Discussion of “numerical limit analysis of three-dimensional slope stability problems in catchment areas” by Camargo et al. (doi:10.1007/s11440-016-0459-3)

A paper recently published by Camargo et al. (Acta Geotech 11(6):1369–1383, 2016) (hereafter identified as “the authors”) presented the numerical limit analysis method (NLA) to compute the safety factor and collapse mechanism of three-dimensional (3D) slopes. For NLA, the authors employed the discrete three-dimensional lower bound formulation with pore water pressure consideration and Drucker–Prager yield criterion, and cast a slope problem as a second-order conic programming problem. The developed program was implemented in MATLAB and validated through three examples of slope problems, and was applied to solve a large-scale 3D slope problem of a failure case study. The discussion of this article focuses on the formulation of the developed 3D NLA and static admissibility of stress field solutions obtained from NLA.     

Numerical limit analysis of three-dimensional slope stability problems in catchment areas

Risk analysis of existing slopes in catchment areas requires quantification of their stability. This quantification becomes particularly difficult when dealing with larger areas under 3D conditions and including saturated and unsaturated water flow. This paper proposes the use of an effective numerical procedure to solve three-dimensional slope stability problems in large areas subjected to pore pressure effects. This numerical approach, numerical limit analysis, utilizes the finite element method and mathematical programming techniques. Mathematical programming is needed because the basic plasticity theorems for limit analysis can be cast as optimization problems. The generated optimization problem is formulated under a second-order cone programming framework, which is known to solve large-scale problems with great computational efficiency. The main objective of this work was to determine the slope safety factor and the collapse mechanism of soils governed by the Drucker–Prager yield criterion for large-scale 3D problems including pore pressure effects. This approach is applied to an experimental catchment in the Oregon Coast Range that failed after an intense rainfall. The results were compared with a previous stability analysis of the area available in the literature that used a novel 3D limit equilibrium method.     

基于地质模型快速构建的边坡三维稳定性分析

目前边坡稳定性分析多以二维剖面为主,随着各种数值分析软件的应用,边坡的三维稳定性分析技术越来越成熟,但是对于三维边坡的地质模型快速构建的方法不多,且建模精度难以得到保证。本文借助Geobim软件建立工程区的三维地质模型,提出借助surfer、Ansys等软件快速建立flac3d能够识别的类型文件的方法,此方法能够大大简化建模过程,提高建模精度。根据工程区的主要工程地质问题,对工程区的三维地质模型进行概化,结合相关实例对其进行三维地质建模,并借助flac3d软件计算不同工况下边坡的稳定性情况。根据计算结果可知,在暴雨工况下,边坡的稳定性降低,填方工况下,边坡的侧向位移变大,但是其稳定性相对较好,边坡的主要破坏模式为沿着填筑体及边坡的浅表层全、强风化层发生剪切破坏。     

基于有限元强度折减法的边坡动力稳定性可靠性分析

传统有限元强度折减法在边坡稳定性数值分析中取得了一定的成功,但由于未考虑岩土体材料参数的变异性等不确定性因素,尚不能直接应用于边坡稳定性特别是动力稳定性可靠性分析问题。为此提出了基于有限元强度折减法的地震边坡动力稳定性可靠性分析方法。将有限元极限分析法、动力分析法和可靠性分析法三者耦合,分析求解边坡在地震作用下的动力稳定性可靠性问题,并将这一过程在数值计算程序中得以实现。在计算分析过程中,克服了原方法需不断人工试算才能得到边坡安全系数而无法量化处理问题,并对边坡动力失效准则进行了适用于程序化的改进,使其计算过程完全实现自主运行。结合典型算例分析结果表明,该方法显著的特点是能较全面地反映岩土体的动力特性和边坡岩土体材料强度参数的变异性及相关性,所得结果相对更加合理且更符合工程实际。该方法既是对有限元强度折减法的应用范围的有益推广,也为边坡动力稳定性可靠性问题研究提供了一条新的有效途径。     

滑带土抗剪强度参数的三维反分析

以滑面正应力修正模式的三维边坡稳定性分析方法为基础,建立了确定滑带土抗剪强度参数的三维反分析模型。先对滑坡体的稳定性进行调查评估,确定滑坡的安全系数与三维滑面的几何尺寸;然后,通过对滑面正应力进行合理假定与修正,建立了平衡方程,以强度参数为未知数,利用数学方法并结合试验、经验等手段求解出抗剪强度参数。该模型克服了二维反分析将会高估滑带土的抗剪强度参数的缺陷,而且计算过程简单,易于编程实现。两个简单算例表明,三维反分析模型可以考虑三维滑体的空间效应,得到的抗剪强度参数更真实、更符合实际情况,为滑坡治理提供参考。     

