弹性压扭直杆的Greenhill公式对精确模型的推广

将圆截面Kirchhoff弹性压扭直杆的Greenhill公式推广到精确模型.基于平面截面假定,在弯扭的基础上增加了拉压和剪切变形,将弹性杆的位形表达为截面的弧坐标历程.由弹性杆精确模型的平衡微分方程,得到了两端受力螺旋作用时对应于直线平衡状态的特解,导出了线性化扰动方程及其通解,再根据两端为铰支时的边界条件以及积分常数存在非零解的条件导出弹性直杆精确模型的Greenhill公式.结果表明,由力螺旋表示的稳定域为一对称的封闭区域,拉压和剪切对稳定性的影响取决于拉压柔度与剪切柔度之差、抗弯刚度和杆长这三个因素.

Greenhill formula for an exact model of elastic rod

Greenhill formula for Kirchhoff elastic rod is extended to that of exact model of the rod. Under the assumption of the plane cross section, the configuration of an extensible and shearable elastic rod is expressed as a history of the cross section with arc coordinate. A special solution which describes equilibrium in straight line state of the rod is obtained from a differential equilibrium equation. A linear perturbation equation is derived and its general solution is obtained in which the integral constants are determined by constrained conditions at two ends of the rod. The condition for a non zero solution of the integral constants to exist leads to the Greenhill formula of exact elastic rod model, which shows that the boundary of stable area of the force screw is a closed curve and of symmetry and the inference of extensible and shearable to stability of the rod is dependent on three factors: the difference in flexibility between shear and extension of a section of the rod, the bending stiffness, and the length of the rod.