基于非线性规划的有限元塑性极限分析下限法研究

在分析Sloan建立的有限元塑性极限分析线性规划数学模型存在的局限性基础上,提出了基于非线性规划的有限元塑性极限分析下限法数学模型。采用非线性屈服条件构建了下限法静力容许应力场,建立了求解超载系数、强度储备系数的下限法数学模型,并提出了针对塑性极限分析非线性规划数学模型的求解策略;最后对一个经典算例进行了深入分析,验证了方法的正确性。     

基于刚体平动运动单元的上限有限元研究

刚性块体极限分析上限法常应用于岩土工程稳定性研究,然而应用时需假定刚性块体破坏模式并递推繁琐的几何关系。为此,提出一种适应性更广的基于非线性规划模型的刚体平动运动单元上限有限元法,并解决了其优化模型初始值的确定问题。通过引入有限单元思想,将计算区域离散成刚体单元,同时以单元速度和节点坐标作为决策变量,由上限定理建立非线性规划模型获得上限解。利用编制的上限有限元程序进行边坡和浅埋隧道稳定性算例验证,表明运动单元上限有限元法能调整速度间断线至较优方位,所得破坏模式特征鲜明,上限解精度高,可广泛应用于边坡、隧道等稳定性分析研究。     

Stability analysis of slopes reinforced with piles

In this paper, the kinematic approach of limit analysis is used to analyse the stability of earth slopes reinforced with piles. First, the case of slope without piles is considered and a procedure is developed to calculate the safety factor for the slope. Results are compared with those obtained using both the limit equilibrium method and more complex upper and lower bound limit analysis solutions. Then, the stability of slopes reinforced with piles is analysed. Expressions are derived allowing the force needed to increase the safety factor to a desired value and the most suitable location of piles within the slope to be evaluated. A study is carried out to illustrate the effect of piles on slope stability.     

Limit equilibrium method based on an approximate lower bound method with a variable factor of safety that can consider residual strength

In this paper, the factor of safety for a prescribed slip surface will be determined from an equivalent lower bound method (extremum principle), which can satisfy all equilibrium conditions without an interslice force function. This approach will give an overall factor of safety close to that from the classical methods for normal problems, while the thrust line, the local factor of safety for individual slice/block and the progressive yielding phenomenon can be estimated, which will be useful for some special cases. The force and moment equilibrium of every slice will be satisfied, while the location of the thrust line will always be acceptable in the present formulation. To solve the difficult optimisation problem, an innovative coupled particle swarm/harmony search algorithm is proposed in this paper, and a practical engineering problem for which the factor of safety is close to 1.0 is used to illustrate the consideration of the residual strength in the limit equilibrium slope stability analysis.     

基于下限原理有限元的强度折减法

对于岩土工程中常用的强度折减系数,其规划问题是非线性的,不能直接利用线性规划进行求解。基于四边形单元的下限原理有限元法,根据强度折减系数与超载系数近似符合双曲函数的特点,通过调整强度参数使得超载系数逼近于1[1],可将边坡稳定性分析中常用的强度折减系数的非线性规划求解转化为线性规划求解问题。分析表明,采用拟合双曲线插值法求解强度折减系数的计算效率高于常规的二分法及割线法,且具有较好的收敛性;该方法能够充分利用当前高效的线性规划算法,便于工程应用。     

三维塑性极限分析下限法原理及应用

介绍了三维塑性极限分析下限法的基本原理。将计算区域离散为有限个三维空间单元,利用极限分析中的下限定理,借助有限元思想建立数学规划模型,并引入非线性规划的数值方法寻求问题的下限解。以均匀土质边坡的极限荷载作为算例,比较了所得数值解与经典塑性力学理论解,论证了该方法的正确性。     

非均质土坡的有限元塑性极限分析

在评价现有边坡稳定分析方法的基础上,提出了边坡稳定的有限元塑性极限分析方法。借助有限单元思想和线性规划,建立了边坡稳定的数学规划模型,由此可以求出安全系数的上下限解,从而,界定了边坡的安全范围,同时,可以给出下限状态下的应力场和上限状态下的速度场。以一个经典非均质土坡的边坡稳定作为算例,比较了多种方法的分析结果,论证了本方法的正确性。     

考虑孔隙水压力的土坡稳定性的有限元下限分析

以极限分析下限法理论为基础,应用有限单元思想离散结构物,建立了同时满足平衡条件、应力边界条件、屈服条件和应力间断条件的静力许可应力场,其中孔隙水压力被当作一种类似于重力的外力荷载。引入线性数学规划手段后,得到了考虑孔隙水压力的边坡稳定的下限法数学规划模型,由此可以求出安全系数的下限解及其对应的应力场。最后,以2个经典的土坡为算例,与多种方法的分析结果比较,论证了该方法的正确性。     

Large static problem in numerical limit analysis: A decomposition approach

A general decomposition approach for the static method of limit analysis is proposed. It is based on piecewise linear stress fields, on a partition into finite element sub‐problems and on a specific coordination of the subproblem stress fields through auxiliary interface problems. The final convex optimization problems are solved using nonlinear interior point programming methods. As validated for the compressed bar with Tresca/von Mises materials in plane strain, this method appears rapidly convergent, so that very large problems with millions of constraints and variables can be solved. Then the method is applied to the classical problem of the stability of a Tresca vertical cut: the static bound to the stability factor is improved to 3.7752, a value to be compared with the recent best upper bound 3.7776. Copyright © 2010 John Wiley & Sons, Ltd.     