Slope stability analysis by SRM-based rock failure process analysis (RFPA)

The fundamental principles of the strength reduction method (SRM) are incorporated into the rock failure process analysis (RFPA) code to produce an RFPA–SRM method for analysing the failure process and stability of rock and soil slopes. The RFPA–SRM method not only satisfies the global equilibrium, strain-consistent, and non-linear constitutive relationship of rock and soil materials but also takes into account the heterogeneous characteristics of materials on the micro- and macro-scales. When the proposed method is used for slope stability analysis, both the critical failure surface and the safety factor can be obtained directly without any assumptions regarding the shape and location of the failure surface. The numerical results agree well with those obtained using conventional limit equilibrium and other FEM strength reduction methods. The proposed technique is applied to a number of more complex cases, including slopes in mixed rock–soil formations, rock layer formations, and highly jointed rock masses. It is shown that the RFPA–SRM method can describe the mechanism of failure of slopes and has potential applications in a large range of geoengineering problems.     

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.     

工程荷载作用下缓倾角反倾似层状岩质边坡变形稳定性分析

在反倾层状边坡的破坏机制和稳定性分析方面,由于其稳定性一般较好,系统深入地研究其破坏机制和稳定性的反而不多。以火山岩地层多次火山喷发旋回形成的缓倾角反倾似层状岩质边坡为研究对象,利用有限元分析其位移矢量图,揭示边坡的变形破坏机制和稳定性控制要素,利用极限平衡法、矢量和法,评价其稳定性,利用强度折减法分析塑性区的扩展规律,揭示滑动面发展的时间空间规律,结果表明,（1）类似层状反倾岩质边坡的潜在破坏模式为剪切–张拉破坏,滑面形态表现为近似折线状：顺坡向穿过凝灰岩夹层和强风化岩体,后缘通过陡倾结构面;（2）极限平衡法、矢量和法的边坡稳定性综合评价表明边坡浅层稳定性不满足工程稳定性要求,深层稳定性满足工程稳定性要求,且三维矢量和安全系数大于二维结果,是三维效应造成;（3）塑性区扩展揭示滑面的空间发展序列为滑面4、5、3。文中的技术路线和分析方案可用于缓倾角反倾层状岩质边坡稳定性评价。     

基于软化特性的三维边坡强度折减有限元分析

基于现有边坡强度折减有限元的基本原理,建立了一套新的强度折减算法。该算法把强度参数降低的过程看成脆塑性应力跌落的过程,并设置一折减增量 ,当 充分小时认为系统达到收敛条件而退出循环,最终的折减系数就是边坡的安全系数。该算法很容易扩展到三维边坡的稳定性分析,得到边坡滑动面滑动和破坏的发展趋势。通过2个算例,分析了进行三维模型计算的重要性,建议当地质条件复杂时宜从三维角度分析计算边坡,不能简单地将问题当作平面应变处理。     

用有限元强度折减法对某岸坡进行稳定性分析

有限元强度折减法是近年来岩土工程界广受关注的一种边坡稳定性分析方法。通过对岩体结构面强度折减,使斜坡达到不稳定状态,有限元计算不再收敛,此时的折减系数就是斜坡的安全稳定性系数。本文对某岸坡进行非线性有限元接触分析,计算表明,与传统刚体极限平衡法计算的稳定性系数很接近,而且有助于对滑体破坏的机制的理解。工程实例证明,有限元强度折减法在斜坡的稳定性评价中是切实可行的。      

锦屏一级水电站引渠内侧自然边坡的稳定性

层状岩体顺层滑动但却存在反倾的后缘结构面的边坡稳定性研究尚未见到具体讨论。本文采用了数值模拟方法对锦屏一级水电站引渠内侧边坡的稳定性进行了分析,通过结构面参数折减法,求得该边坡稳定性系数为1.191,后端岩体对前端不稳定体不存在推力。通过数值分析方法和极限平衡法对结构面相关参数的敏感性进行了分析对比,对比表明采用参数折减法的数值分析方法结果可靠。本文为类似复杂岩体边坡稳定性判定提供参考。     

