考虑剪切变形的变截面欧拉梁单元刚度矩阵

ANSYS中的Beam44单元是考虑剪切变形的变截面欧拉梁单元,为明确其单元刚度矩阵推导方法,以形函数为基础,根据虚功原理,系统给出了考虑剪切变形的变截面欧拉梁单元刚度矩阵推导过程。以矩形、圆形、圆环和箱形截面梁为例,经与ANSYS中Beam44单元刚度矩阵对比,明确了其单元刚度矩阵的推导方法、相关假定和使用要求。研究发现:变截面欧拉梁因形函数本身的近似性和截面参数随截面位置变化的复杂性,难以给出普适性的单元刚度矩阵理论表达式。ANSYS对Beam44单刚矩阵推导时,采用了以下3种处理方式以简化计算:采用与等截面梁相同的厄米特形函数,在纯弯模式单刚矩阵的积分计算中对几何矩阵和截面参数分别积分,不考虑截面剪切系数k随截面位置的变化,由此所得单刚矩阵形式与等截面梁Beam4一致。ANSYS对Beam44单元所给等效截面参数,如面积、抗扭惯矩和抗弯惯矩的表达式,源于梁单元两端截面形状为相似形且截面尺寸随杆件线性梯度变化的情况,应用于其他情况时会与理论值有一定偏差。由此可知,在使用Beam44单元时为提高计算精度,应尽量减小单元长度来控制两端截面参数比值,同时对剪切系数k宜取单元中截面的k值或左右截面的均值。

Element Stiffness Matrix of Tapered Euler Beam Element Including Shear Deformation

Element Beam44 in ANSYS is a tapered Euler beam element including shear deformation and the study was initiated for its element stiffness matrix.The derivation process was presented based on the shape functions and the principle of minimum potential energy.Then the theoretical element stiffness matrixes were compared with those obtained from Beam44,taking four tapered-section beams with rectangular,circular,tubular and box sections as examples.Following the comparison,the derivation method,hypotheses and application suggestions of element Beam44 were deduced.It was found that the universal expression of element stiffness matrix of tapered Euler beam element couldn’t be obtained theoretically for two reasons.First,variations of section parameters along element are complicated,even couldn’t be expressed by formula;second,the exact shape functions are also not available.The element stiffness matrix of Beam44 shares the same form of that of Beam4 by virtue of three simplifications:employing the shape functions of constant-section Euler beam element directly;splitting the single integration of element stiffness matrix for pure bending mode into two integrations;and regarding the shear factor k as a constant.The equivalent section parameters of Beam44 such as area,torsion and bending moments of inertia are just obtained from a tapered beam element whose two end-sections are proportional,and would show some differences for other tapered beam elements.Consequently,the element mesh should be refined to control the ratios of end-section parameters and the shear factor k should be acquired from the middle section or the average value of the two end-sections:both of the suggestions are aiming at high calculation accuracy.