集中力作用下悬臂梁几何中轴的弹性大挠度分析

以平截面假定为基础,并以几何中轴作为梁变形的参考轴,推导了大挠度情况下,弹性梁在轴力和弯矩联合作用下几何中轴的曲率——弯矩和轴力方程,并建立了求解几何中轴变形曲线的方程组,这样的方程组考虑了梁轴向变形的影响,克服了以中性轴作为梁变形参考轴时不能考虑轴向变形的不足。利用所建立的方程组,分析了受集中力作用的弹性悬臂梁大挠度问题,并同以中性轴作为梁变形参考轴的部分结果进行了对比。结果表明,当载荷较小时,2种理论计算出的弹性大挠度悬臂梁的变形曲线差别很小;而当载荷较大时,2种理论计算出的弹性大挠度悬臂梁的变形曲线存在比较明显的差异。

Analysis of Large Deflection of Geometric Central Axis of an Elastic Cantilever Subjected to Concentrated Force

Based on the hypothesis of planar cross-section and the geometric central axis taken as the reference axis,the curvature-the axial force and the moment equation of the geometric central axis subjected to both a axial force and a moment is derived on the occasion of the large deflection,and the system of equations for solving the deformation of the geometric central axis is established.Because the influence of the axial deformation can be considered in the system of equations,the disadvantage that the axial deformation cannot be considered when the neutral axis is as the reference axis is overcome.By use of the system of the equations,large deflection of an elastic cantilever subjected to a concentrated force at its tip is analyzed and some comparison between the system of equations of the geometric central axis and those of the neural axis is made.The comparison shows that the difference between the two systems is quite small when the beam is subjected to a small force and that the difference is quite obvious when the beam is subjected to a larger force.