经典梁热过屈曲问题的解析解

Analytical solutions for thermal postbuckling of classical beams

基于非线性经典梁理论,建立了控制轴向和横向变形的基本方程,将两个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。对于本文所考虑的三类边界条件,该方程与相应的边界条件构成了微分特征值问题;直接求解该问题,得到热过屈曲构形的解析解,该解是外加热载荷的函数。为考察热载荷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,讨论了梁过屈曲行为的性质。本文得到的解析解可用于验证或改进各类近似理论和数值方法。

An exact,closed form,solution is obtained for the nonlinear static responses of beams subjected to uniform in-plane thermal loading.The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear classical beam theory(CBT).The two 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 differential eigenvalue problem.The nonlinear equation is directly solved without any use of approximation and a closed-form solution for thermal postbuckling deformation is obtained as a function of the applied thermal load.To show the influence of in-plane loading and boundary conditions,numerical examples based on the analytical solutions are given,and some properties of the beams for the postbuckling responses are discussed.The analytical solutions obtained herein can serve as benchmarks to verify and improve various approximate theories and numerical methods.