剪切可变形梁非线性静态响应的精确解

发布时间：2018-10-15 13:14

【摘要】：本文给出了纵横向载荷作用下,梁非线性静态问题的精确解。基于非线性一阶剪切变形梁理论,导出了梁非线性静态问题的基本方程。将三个非线性方程化简为一个关于横向挠度的非齐次四阶非线性积分-微分方程,当只有轴向载荷作用时,该方程和相应的边界条件构成微分特征值问题。直接求解该方程,得到了梁非线性静态变形闭合形式的解,这个解显式地给出了梁的变形与外载荷之间的非线性关系,描述了梁变形后的非线性平衡路径。利用这个解,得到了梁临界屈曲载荷的一阶结果与经典结果。为考察载荷、长高比以及边界条件的影响,根据得到的解析解给出了一些数值算例,并讨论了梁不同阶屈曲模态下非线性静态响应的一些性质。结果表明:对应于方程特征参数λ的不同取值区间,梁的轴向载荷-挠度曲线有不同的解支;而对应于参数λ的同一取值区间,梁分别对应两个不同的屈曲模态。
[Abstract]:In this paper, the exact solution to the nonlinear static problem of beams under longitudinal and transverse loads is given. Based on the theory of nonlinear first order shear deformed beam, the basic equation of nonlinear static problem of beam is derived. Three nonlinear equations are simplified into a non-homogeneous fourth order nonlinear integro-differential equation with respect to transverse deflection. When only axial loads are applied, the differential eigenvalue problem is formed by the equation and the corresponding boundary conditions. By solving the equation directly, the closed form of nonlinear static deformation of the beam is obtained. The nonlinear relationship between the deformation of the beam and the external load is given explicitly, and the nonlinear equilibrium path after the deformation of the beam is described. By using this solution, the first order and classical results of the critical buckling load of a beam are obtained. In order to investigate the effects of load, ratio of length to height and boundary conditions, some numerical examples are given based on the analytical solutions obtained, and some properties of nonlinear static response of beams under different buckling modes are discussed. The results show that the axial load-deflection curve of the beam has different solution support corresponding to the different value interval of the characteristic parameter 位 of the equation, and corresponding to the same value interval of the parameter 位, the beam corresponds to two different buckling modes respectively.
【作者单位】：兰州理工大学理学院;
【基金】：国家自然科学基金(11472123) 西安交通大学机械结构强度与振动国家重点实验室开放课题(SV2014-KF-04)
【分类号】：O344.1

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