Lower bound limit analysis using finite elements and linear programming
Lower bound limit analysis using finite elements and linear programming
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DOI：
10.1002/nag.1610120105
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发表时间：
1988
影响因子：
4
通讯作者：
S. Sloan
中科院分区：
文献类型：
--
作者：
S. Sloan
本文介绍一种在平面应变条件下计算土力学下限极限荷载的方法。为了调用经典塑性理论的下限定理，假设一个完全塑性土模型，它可以是纯粘性或粘性摩擦，连同相关的流动规则。使用一个合适的线性近似的屈服面，该程序计算一个静态容许应力场通过有限元和线性规划。应力场采用线性3节点三角形建模，每个三角形的边缘可能会出现静态容许应力不连续。施加的应力边界，平衡和屈服条件导致一个表达式的崩溃负载是最大化的节点应力的线性约束。由于完全满足静力容许解的所有要求（除了优化计算中的小舍入误差），因此获得的解是真实倒塌荷载的严格下限，因此是“安全的”。该技术的一个主要缺点，首先描述的Lysmer，1是大量的计算机时间需要解决线性规划问题。本文表明，这种限制可以避免使用有效集算法，而不是传统的单纯形或修改单纯形策略，解决由此产生的优化问题。这是由于约束矩阵的性质，它总是非常稀疏，通常具有比列多得多的行。它也证明了该程序，无需修改，可以用来获得严格的下限为纯粘性土，具有增加的强度随深度。这类重要的问题很难用常规方法解决。给出了一些例子来说明该程序的有效性。
This paper describes a technique for computing lower bound limit loads in soil mechanics under conditions of plane strain. In order to invoke the lower bound theorem of classical plasticity theory, a perfectly plastic soil model is assumed, which may be either purely cohesive or cohesive-frictional, together with an associated flow rule. Using a suitable linear approximation of the yield surface, the procedure computes a statically admissible stress field via finite elements and linear programming. The stress field is modelled using linear 3-noded traingles and statically admissible stress discontinuities may occur at the edges of each triangle. Imposition of the stress-boundary, equilibrium and yield conditions leads to an expression for the collapse load which is maximized subject to a set of linear constraints on the nodal stresses. Since all of the requirements for a statically admissible solution are satisfied exactly (except for small round-off errors in the optimization computations), the solution obtained is a strict lower bound on the true collapse load and is therefore ‘safe’. A major drawback of the technique, as first described by Lysmer,1 is the large amount of computer time required to solve the linear programming problem. This paper shows that this limitation may be avoided by using an active set algorithm, rather than the traditional simplex or revised simplex strategies, to solve the resulting optimization problem. This is due to the nature of the constraint matrix, which is always very sparse and typically has many more rows that columns. It also proved that the procedure can, without modification, be used to derive strict lower bounds for a purely cohesive soil which has increasing strength with depth. This important class of problem is difficult to tackle using conventional methods. A number of examples are given to illustrate the effectiveness of the procedure.