采用矢量型转动变量的二维协同转动梁元法  

Advanced co-rotational procedure for large displacement analysis of 2D beam element using vectorial rotational variables

作　　者：李忠学[1] 

机构地区：[1]浙江大学土木工程学系,浙江杭州310027

出　　处：《浙江大学学报（工学版）》2006年第7期1219-1223,共5页Journal of Zhejiang University：Engineering Science

基　　金：国家自然科学基金资助项目(50408022);教育部及浙江省留学回国人员科研启动基金资助项目

摘　　要：基于先进的协同转动法,建立可以考虑大位移大转角问题的二维三节点等参梁元公式.首先将协同转动坐标系固定在单元中部节点上,并随节点的刚体转动和平动而运动,然后计算单元节点坐标和位移,便可排除单元刚体位移的影响,从而显著降低建立单元切线刚度矩阵的复杂性.在该梁元中定义了一组矢量型转动变量,以精确描述梁单元的几何变形,这种变量为单元节点方向矢量分量的较小值.采用这种变量,可以通过简单的矢量变换即可建立起局部变量与整体变量之间的关系,从而非常方便地将局部单元切线刚度矩阵转换到整体座标系下.此外,引入Lagrangian插值函数来描述单元几何形状和位移场,采用Green应变和第二Piola-Kirchihoff应力张量来计算应变能,再通过对单元应变能进行一阶和二阶微分,得到单元内力矢量和对称的切线刚度矩阵.通过几个算例分析,验证了该有限元法的可靠性和计算效率.A 3-noded iso-parametric element formulation for large displacement analysis of 2D beam was presented on the basis of an advanced co-rotational framework. First, a co-rotational framework was fixed at the interior node, and it translated and rotated rigidly with the element. Then the nodal coordinates and displacements were calculated against this framework, and the effect of rigid-body motion on the element tangent stiffness matrix can be excluded in advance. Second, a set of vectorial rotational variables were defined to describe accurately the element deformation. The rotational variable was the smaller component of the orientational vector at each node, and this resulted in a simple relationship between the local variables and the global variables, and lead to great convenience in transforming local element tangent stiffness matrix into global system. Furthermore, Lagrangian shape functions were adopted to depict the geometry and the element deformation, and Green strain and the second Piola-Kirchhoff stress tensors were introduced in calculating the strain energy of element, from which the internal force vector and the element tangent stiff- ness matrix were derived as its first partial derivative and its second partial derivative with respect to local variables, respectively. Finally, the reliability and efficiency of the proposed procedure were illustrated through several elastic examples.

关 键 词：协同转动法 矢量型转动变量 大位移 大转角 二维梁单元 

分 类 号：TU311[建筑科学—结构工程] O343.1[理学—固体力学]