Limit analysis of the stability of slopes reinforced with anchors

An approach based on the category of upper bound theorem of limit analysis is proposed in this article to consider the reinforcing effect of one row of anchors on slope stabilization. The shear strength reduction technique is used in the determination of the safety factor of the slope. The effect of anchor reinforcement is assumed to be an external axial force applied on the slope, and in the formulas of kinematic limit analysis, the work rate done by the anchor can be calculated. So, the stability analysis can be conducted without any assumptions on the acting position and decomposition of the axial force of anchors. Results were compared with those obtained using both the limit equilibrium method and numerical method. A parametric study was carried out to illustrate the effect of anchor orientation and position on slope stabilization. Copyright © 2011 John Wiley & Sons, Ltd.     

Three-dimensional finite element analysis for soil slopes stabilisation using piles

This paper presents the analytical methods of slope-stabilising piles using the three-dimensional (3-D) finite element (FE) analysis with the strength reduction method (SRM). This 3-D FE model is employed to overcome the limitations observed in two-dimensional (2-D) FE analysis. The solutions obtained from 3-D FE analyses are verified to be less conservative in this paper. The 3-D analysis is considered to be of particular importance to pile-slope problems. The soil that flows between piles cannot be taken account properly in the 2-D FE analysis. The method adopted in this paper can avoid the assumption of soil movement and the pressure distribution along the piles subjected to soil movement. The numerical analysis employs the Mohr–Coulomb failure criterion with the strength reduction technique for soil and an elastic member for piles. The spacing effect of the pile is considered in the 3-D model, the S/D (S: centre to centre, D: diameter of pile) ratio, equal to 4.0, is found to be equivalent to the single pile stabilisation. The middle portion of the slope is identified as the optimal location to place the piles. The proper length of the pile, which can be used to stabilise the slope, is also examined using 3-D FE analyses. It is concluded that L/H greater or equal 0.70 is recommended (L: pile length, H: slope height). The numerical analyses are conducted based on a coupled analysis, which simultaneously considers both the slope stability and the pile response. The failure mechanisms of the pile-slope system subjected to the pile locations, pile head conditions and pile length are each discussed. The contact pressure, shear force and moment along the piles are presented to illustrate the pile stabilising mechanism herein.     

Implications of variationally derived 3D failure mechanism

This paper revisits the variational limit equilibrium (LE) analysis of three‐dimensional (3D) slope stability in the context of limit analysis (LA). It proves the kinematic admissibility of the 3D mechanism in LA, although it was derived from LE variational extremization. It also includes algorithms in the realm of LA that are associated with the variational mechanism. A comparison between the variational results and reported LA upper‐bound or LE closed‐form results is conducted. It demonstrates that the variationally derived mechanism consistently yields upper‐bound solutions for 3D symmetrical slopes that are as accurate as those produced by postulated mechanisms in LA. However, the results are more critical than those derived from spherical failure mechanism in LE. The generalized log spiral 3D mechanism rigorously legitimizes the variational slope stability analysis in both frameworks of mechanics LE and LA. Stability charts were produced where the 3D factor of safety can be assessed for a constrained length of failure, while including factors like pore water pressure and seismic loading. The results presented within this study demonstrate the capabilities of the variational 3D solution and can be used to evaluate approximate methods, numerical or closed‐form, developed in 3D slope stability analyses. Copyright © 2016 John Wiley & Sons, Ltd.     

复杂荷载条件下边坡稳定性的上限极限分析

基于极限分析上限定理与土的抗剪强度折减技术,考虑边坡顶面超载、坡体内孔隙水压力与地震惯性力等复杂荷载,建立了边坡稳定的极限平衡状态方程,采用优化方法求解边坡稳定安全系数及相应的潜在破坏模式。其中水与地震对边坡的影响作用通过虚功方程来体现,将孔隙水压力与地震惯性力当作外力荷载做功,将强度折减系数作为评价边坡稳定性的定量指标。通过典型算例的对比分析,验证了方法的合理性,并探讨了孔隙水压力、地震加速度系数对边坡稳定性的影响。     

