A relation between safety factors with respect to strength and height of slopes

Conventional slope stability analysis is usually based on the linear Mohr–Coulomb failure criterion utilizing the notion of safety factors with respect to shear strength, and one of the available slice methods. Failure criteria of most soils are not linear, and it is possible to show that this non-linearity has a very significant effect on calculated safety factors. The present work is based on a non-linear failure criterion, which appears to fit the experimental information better than Mohr–Coulomb. All slice methods utilize various kinematical and static assumptions, which cannot be rationally justified. The present work is based on rigorous variational approach to slope stability analysis, which does not employ any kinematical and static assumptions. Safety factors with respect to shear strength are useful abstractions, but physical significance of results based on them is clear only at failure when they are equal to 1 (at any other value of the safety factor with respect to strength results of the analysis correspond to fictitious material with a modified shear strength function). In the present note, we use the variational analysis in order to establish a simple analytical relation between safety factors with respect to strength and height. These two safety factors provide alternative measures for the stability of a given slope; but the safety factor with respect to height appears to have clearer physical interpretation.     

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.     

堆石非线性强度特性对高土石坝稳定性的影响

抗剪强度指标的选择对大坝坝坡稳定性评价有着十分重要的影响。堆石料抗剪强度具有明显的非线性,在高坝中这一特点更为显著。因此,计算堆石坝坝坡稳定安全系数时应采用非线性指标,而不是传统的线性指标。讨论了不同的堆石料抗剪强度模型、计算参数取值标准、非线性指标的稳定分析方法以及允许安全标准问题。统计分析了171组822个三轴试验结果发现,邓肯非线性指标对堆石料抗剪强度的模拟误差要小于线性指标和德迈洛非线性指标。采用不同的抗剪强度指标计算261.5 m高的云南糯扎渡大坝坝坡的稳定性。结果表明：各种工况下线性有咬合力、邓肯非线性指标与德迈洛非线性指标3种方法得到的安全系数和滑裂面位置都十分接近,非线性指标计算的结果并不比线性指标高。因此,进行非线性分析时,现有规范规定的基于线性指标的允许安全系数标准无需改变。     

Non-linear shear strength reduction technique in slope stability calculation

To exactly implement the non-linear Hoek–Brown shear strength reduction in slope stability calculations, three aspects of the problem are considered. Firstly, the normal and shear stress relationship of the generalized Hoek–Brown criterion (2002 edition) is derived by introducing the Mohr–Coulomb instantaneous friction angle as a variable. Secondly, the instantaneous Mohr–Coulomb friction angle and cohesive strength are derived for a micro-unit at a given stress state and they can be used to describe the shear strength of the rock mass under different stress states. Finally, the implementation of the non-linear shear strength reduction is described and a slope example is selected to verify our method. This technique can be also applied to other non-linear failure criteria.     

基于非线性破坏准则的边坡稳定性极限分析

上限定理是求解岩土工程问题的有效工具。以上限定理为理论基础,分析边坡的稳定性问题,并考虑了岩土材料破坏准则的非线性特性。在非线性Mohr-Coulomb破坏准则下,采用条分法与极限分析上限法相结合的方法,对边坡稳定性进行分析。通过切线法引入非线性强度参数 、 ,推导了岩土材料在非线性破坏准则下的相关联流动法则,建立功能方程,推导了边坡直线滑裂面、折线滑裂面和光滑曲线滑裂面安全系数F的计算方程。采用数学规划法计算后与工程实际中常用的边坡稳定性分析方法进行对比,并获得安全系数F =1.0时的稳定性系数Ns。与已有的研究成果进行比较分析,结果表明了该方法的正确性及优越性     

