An equilibrium finite element model for buckling analysis of plates
An equilibrium finite element model for buckling analysis of plates
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DOI：
10.1002/nme.1620111108
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发表时间：
1977
影响因子：
2.9
通讯作者：
B. Tabarrok;A. Simpson
中科院分区：
文献类型：
--
作者：
B. Tabarrok;A. Simpson
对于结构系统的平衡问题，众所周知，使用协调位移有限元模型产生的能量水平高于精确值。另一方面，使用平衡元素（到目前为止仅限于线性问题）产生的能量水平低于确切的。因此，这两种互补的方法提供了一个有价值的洞察力的结构的响应和状态的收敛的解决方案，作为有限元网格细化。这两种分析方法的特点，推动了位移和平衡模型的同时发展。然而，平衡模式的发展进度落后于位移模式。最初的几个平衡模型是由Fraeijs de Veubeke使用平衡应力场作为插值函数开发的。在这种方法中，人们发现，与位移模型相比，平衡模型的发展非常有限。后来人们认识到，使用应力函数而不是应力场，消除了平衡模型发展中的主要困难，实际上它使平衡模型的发展有点类似于位移模型的发展。对于板的弯曲问题，Morley [3]和Sanders [4]根据Southwell应力函数建立了平衡模型^。“^埃利亚斯使用艾里和索斯韦尔应力函数，在有限元的背景下讨论了板的弯曲和拉伸中众所周知的对偶性。这里值得一提的是应力函数公式化的两个特点。首先，在多连通域的情况下，必须对应力函数施加特殊约束，以确保其单一值6。其次，如果应力边界条件具有复杂的变化，则可能难以选择适当的应力函数来完全满足它们。在结构分析的线性问题中，经常出现两个特征值问题。这就是自由振动和弹性屈曲问题。在这些本征值问题中，人们关注的是两个能量量之差的平稳性，因此这些问题的特征不在于最小/最大能量属性。然而，这将是非常有用的，分析这样的特征值问题的两种方法，因为在这些方法中的近似是不同的物理性质。t教授。f读者。
For the equilibrium problems of structural systems it is well known that the use of compatible displacement finite element models yields energy levels which are higher than the exact values. On the other hand use of equilibrium elements (until now restricted to linear problems) yields energy levels lower than the exact. As such these two complementary approaches provide a valuable insight into the response of the structure and on the state of convergence of solution as the finite element mesh is refined. This feature of the two methods of analysis has given an impetus for the simultaneous development of displacement and equilibrium models. However progress in the development of equilibrium models has lagged behind that of the displacement models. The first few equilibrium models were developed by Fraeijs de Veubeke using equilibrating stress fields as interpolation functions.',* In this approach it was found that, in contrast to displacement models, the development of equilibrium models was very restrictive. Later it was recognized that the use of stress functions, instead of stress fields, removes the major difficulties in the development of equilibrium models and indeed it renders the development somewhat analogous to the development of displacement models. For the plate flexural problems Morley3 and Sanders4 developed equilibrium models in terms of Southwell stress function^."^ Elias, using the Airy and Southwell stress functions, discussed the well known duality in flexure and stretching of plates, in the context of finite element^.^ Two features of stress function formulation are worth mentioning here. First, in the case of multiply connected domains special constraints have to be imposed on the stress functions to ensure their single valuednes6 Second, if the stress boundary conditions have a complex variation it may be difficult to select appropriate stress functions to satisfy them identically. Now amongst the linear problems of structural analysis two eigenvalue problems arise frequently. These are the free vibration and the elastic buckling problems. In these eigenvalue problems one is concerned with the stationarity of differences of two energy quantities and hence these problems are not characterized by minimum/maximum energy properties. Nevertheless it would be very useful to analyze such eigenvalue problems by the two approaches since the approximations in these approaches are of physically different nature. t Professor. f Reader.