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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。
This paper derives the exact solution of the beam over-buckling under in-plane thermal loading. First, based on the nonlinear first-order shear deformation beam theory, the basic equations governing axial and transverse deformation are deduced. Then, the three nonlinear equations are reduced to a fourth-order nonlinear integro-differential equation with respect to transverse deflection. The equation and the corresponding boundary conditions constitute the differential eigenvalue problem. Solving this problem directly gives a closed solution to the thermal over-flexion configuration, which is a function of external heating load. Using the exact solution, the analytical relationship between the first-order results of critical buckling loads and the classical results is obtained. In order to investigate the effects of thermal load, transverse shear deformation and boundary conditions, numerical examples are given for the two ends fixed, the simple ends at one end and the simple supported one end fixed boundary under exact solution. The buckling behavior under the in-plane thermal load is compared with the classical results. The exact solutions obtained in this paper can be used to verify or improve various approximate theories and numerical methods.