On the local stability condition in the planar beam finite element
On the local stability condition in the planar beam finite element
复制标题
平面梁有限元中的局部稳定条件
DOI：
10.12989/sem.2001.12.5.507
复制
发表时间：
2001
影响因子：
2.2
通讯作者：
Bojan Cas
中科院分区：
文献类型：
--
作者：
I. Planinc;M. Saje;Bojan Cas
在标准有限元算法中，切向刚度矩阵的公式中没有考虑局部稳定条件。因此，局部稳定的丧失与全球不稳定的开始没有充分的联系。这种现象通常出现在材料类型的局部化，如剪切带和塑料铰链。本文在平面、有限应变、速率无关、材料非线性梁理论的背景下解决了这个问题，尽管提出的技术原则上不限于梁结构。首先提出了Reissner有限应变梁理论的弱公式，其中变形轴的伪曲率是唯一的未知函数。我们进一步推导了大变形情况下的局部稳定条件，并提出了各种可能的插值和数值积分方案组合，这些方案可以触发静定梁的局部和全局失稳同时损失。对于实际应用，我们建议使用一种特殊的数值积分规则，其中插值节点和积分点在数量上相等，但在位置上不相等，除了局部不稳定点，插值节点和积分点在那里合并。如果失稳点是梁的端点（这种情况在工程实践中经常遇到），则程序大大简化；其中一种算法结合了拉格朗日插值法和洛巴托积分法。本文采用伽辽金有限元离散，但概念上类似的技术可以推广到其他离散方法。
In standard finite element algorithms, the local stability conditions are not accounted for in the formulation of the tangent stiffness matrix. As a result, the loss of the local stability is not adequately related to the onset of the global instability. The phenomenon typically arises with material-type localizations, such as shear bands and plastic hinges. This paper addresses the problem in the context of the planar, finite-strain, rate-independent, materially non-linear beam theory, although the proposed technology is in principle not limited to beam structures. A weak formulation of Reissner`s finite-strain beam theory is first presented, where the pseudocurvature of the deformed axis is the only unknown function. We further derive the local stability conditions for the large deformation case, and suggest various possible combinations of the interpolation and numerical integration schemes that trigger the simultaneous loss of the local and global instabilities of a statically determined beam. For practical applications, we advice on a procedure that uses a special numerical integration rule, where interpolation nodes and integration points are equal in number, but not in locations, except for the point of the local instability, where the interpolation node and the integration point coalesce. Provided that the point of instability is an end-point of the beam-a condition often met in engineering practice-the procedure simplifies substantially; one of such algorithms uses the combination of the Lagrangian interpolation and Lobatto`s integration. The present paper uses the Galerkin finite element discretization, but a conceptually similar technology could be extended to other discretization methods.