一种考虑截面完整变形特征的空间三维梁有限元模型

Finite element model of the three dimensional beam for accurately describing arbitrary deformation of the beam′s cross-section

提出了一种基于截面插值的三维空间梁模型。首先引入截面插值函数——拉格朗日函数描述截面形状,以位移向量为未知变量描述截面位移,在此基础上依据插值理论构造梁单元位移场,不同于传统梁单元通过假定的中性轴挠度和转角来确定梁截面各点位移,该梁模型摒弃了中性轴假设与平截面假设,通过截面插值函数得到梁截面面内、面外变形;然后通过有限元理论推导了梁单元刚度矩阵与质量矩阵,并采用MATLAB编制了相应的有限元程序;最后通过几个典型算例展开分析,并将该单元的计算结果与经典梁理论和商用有限元软件数值计算结果进行对比,算例结果表明该单元有着更好的适用性和更高的精度。

A type of space beam model based on cross-section interpolations is proposed. Firstly, the Lagrange functions are introduced as the interpolation function to describe the shape of beam cross-section, and the displacement vectors are used as unknown variables to describe the displacements of the cross-section. On this basis, the displacement field of the beam element is constructed according to the interpolation theory. The displacements of the beam in the conventional beam element are determined by the deflection and rotation of the assumed neutral axis, while the new beam element rejects the neutral axis hypothesis and flat section hypothesis, and the deformation of the beam cross-section is obtained by the interpolation functions. Then the stiffness matrix and the mass matrix of the beam element are derived by the finite element theory, and the finite element program is compiled by MATLAB. Finally, the analysis of several typical examples is carried out in this paper, and the results are compared with those of the classical beam element and commercial finite element software. The numerical results show that the new beam element has better applicability and higher precision.