受扭圆轴弹性屈曲分析时边界条件的不一致性

受扭圆轴广泛存在于机械结构中。对于细长圆轴,当其两端受扭矩作用时将发生弹性屈曲。在两端简支条件下,文献中已有的解析解和基于能量法的数值近似解结果不吻合,原因是人们没有注意到边界条件的不一致性。笔者用有限元法进行了分析,给出了单元的线性刚度矩阵和增量刚度矩阵。分析发现:由能量变分方程所得到的应自动满足的自然边界条件———力边界条件和得到解析解的二阶平衡微分方程所应满足的力边界条件两者在简支情况下是不一致的。考虑到与解析解的力边界条件的等效性后用有限元数值分析方法得到的结果与解析解极为吻合。有限元解与解析解间的差异,有时并非单元性能的原因或计算误差造成的。

Discrepancy of Boundary Conditions in Flexural Buckling Analysis for Shafts Under Torque Loadings

There are many shafts with circular cross-sections under the action of torque in mechanical structures. The torque may lead thin shafts to flexural buckling in some cases. This paper analyzes the elastic flexural buckling of circular shafts under the action of torque. Finite element method is used in the analysis. The linear incremental and stiff matrices are derived. Attention is focused on the flexural boundary conditions the case of shafts with both ends simply supported, since previous research has not noticed the inconsistency in boundary conditions used to obtain analytical solutions and finite element data. It is demonstrated that the natural boundary conditions by the calculus variations are inconsisfent with the boundary conditions satisfied by the second order differential equations. Numerical result agrees well with existing analytical solutions when in the finite element onethod is used for equivalent boundary conditions. Therefore, not all the differences between numerical results and analytical results are caused by the element performance or computational error.