考虑剪切变形的欧拉梁单元一致质量矩阵

Consistent Mass Matrix of Euler Beam Element including Shear Deformation

为明确考虑剪切变形的欧拉梁单元一致质量矩阵推导方法,以形函数为基础,基于虚功原理,区分伸缩、扭转以及是否考虑剪切变形的弯曲受力状态,给出了欧拉梁单元一致质量矩阵表达式。研究表明:不考虑剪切变形时,欧拉梁受弯状态的一致质量分析中一般不计单元上微元体水平方向的惯性力,此时仅需弯曲变形引发的竖向位移形函数即可;考虑剪切变形时,则需计入水平方向的惯性力,且同时需要完整的竖向位移形函数和纯弯曲转角位移形函数;对于变截面欧拉梁单元,其一致质量矩阵表达式过于复杂,可在等截面梁的基础上,根据元素位置对矩阵元素匹配左右端或平均截面面积及截面极惯性矩近似给出实用的质量矩阵表达式;考虑剪切变形时,欧拉梁的刚度矩阵亦可经竖向位移和纯弯曲转角位移形函数计算得到,这一过程与使用纯弯曲竖向位移和纯剪切竖向位移形函数的过程在本质上是一致的。

The study was initiated for the consistent mass matrix of Euler beam element including shear deformation.The consistent mass matrix of uniform element was got separately for the uncoupled tension,torsion,and bending conditions,with the shear deformation included or not,based on the shape functions and the virtual work.It was shown that the inertia force along the axial direction was always ignored in the mass matrix derivation for the bending condition if the shear deformation was not included,so only the shape functions for vertical deformation were needed for the bending condition.When the shear deformation was included,the inertia force along the axial direction must be considered and the shape functions for the section rotation angle due to bending were also required besides the complete shape functions for vertical deformation due to the bending and shear forces.For tapered Euler element,the theoretical expression for the consistent mass matrix would be quite complicated and a simple expression was proposed following an approximate strategy,matching the ending or average section areas or polar moments with the elements in the mass matrix according to their positions.Additionally,the stiffness matrix could also be deduced on the foundation of the complete shape functions for vertical deformation and the shape functions for the section rotation angle.This derivation procedure was different with the traditional manner superficially but they shared the same principle essentially.