纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程，并应用微分变换法（DTM）对控制微分方程及边界条件进行变换，计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁，并与已有文献结果进行对比，验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。

Analysis of Free Vibration and Critical Buckling Load of Fiber Reinforced FGM Beams

Based on the classical beam theory (CBT), the transverse free vibration and critical buckling load of fiber reinforced functionally graded material (FGM) beams subjected to axial force are studied. Firstly, the mixing law model is considered to characterize the material properties of the fiber reinforced FGM beam. Secondly, the governing differential equations of transverse free vibration and critical buckling load of the fiber reinforced FGM beam subjected to axial force are derived by Hamilton principle, and the governing differential equations and boundary conditions are transformed by differential transformation method (DTM). The dimensional natural frequencies for transverse free vibration and dimensionless critical buckling loads of the fiber reinforced FGM beam are calculated under three boundary conditions of clamped-clamped (C-C), clamped-simply supported (C-S) and simply supported-simply supported (S-S). The problem is reduced to homogeneous material beams and FGM beams and compared with the existing literatures to verify its effectiveness. Finally, the effects of stiffness ratio, fiber volume fraction and dimensionless compressive load on the dimensionless natural frequency and all parameters on the dimensionless critical buckling load of the fiber reinforced FGM beam under different boundary conditions are discussed.