高土石坝坝坡稳定非线性分析

在高应力状态下堆石料的抗剪强度具有明显的非线性。强度非线性有指数和对数两种描述形式。对几座高土石坝的坝坡稳定分析表明，使用对数形式的非线性强度准则得到的坝坡计算安全系数比按线性强度准则计算的安全系数要高。现行规范所规定的许可安全系数，是与许多年的大量的用偏低的线性强度所作的稳定计算相适应的，偏于保守，改用非线性强度后应作适当的提高，对其提高的程度进行了统计分析，并给出了用非线性强度指标稳定分析时安全系数取值标准的建议值。     

A reliability-based approach to evaluating the stability of high rockfill dams using a nonlinear shear strength criterion

In recent years, many high rockfill dams have been constructed in China for the purpose of hydropower generation. One of the critical aspects of rockfill dam design is the analysis of slope stability. Triaxial compression tests show that the failure envelopes of rockfills are curved and that nonlinear shear strength criteria yield better predictions of the shear strengths of rockfills than the Mohr–Coulomb criterion. Because the determination of shear strength parameters involves uncertainties, a reliability-based methodology was developed for use in evaluating the failure probability of rockfill dam slopes by integrating Bishop’s simplified method, Duncan’s nonlinear strength criterion and the first-order second-moment reliability method presented in this paper. A computer program, SCU-SLIDE, was developed and its outputs validated by comparison with Monte Carlo simulation results. The approach developed was used to study the stability of the Shuang Jiang Kou rockfill embankment dam, which when completed will be the tallest dam in the world. The results of the stability analysis are discussed and conclusions are presented in this paper.     

Cauchy problem of three‐dimensional critical slip surfaces of slopes

Using the continuum‐based finite element approach in conjunction with the strength reduction technique, one can easily find out the factor of safety of a slope with rather high precision, but will encounter some obstacles in the accurate location of the corresponding three‐dimensional critical slip surface (CSS). On the basis of the Mohr–Coulomb failure criterion and the stress field in the limit equilibrium state of the slope, it is deduced that the three‐dimensional CSS is the solution of the Cauchy problem of a quasi‐linear partial differential equation (PDE) of first order. By means of the method of characteristics for the problem, the three‐dimensional CSS that may take any shape can be determined without the specification of its geometry. Copyright © 2010 John Wiley & Sons, Ltd.     

Seismic rock slope stability charts based on limit analysis methods

Earthquake effects are commonly considered in the stability analysis of rock slopes and other earth structures. The standard approach is often based on the conventional limit equilibrium method using equivalent Mohr–Coulomb strength parameters (c and ?) in a slip circle slope stability analysis. The purpose of this paper is to apply the finite element upper and lower bound techniques to this problem with the aim of providing seismic stability charts for rock slopes. Within the limit analysis framework, the pseudo-static method is employed by assuming a range of the seismic coefficients. Based on the latest version of Hoek–Brown failure criterion, seismic rock slope stability charts have been produced. These chart solutions bound the true stability numbers within ±9% or better and are suited to isotropic and homogeneous intact rock or heavily jointed rock masses. A comparison of the stability numbers obtained by bounding methods and the limit equilibrium method has been performed where the later was found to predict unconservative factors of safety for steeper slopes. It was also observed that the stability numbers may increase depending on the material parameters in the Hoek–Brown model. This phenomenon has been further investigated in the paper.     

雨水入渗对非饱和土坡稳定性影响的参数研究

很多国家和地区的斜坡失稳与雨水入渗有密切关系。通过参数分析研究可以深化对这种关系的认识和理解,因而对滑坡灾害的预测和预防有重要意义。针对香港地区一种典型非饱和土斜坡,用有限元法模拟雨水入渗引起的暂态渗流场,然后将计算得到的暂态孔隙水压力分布用于斜坡的极限平衡分析。计算中采用延伸的摩尔－库伦破坏准则以便考虑基质吸力对抗剪强度的贡献,研究了降雨特征、水文地质条件及坡面防渗处理等因素对暂态渗流场和斜坡安全因数的影响。数值模拟结果表明：降雨强度、降雨历时和雨型对暂态渗流场及斜坡稳定性有明显的影响;土体的渗透系数,尤其是渗透系数各向异性的影响特别显著;斜坡中相对隔水层的存在以及斜坡防渗护面的效果等因素的影响均不容忽视。     

