非均匀薄板弯曲的精确元法

An Exact Element Method for Bending of Nonhomogeneous Thin Plates

摘要: 本文在阶梯折算法的基础上,提出构造有限元的新方法——精确元法.它不用一般变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法,得到薄板弯曲一个非协调三角形单元,它具有6个自由度.文中给出证明,位移和内力均收敛于精确解,并有很好的精度.文末给出算例.算例表明利用本文的方法,内力和位移均可获得满意的结果.
- 算法 /
- 非均匀薄板 /
- 弯曲 /
- 精确元法
Abstract: In this paper, based on the step reduction method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a triangle noncompatible element with 6 degrees of freedom is derived to solve the bending of nonhomogeneous plate. The convergence of displacements and stress resultants which have satisfactory numerical precision is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.
- algorithm /
- nonhomogeneous thin plate /
- bending /
- exact element method

[1]	Cowper, G. R., E. Kosko, G. M. Lindberg and M. D. Olson, Statics and dynamics of high-precision triangular plate bending element, AIAA Journal, 7, 10(1969), 1957.
[2]	Argyris, J. H. and K. E. Buck, A sequel to technical note 14 on the TUBA family of plate element, Aeronautical Journal, 72(1968), 977.
[3]	Irons, B. M., A conforming quartic triangular element for plate bending, International Journal for Numerical Methods in Engineering, 1, 1(1969), 29.
[4]	Hermman, J. R., A bending analysis for plates, AD 646300(1966), 577.
[5]	Hermman, L. R., Finite element bending analysis for plates, Journal of the Engineering Mechanics Division, Proceedings of the ASCE, 93, EM5(1967), 13.
[6]	Morley, L. S. D., On the constant moment plate bending element, Journal of Strain Analysis,6, 1(1971), 20.
[7]	Zienkiewicz, O. C., The Finite Element Method, 3rd Ed., McGraw-Hill Book Company (1977).
[8]	Irons, B. and S. Ahmad, Techniques of Finite Elements, Ellis Horwood, Ltd., Chichester, UK (1979).
[9]	叶开沅,非均匀变厚度弹性力学的若干问题的一般解,Ⅳ.非均匀变厚度梁的弯曲、稳定和自由振动,兰州大学学报,力学专号,(1) (1979), 133-157.
[10]	纪振义、叶开沅,任意变系故微分方程的精确解析法,应用数学和力学,10 (10) (1989), 841-852.
[11]	Timoshenko, S. and S. Woinowsky-Krieger, Theory of Plates and Shell, 2nd Ed., McGraw-Hill Book Company, Inc. (1959).