2D) analysis to analyse a 3D problem will lead to a significant difference in the factors of safety depending on the slope geometries. This is of particular relevance to any back analyses of slope failure as it will lead to an unsafe estimation of material strengths.     

基于极限分析上限方法的海底斜坡地震稳定性

极限分析上限方法在海底斜坡稳定性评价中受到了广泛关注,但已有成果未考虑地震荷载以及多土层海底斜坡可能出现的局部破坏机制。基于上限定理,推导了拟静力水平地震条件下多土层海底斜坡外力功率与内能耗散率平衡方程;结合强度折减技术和最优化方法,求解了海底斜坡整体和局部地震稳定性安全系数,并实现了多土层海底斜坡的局部滑动面搜索;通过典型算例分析,验证了本文方法的有效性。在此基础上,探讨了不同水平地震条件下两种组合土层海底斜坡的整体和局部稳定性,通过与数值解对比,其结果可以较准确地评价海底斜坡稳定性并有效预测滑移面位置。最后,将极限分析上限方法应用于一海底斜坡工程实例。     

水力和超载条件下锚固岩石边坡动态稳定性拟静力分析

基于极限平衡理论,综合考虑水力条件、坡顶超载、地震荷载效应和锚固效应对岩石边坡进行了全面的稳定性分析。计算给出了多影响因素条件下岩石边坡稳定性安全系数的表达式,并重点分析了几种相关参数组合对岩石边坡稳定性的影响。分析表明,坡顶张拉裂缝积水、地下水渗流作用、滑面出流缝被堵塞、地震影响效应不利于岩石边坡抗滑稳定性,而锚索锚固效应则对提高边坡抗滑稳定性有积极作用;坡顶张拉裂缝积水、滑面出流缝被堵塞、水平向地震影响效应都不利于岩石边坡抗倾覆稳定性,但锚索锚固效应、坡顶超载、与竖直方向地震效应则对提高边坡抗倾覆稳定性有益。最后针对工程实际,提出了相应的工程建议。     

Stability design charts for homogeneous slopes under typical conditions based on the double shear strength reduction technique

Based on a brief review of the existing shear strength reduction (SSR) techniques, the objective function of the comprehensive safety factor for simple homogeneous slopes is established by combining the double SSR technique (considered the shortest pathway of the strength reduction) with the upper bound limit analysis theorem, leading to a strict upper bound on the safety factor. Combining nonlinear sequential quadratic programming (SQP) with the random walk method, the value of the comprehensive safety factor can be optimized, avoiding the trap of a local minimum. Compared with classical examples, the present method is a conservative and effective method for slope stability evaluation. A set of design charts for homogeneous slopes under simple and typical conditions, such as surcharge load, pore water pressure, and seismic loading are produced by the analysis of substantial data, which can eliminate the necessity for iterations when calculating the safety factor. These stability charts are presented in a convenient manner to determine the comprehensive safety factors and corresponding failure patterns under different typical conditions, which might be preferred by engineers for performing the preliminary evaluations of slope safety. Several examples are used to illustrate the application of these stability charts under different conditions.     

非线性破坏准则下边坡稳定性极限分析斜条分法

考虑坡顶均布荷载和地震效应典型情况下,将边坡滑体进行任意斜条块划分,建立了具有倾斜界面的多块体破坏模型。基于极限分析上限法和非线性摩尔-库仑破坏准则,考虑岩体内正应力的不均匀性,引入多点切线法和强度折减法推导得出边坡临界破坏状态下的安全系数Fs通用计算公式。采用序列二次规划法对安全系数Fs的目标函数进行最优化计算,并与既有研究成果进行对比分析,其结果具有较好的一致性,相对误差不超过3.565%,表明了该方法的正确性。同时对比传统单点切线法计算结果,多点切线法较单点切线法获得的边坡安全系数值偏小,表明了多点切线斜条分法偏于保守,是安全的。参数分析表明坡顶均布荷载、地震效应和非线性参数均对边坡安全系数及潜在临界滑裂面有重要影响。多点切线法引入非线性摩尔-库仑破坏准则对边坡进行稳定性极限分析,为相关研究人员提供了一种新的思路与方法。