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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >On the Finite Element Formulation of Elastic-Plastic Frames
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On the Finite Element Formulation of Elastic-Plastic Frames

机译：关于弹塑性框架的有限元公式化

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In this paper I am introducing, in a condensed way, a family of new finite elements for the kinematically exact analysis of elastic-plastic planar frames. The finite elements are founded on the nonlinear REISSNER theory, yet with shear strains taken to be negligibly small. The Reissner theory allows the displacements, the rotations and even the strains to be arbitrary large. Due to physical reasons, however, the Reissner theory, which assumes that the plane cross-sections remain plane during the deformation, is not valid for arbitrary large strains. An adequate classification of the present approach would therefore be "a theory of arbitrary large displacements and rotations lith moderate strains". There is a variety of finite element formulations using the Reissner beam theory and an endless set of those which employ simplified versions of Reissner's equations. The common characteristic of all these formulations is the use of multi-function variational principles with two and sometimes three functions, most often the displacements u(x) and w(x) of the centroidal axis of the beam, and its rotation φ(x). In such cases it is - in a nonlinear setting - very difficult, if not impossible, to assume appropriate set of field-consistent approximating functions of unknown functions. The inconsistency results in poor convergence, stress oscillations and several kinds of locking see PRATHAP [2]). These represent serious deficiencies of the formulations. In contrast, the present formulation employs a newly derived variational principle, in which the only unknown function is rotation φ(x) of the centroidal axis of the beam. Thus the problem of the field-consistency does not arise, and the phenomena mentioned above are absent. Besides, for a comparable number of degrees of freedom, the present formulation is always more accurate than other known formulations. Such a formulation has already been derived for planar and spatial elastic and hyperlastic beams. The formulation presented here for elastic-plastic beams is in a close relation to these works. Many details not presented here can be found in these references.
机译：在本文中，我将以简明的方式介绍一系列新的有限元，用于弹性塑性平面框架的运动学精确分析。有限元是基于非线性REISSNER理论建立的，但剪切应变可以忽略不计。 Reissner理论允许位移，旋转，甚至应变任意大。然而，由于物理原因，假设在变形过程中平面横截面保持平面的Reissner理论不适用于任意大应变。因此，本方法的适当分类将是“任意大位移和自转中等应变的理论”。有多种使用Reissner束理论的有限元公式，以及无数的采用Reissner方程简化形式的公式。所有这些公式的共同特征是使用具有两个函数，有时三个函数的多功能变分原理，最常见的是光束质心轴的位移u（x）和w（x），以及其旋转φ（x ）。在这种情况下，在非线性情况下，如果不是不可能的话，很难假设未知函数的适当的场一致近似函数集。不一致导致收敛性差，应力振荡和几种锁定，请参见PRATHAP [2]。这些代表了制剂的严重缺陷。相反，本发明采用了新推导的变分原理，其中唯一未知的函数是光束质心轴的旋转φ（x）。因此，不会出现场一致性的问题，并且没有上述现象。此外，对于相当数量的自由度，本发明的配方总是比其他已知的配方更准确。已经针对平面和空间弹性和超弹性梁导出了这种公式。这里介绍的弹塑性梁的配方与这些工作密切相关。在这些参考文献中可以找到许多此处未介绍的细节。

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