This paper deals with the linear buckling problem for inhomogeneous Euler–Bernoulli column having both mechanical and geometrical properties variable along its length. Four classes of longitudinally functionally graded material columns with variable cross-sections are considered. The solutions of the relevant differential equations are obtained in terms of both hypergeometric functions and elementary functions. The normalized buckling loads are computed for five typical boundary conditions and they are validated by a comparison with approximate numerical results available in literature. The proposed formulation may provide a further benchmark for the accuracy assessment of numerical and approximated solutions.

中文翻译：

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一类变截面 FGM 柱的线性屈曲精确解

本文处理非均匀 Euler-Bernoulli 柱的线性屈曲问题，该柱具有沿其长度变化的力学和几何特性。考虑了四类具有可变横截面的纵向功能梯度材料柱。相关微分方程的解是根据超几何函数和初等函数获得的。计算了五种典型边界条件的归一化屈曲载荷，并通过与文献中可用的近似数值结果进行比较来验证它们。所提出的公式可以为数值和近似解的准确性评估提供进一步的基准。