Controlling the safety of geotechnical structures: a proposed approach

Summary This paper considers the safety of geotechnical structures using various design philosophies which include the global safety method and a number of limit state design methods. The methods are considered individually and their treatment of uncertainties of design briefly discussed in terms of parameter and system uncertainty. Each method is not designed to produce a definitive measure of safety, but should be viewed as an aid in the process of controlling or managing safety. Crucial to all design methods is the idea of a design parameter. However, in certain instances it is often not clear whether a design parameter should be an upper or lower bound on its possible value. A proposed limit state design method for overcoming this problem is outlined, based on putting bounds on parameters. An example of a reinforced concrete cantilever retaining wall is used to demonstrate its advantages over current methods. The proposed method offers a more rational approach to the design of retaining structures.     

线性与非线性强度的土石坝坝坡稳定分析下限法

土石坝的坝坡稳定是影响土石坝安全的重要因素,传统的土石坝坝坡稳定采用的是瑞典圆弧法或者毕肖普法,其计算结果既不是下限解也不是上限解。在Sloan的工作基础上,基于有效应力的方式, 用有限单元思想离散结构物,建立满足平衡条件、间断条件、应力边界条件以及屈服条件的极限分析下限法的非线性规划模型,并且编制了相应的程序,应用到土石坝坝坡稳定性的计算中。考虑了地震荷载和渗流作用,采用迭代算法对土石坝进行非线性强度指标的坝坡稳定计算。最后,以几个典型土坡和具体的土石坝工程为算例,与多种方法的分析结果比较,表明了该方法的可行性。     

Optimal lower bound of passive earth pressure using finite elements and non-linear programming

The present study pertains to the finding of the lower bound solution, formulating it as a non-linear programming problem using the generalized method developed by Lysmer with certain variations to incorporate the non-linear no-yield condition constraints directly in the analysis. The method considers the family of plane stress fields having the property that all stresses vary linearly within each triangular element of some mesh which covers the soil mass under study. For this type of stress field it is possible to express all equilibrium conditions as a set of linear constraints and the no-yield as a set of non-linear constraints. The boundary condition constraints may be of linear equality or inequality type. By expressing some of the design variables in terms of the remaining variables the linear equality constraints are implicity satisfied. Such a technique minimizes the complexity of the problem by eliminating the equality constraints and reduces the dimensionality of the problem, saving much, computational effort. The optimal lower bound is isolated by formulating it as a non-linear programming (NLP) problem subjected to both linear and non-linear inequality constraints. The sequential unconstrained minimization technique using the extended penalty function method as suggested by Kavlie has been used to isolate the optimal lower bound. The method has successfully been applied to the passive earth pressure and bearing capacity problem. Numerical results are obtained and compared with Lysmer's solution to show the effectiveness of the present approach.     

基于方位离散线性化的上限原理有限元法

上限原理有限元法不仅可以得到边坡的安全系数,还可以给出临界滑动面,且具有比极限平衡法更严谨的理论基础,因此,拥有更广阔的应用前景。针对传统的上限有限元法不能考虑强度各向异性的问题,提出了一种新的摩尔-库仑屈服面线性化方法。该方法在对方位角离散化的基础上,建立了线性化的方位离散塑性流动约束方程,丰富了基于线性规划的上限法理论。两个算例结果表明：该方法可以稳定地从极限解的上方收敛;且对边坡进行稳定性分析,若忽略了边坡的强度各向异性,则会高估边坡的稳定性,得到较大的安全系数。     

考虑非均质各向异性效应的阻滑桩加固土坡稳定性分析

基于极限分析上限定理与土的抗剪强度折减系数概念,考虑土的强度分布的非均质性与各向异性, 建立了土坡的极限平衡状态方程,由此确定土坡的稳定安全系数及其相应的潜在破坏模式。对于在给定的荷载条件下不能满足抗滑稳定性要求的土坡,考虑采用阻滑桩加固方式,根据桩侧有效土压力的合理分布模式确定桩体与滑动面相交的截面上等效抗滑力和抗滑力矩,考虑土的强度非均质性与各向异性的条件,利用极限分析上限定理建立阻滑桩加固土坡的极限平衡状态方程,将桩侧土压力作为目标函数,运用数学规划方法确定极限平衡状态时的临界桩侧有效土压力。通过大量的变动参数对比计算,探讨了土的强度的非均质性与各向异性等因素对阻滑桩桩侧极限抗力及最优加固位置的影响。     

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.     

On Improving the Solution by Using Lower Bound Finite Elements Limit Analysis and Linear Programming

This study presents an improved technique for obtaining the collapse loads for any geotechnical stability problem with the application of the lower bound finite elements limit analysis in combination with linear programming. In the proposed method, a lower order polygon is initially used throughout the problem domain to model the Mohr–Coulomb yield function; the order of the polygon refers to its total number of sides. After obtaining the initial solution, the problem domain is then discretized into a number of zones in which a different order polygon is used to model the yield surface. It is noted that a higher order polygon needs to be chosen only in a region, not everywhere, where the stress state approaches towards the yield. The total memory usage as well as the computational time needed to solve the problem with the proposed technique becomes significantly smaller. In order to check the validity of the method, the bearing capacity factor N_γ was computed for smooth as well as rough strip footing and the obtained computational results were found to be quite encouraging.     

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.     

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.     

基于平动加转动运动场的边坡稳定上限分析

基于塑性力学上限理论的边坡稳定安全系数上限解法是边坡稳定性分析的常用方法之一,其关键在于构造合适的几何容许运动场。基于摩尔-库仑屈服准则和相关流动法则,将滑动体进行条分,提出了一种同时考虑条块平动和转动的容许运动场及相应的稳定安全系数求解方法。该方法不仅具有适用性好、方法简单的特点,数值分析还表明,该方法能给出比基于纯平动运动场更小、非常接近Morgenstern-Price法的稳定安全系数计算结果。     

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。