中文摘要：

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。

英文摘要：

An exact, closed form, solution is obtained for the nonlinear static responses of beams subjected to transversally and axially loading. The nonlinear first-order shear deformation beam theory is employed to derive the basic equations for the nonlinear static responses of beams and three equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations; the equation and the corresponding boundary conditions lead to a differential eigenvalue problem when the beam is subjected to axially loading only. And the nonlinear equation is directly solved without any use of approximation and a closed-form solution for the nonlinear static responses is obtained. The exact solution explicitly expresses the nonlinear relation between the applied load and deformation of the deformed beam and describes the nonlinear equilibrium paths of the deformed beam. By using the solution, one can obtain critical buckling load for the shear deformable beams or classical ones. Based on the exact solutions obtained, the numerical analyses are carried out to investigate some properties of the nonlinear static responses for the beams with different buckling modes. The results show that the load-deflection curve for beams has many different branches corresponding to different value intervals of the parameter, λ, while for same value interval of the parameter, λ, beam has two different deformed configurations for a given load.