纵横向载荷作用下经典梁非线性静态响应的精确解

摘要

An exact solution is obtained for nonlinear static problem of beams subjected to transversal and axial loading.The nonlinear classical beam theory is employed to derive the governing equation for the nonlinear static problem of the beam,which is a nonlinear integral-differential equation about beam deflectior.And this equation is directly solved and a closed-form solution for the nonlinear static deflection of the beam is obtained.This exact solution explicitly gives out the nonlinear relation of beam deformation to external load.In order to examine the influence of load and boudary condition,some numerical computation examples are given out according to the exact solution obtained and some properties of the nonlinear static response of the beam with different-order buckling modalities discussed.The result indicates that the deflection vs load curve of the beam will different branch for different value assignment intervals of the parameter λ and two different buckling modalities will exist for an identical value assignment interval of the parameter λ.%导出了纵横向载荷作用下,梁非线性静态问题的精确解.利用非线性经典梁理论,推导出梁非线性静态问题的基本方程,得到的控制方程是一个关于挠度的非线性积分-微分方程.直接求解该方程,得到梁非线性静态变形闭合形式的解,这个解显式地给出梁的变形与外载荷之间的非线性关系.为考察载荷以及边界条件的影响,根据得到的解析解给出一些数值算例,并讨论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;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.