集中力作用下可伸长悬臂梁的大挠度分析

Large Deflection Analysis of a Cantilever with Extensible Axis Under the Action of a Concentrated Force

悬臂梁模型在诸多工程领域有广泛应用,例如土木工程、机械工程、海洋结构物、航空航天、传感装置等,而其几何非线性和伸长(缩短)量对于其力学行为有重要影响.本文对轴线可伸长的悬臂梁在集中力作用下产生的大变形现象进行了全面研究.首先定义了轴线伸长量,然后将其引入到应变能中,建立了系统的能量泛函;再对泛函进行变分,推导出控制方程组和边界条件;最后采用打靶法求解非线性方程组,得到了悬臂梁的挠曲线、转角和伸长量,并与小变形模型和轴线不可伸长的大变形模型进行了对比.这些分析结果有助于深刻理解结构的非线性变形,同时为精确设计变形较大的工程结构物提供了理论参考.

The cantilever model holds great implications in such areas as civil engineering,mechanical engineering,marine structures,aerospace and aeronautics and sensor devices,and its geometric nonlinearity and extensible or compressive value have a great impact on the mechanical behaviors.In the present work,a comprehensive study was carried out on the large deformation of the cantilever with its axis being stretchable under the action of a concentrated force.Firstly,the extensible ratio of the axis was defined,and then it was introduced into the strain energy,thus the energy functional of the system was established.Taking variation on the energy functional,the governing equation group and boundary conditions were derived.Then the shooting method was used to solve this nonlinear equation group,and the deflection,rotation and elongation value of the cantilever were obtained.These results have been compared with those from the infinitesimal deformation model,and large deformation model with the axis being inextensible.These analyses may be beneficial for us to deeply understand the nonlinear deformation of structures,and provide some theoretical references for engineering structures with strong deformation.