A generalized nonlinear failure criterion for frictional materials

A generalized nonlinear failure criterion formulated in terms of stress invariants is proposed for describing the failure characteristics of different frictional materials. This failure criterion combines a power function and a versatile function in the meridian and deviatoric plane, respectively, which is a generalization of several classic criteria, including the Tresca, Drucker–Prager, Mohr–Coulomb, Lade–Duncan and Matsuoka–Nakai failure criterion. The procedure for determination of the strength parameters was demonstrated in detail. Comparisons between the failure criterion and experimental results were presented for uncemented/cemented Monterey sand, normally consolidated Fujinomori clay, rockfill, concrete, Mu-San sandstone and granite, which reveal that the proposed failure criterion captures experimental trend quite well.     

Numerical treatment of progressive localization in relation to borehole stability

Results of large-scale finite element computations are presented for the problem of borehole failure due to the high stresses at great depths. Rock is modelled by an extension of the Mohr–Coulomb flow theory of plasticity for material with Cosserat micro-structure. The introduction of internal length (grain size) improves the computational stability and allows for robust post-localization computations. The presented results show clearly a progressive failure mechanism and the computed failure modes are in a good qualitative agreement with laboratory and field observations.     

Kinematical analysis of 3D passive earth pressure with nonlinear yield criterion

Conventional methods for calculation of passive earth pressure were mainly based on the assumptions of the linear Mohr‐Coulomb yield condition and plane strain failure mechanism. However, both theoretical and experimental studies have shown that such assumptions are not satisfied in some geotechnical projects. Herein, a novel method incorporating a kinematically admissible 3‐dimensional (3D) rotational failure mechanism and the nonlinear power‐law yield criterion is proposed to compute the passive earth pressure acting on the inclined retaining walls. Instead of using the nonlinear yield criterion directly, a straight line tangential to the nonlinear yield curve is employed to represent the strength of soils, and therefore, the nonlinear problem is transformed into the traditional linear problem. The 3D failure mechanism is generated through rotating a circle defined by 2 log‐spirals, and a plane strain block is inserted into the mechanism to consider the retaining walls with different widths. Earthquake effects are taken into account by using quasi‐static representation, and the horizontal seismic coefficient concept is adopted for the estimation of passive earth pressure under seismic conditions. An analytical expression about the 3D passive earth pressure is educed by means of the upper bound theorem of limit analysis. Numerical results for different practical parameters are obtained from an optimization scheme where the minimum of passive earth pressure is sought. Compared with available 2‐dimensional and 3D solutions, the proposed method is validated. A parametric study is conducted to investigate the effects of different parameters on the 3D static and seismic passive earth pressure.     

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.     

基于破坏概率的岩土试件剪切破坏角分析

岩土体剪切破坏时剪切破坏角是一个一直在探讨的问题,传统理论解为Mohr-Coulomb解和Roscoe解,基于试件的宏观破坏现象受微观破坏概率的分布制约这一前提,从岩土体试件内部破坏概率分布函数的角度重新研究了这一问题。通过对试件内部微观破坏概率由微观到宏观的统计分析与坐标变换,发现在形式上破坏概率主轴即相当于主应力,并基此给出了破坏概率破坏莫尔圆。还对Mohr-Coulomb破坏准则与微观破坏概率统计之间的关系开展了探索分析,结果表明Mohr-Coulomb准则是描述试件剪切破坏时大量微破坏行为共同作用的统计结果,Weibull分布仅是岩土试件内部破坏概率分布形式的一种描述,基于数学概率推导的岩土试件破坏模型能更好的与试件的破坏准则联系起来。     

