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Algorithm for deriving explicitly the analytical expression of geometric stiffness matrix of the 4-node,24 degrees of freedom flat shell element

DOI：10.7511/jslx201306008

 作者 单位 E-mail 文颖 中南大学 土木工程学院, 长沙 410075 ywen_ce@csu.edu.cn 曾庆元 中南大学 土木工程学院, 长沙 410075

几何刚度矩阵的推演是结构几何非线性有限元分析的重点和难点之一。推导几何刚度矩阵显式解析表达式成为简化非线性有限元列式，提高分析效率的关键。本文在协同转动法框架下，基于刚体运动法则对四节点二十四自由度的平板壳单元几何刚度矩阵显式解析式进行了推导和讨论；分析了悬臂梁大转动、不同壁厚条件下简支圆柱形屋顶空间大变位两个经典算例。研究结果表明：（1）几何刚度矩阵的显式计算公式不仅为板壳结构几何非线性列式提供了方便而且具有良好的精度；（2）推导的几何刚度矩阵适用于各类型四边形二十四自由度平板壳单元模型；（3）与数值积分相比，采用解析形式的几何刚度矩阵可以显著提高非线性响应计算效率。

The derivation of geometric stiffness matrix is an essential and difficult stage in conducting the finite element analysis of geometrically nonlinear structural problems.Any attempt to obtaining explicitly the analytical expression of geometric stiffness matrix is of great importance for simplifying the formulation,and in particular for improving efficiency and effectiveness of the overall procedure.In the context of co-rotational formulation,an algorithm analytically leading to the geometric stiffness matrix of the 4-node quadrilateral flat shell element with a total of 24 degrees of freedom was presented based on the rigid body motion rule and consecutively subject to discussion.Two benchmark problems,namely the large rotation problem of a cantilever beam and the large deflection behaviour of a hinged semi-cylindrical roof with two typical thicknesses subjected to a central pinching force,were analyzed for demonstrating the reliability and robustness of the proposed procedure.The results of numerical study reveal that:(1)The explicit formula derived herein provides a great deal of convenience while preserving acceptable accuracy of solution;(2)The derived analytical geometric stiffness matrix is element-type independent and (3)As compared with numerical integration,the analytical approach is featured by a significant improvement in computational efficiency.