基于稳定性耦合分析法的余王扁边坡稳定性分析

边坡稳定性分析的极限平衡法与有限元法(FEM)的耦合分析法,首先利用有限元法分析获得边坡岩土体的整体应力场,在此基础上利用极限平衡法进行边坡的稳定系数求解。该方法既反映了边坡的稳定和变形之间的关系,又克服了极限平衡法与有限元法的不足,使二者的优点相互补充,获得的稳定系数基于极限平衡理论体系,可以同传统的稳定系数评价体系接轨。以西安市雁塔区余王扁削坡后边坡为例,用稳定性耦合分析法对其进行了稳定性分析,并把分析结果与各种极限平衡法的计算结果进行了比较,证明了耦合分析方法的可靠性和可行性。      

二维与三维边坡稳定性分析结果的比较与分析

二维与三维边坡稳定性分析得到的安全系数存在着差异,讨论这种差异与滑体几何尺寸及土体强度参数间的变化规律及其形成机制,可为合理地评价边坡稳定性分析提供理论依据。针对均质三维边坡,利用基于滑面正应力修正模式的极限平衡法分析程序,定性地讨论滑体形态、长高比、坡比、土体强度参数等指标对二维与三维安全系数计算结果差异(以F3/F2表示)的影响,总结F3/F2与影响指标间的变化规律,分析变化规律形成的内在机制。从工程应用的角度,给出需要考虑该差异影响的分界标准。研究表明,F3/F2随着长高比L/H、内摩擦角?及坡比m的增大逐渐减小,随着黏聚力c的增大逐渐增大;对于满足下列条件之一的边坡,宜采用三维安全系数来评价其稳定性,(1) L/H ≤ 5的滑坡体;(2) 5< L/H ≤ 10且 c > 25 kPa(或? < 15°,或m < 1.0) 的滑坡体。研究成果可为合理评价边坡稳定性分析方法提供参考。     

基于三维数值模拟的露天煤矿复合边坡稳定性分析

借助VC及fish语言编程,将AutoCAD、Surfer、ANSYS等软件耦合应用,完成了复杂地质条件下的FLAC3D建模。在地质调查及统计分析试验数据的基础上,应用Hoek-Brown经验破坏准则确定了边坡岩体的综合强度指标。然后,采用FLAC3D,对比分析了不考虑地下水及三维流固耦合两种情况下的位移场、应力场和塑性区分布规律,并采用强度折减法求得了三维边坡安全稳定系数。结果表明,由外排土场和采掘帮坡构成的复合边坡呈现复杂的潜在破坏模式,以及在地下水影响下存在潜在失稳。与传统的边坡稳定性分析方法相比,三维数值分析优势明显。      

LU Kun-lin,ZHU Da-yong,YANG Yang 均质边坡准三维安全系数实用计算曲线

针对三维极限平衡法计算边坡安全系数存在着分析过程繁琐、较难普及推广等诸多不足,建立了一套实用的三维边坡安全系数计算曲线。假设三维滑面为中间圆柱和两端为部分球面的组合滑面,采用基于滑面正应力修正的极限平衡分析方法,计算36 864个均质边坡,根据计算结果绘制该套曲线。依据该套曲线可以得到,（1）根据已知边坡体的相关指标快速获得三维安全系数,能够得到较二维分析方法更准确、更客观的稳定性评价;（2）根据实际坡体的稳定状况反分析获得滑面土体强度参数,有效地克服二维反分析得到滑面土体强度参数偏高的弊端。研究成果可为均质边坡三维稳定性初步评价提供简单、实用的参考依据。     

基于变形加固理论的岩土边坡稳定和加固分析

传统的刚体极限平衡法和强度折减法在分析边坡稳定和加固时存在一定的局限性。基于变形加固理论,建立了基于弹塑性有限元分析的边坡稳定和加固分析方法。以余能范数作为边坡整体稳定评价的指标,通过强度参数降低过程中的余能范数的变化来确定边坡的整体稳定安全系数。加固的关键部位和加固力的大小通过边坡中不平衡力的分布来确定。滑动面则由方向相反的、成对的不平衡力的分界面来确定。针对3个经典边坡算例,分析了强度参数逐渐降低过程中的余能范数的变化和不平衡力的分布,确定了边坡的稳定安全系数以及相应的加固关键部位和加固力的大小。结果表明,变形加固理论基于岩土的三维非线性有限元分析,以此来进行边坡的稳定和加固分析是非常有效的,强度参数降低过程中不平衡力的发展演化过程反映的是边坡的破坏过程