剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3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。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,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。

EXACT SOLUTIONS FOR THERMAL POST-BUCKLING OF SHEAR DEFORMABLE BEAMS

An exact and 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 first-order shear deformation beam theory(FBT).The 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 differential eigenvalue problem.The nonlinear equation is directly solved without any approximation and a closed-form solution for thermal post-buckling deformation is obtained as a function of the applied thermal load.By using the solution,one can obtain the analytical relation of critical buckling load between the shear deformable beams and the classical ones.To demonstrate the influence of in-plane loading,as well as transverse shear deformation and boundary conditions,numerical examples based on the exact solutions are presented,and the post-buckling responses of the beams are discussed.The exact solutions obtained herein can be served as benchmarks to verify and improve various approximate theories and numerical methods.