平面框架几何非线性分析的修正拉格朗日-协同转动联合法

A COROTATIONAL UPDATED LAGRANGIAN FORMULATION FOR GEOMETRICALLY NONLINEAR ANALYSIS OF 2D FRAMES

提出了一种几何非线性有限元分析平面框架的方法。应用修正拉格朗日(UL)法计算单元的增量节点位移;计算增量节点力采用了协同转动法,使用的协同构形(Cr)是由前一个平衡构形(C1)经刚体运动后得到。以单元节点的增量自然变形为参数表示出Cr构形下的虚功方程;推导得出了C1构形中单元增量节点位移向量与Cr构形中节点增量自然变形之间的关系;把得到的关系式代入虚功方程,得Cr构形中计算增量节点力的切向刚度矩阵;所得的切线刚度矩阵能通过刚体检验。考虑了加载变形对后续分析的影响,导致切向刚度矩阵增加了与加载变形有关的刚度矩阵。算例结果表明:提出的方法可以采用较少单元达到UL法同样的计算精度;加载变形产生的刚度矩阵能有效地提高计算效率;但变形较大时,加载变形产生的刚度矩阵会导致膜锁。

A method is presented for geometrically nonlinear analysis of 2D frames. An updated Lagrangian （UL） formulation is used to calculate the incremental nodal displacements, and incremental nodal forces are recovered by a corotational formulation whose configuration （Cr） is derived from a rigid-body motion of the last calculated configuration （C1） in equilibrium. Firstly, the equation of virtual work is expressed in terms of incremental nodal natural deformations in Cr; then, the relations are deduced between the incremental nodal natural deformations in Cr and the incremental nodal displacement vector in C1. Adopting the relations in the equation of virtual work, the tangent stiffness matrix for incremental nodal forces is obtained for Cr, and it can pass a rigid-body test. The analysis has considered the loaded deformations which will result in an additional matrix in the tangent stiffness matrix. The numerical examples show that： with fewer elements, the presented method can achieve similar precision to UL formulation; the matrix with respect to the loaded deformation can effectively improve the efficiency, but it may result in membrane locking if the deformation is large.