Numerical simulation of the influence of small-scale defects on the true-triaxial strength of rock samples

A numerical model is used to investigate the influence of randomly located ‘defects’ on the strength of rock samples with a Mogi–Coulomb three-dimensional failure criterion that are subjected to polyaxial states of stress. A methodology is presented to compute ‘equivalent’ Mohr–Coulomb strength parameters as a function of the ‘instantaneous’ stress, and sensitivity analyses are employed to study the influence of the stress state, of the total percentage of defects, and of the type of spatial distribution of defects within the sample. Results indicate that increments of strength due to the intermediate principal stress are less relevant for samples with defects, hence suggesting that the influence of polyaxial stress states is less significant for rock masses (that incorporate ‘defects’ or ‘discontinuities’ into their structure) than for intact rock. Results also indicate that the strength of rock with defects can be modelled as a ‘homogeneous’ material with reduced strength properties. The reduction factors are a linear function of the percentage of defects, and they tend to be greater for “lumped” defects than for “isolated” defects.     

Numerical simulation of direct shear test of rockfill based on particle breaking

Acta Geotechnica - In the traditional Mohr–Coulomb criterion, the shear strength is determined by cohesion and friction angle. However, the particle breakage that occurs on the shear failure...     

高应力状态下的黄土抗剪强度特性研究

土体抗剪强度包络线在一定的应力范围内可近似看做直线, 一般工程中直剪试验正应力和三轴试验围压常为0～400kPa, 试验可得出以黏聚力c和内摩擦角确定的强度。在黄土地区, 人工和自然边坡可超百米, 坡内应力自低到很高变化范围大, 对该类高边坡需考虑强度参数随应力状态的变化。本文先通过线弹性有限元模拟一黄土高边坡应力场。以潜在滑面上关键点的应力为固结应力进行三轴压缩试验(NCTC)。为了对比分析, 又进行了100kPa、200kPa、300kPa、400kPa围压下的三轴压缩试验(CTC)。结果表明, 考虑黄土高边坡应力状态时, 由总应力莫尔圆得到的强度包络线为曲线, 直线型莫尔-库伦强度准则不适用; 而有效应力莫尔圆的到的强度包络线可为直线, 其有效强度参数高于CTC试验。在-应力空间中, NCTC试验总强度包络线和有效包线均位于CTC试验强度包线之上, 抗剪强度高, 这同固结应力高且为偏压固结有关。     

Deformation Localization-A Review on the Maximum- Effective-Moment (MEM) Criterion

The essential difference in the formation of conjugate shear zones in brittle and ductile deformation is that the intersection angle between brittle conjugate faults in the contractional quadrants is acute(usually ~60°) whereas the angle between conjugate ductile shear zones is obtuse(usually 110°). The Mohr-Coulomb failure criterion, an experimentally validated empirical relationship, is commonly applied for interpreting the stress directions based on the orientation of the brittle shear fractures. However, the Mohr-Coulomb failure criterion fails to explain the formation of the low-angle normal fault, high-angle reverse fault, and the conjugate strike-slip fault with an obtuse angle in the σ1 direction. Although it is ten years since the Maximum-Effective-Moment(MEM) criterion was first proposed, and increasingly solid evidence in support of it has been obtained from both observed examples in nature and laboratory experiments, it is not yet a commonly accepted model to use to interpret these antiMohr-Coulomb features that are widely observed in the natural world. The deformational behavior of rock depends on its intrinsic mechanical properties and external factors such as applied stresses, strain rates, and temperature conditions related to crustal depths. The occurrence of conjugate shear features with obtuse angles of ~110° in the contractional direction on different scales and at different crustal levels are consistent with the prediction of the MEM criterion, therefore ~110° is a reliable indicator for deformation localization that occurred at medium-low strain rates at any crustal levels. Since the strain–rate is variable through time in nature, brittle, ductile, and plastic features may appear within the same rock.