In the present study, an attempt is made to present the governing equations on the post-buckling of two-dimensional (2D) FGP beams and propose appropriate optimization procedure to achieve optimal post-buckling behavior and mass. To this end, Timoshenko beam theory, Von-Karman nonlinear relations, virtual work principle, and generalized differential quadrature method are considered to derive and solve governing equations and associated boundary condition (Hinged–Hinged) for an unknown 2D porosity distribution. Proposed method is validated using the papers in the literature. The optimization procedure including defining porosity distributions (interpolations), post-buckling function and Taguchi method is then proposed to optimize the post-buckling path and minimize the mass of the 2D-FGP beams. Results indicate that, great improvement can be achieved by optimizing the porosity distribution; for an identical mass, the post-buckling paths of optimum points are closer to desired path (dense structure). The difference between uniform and non-uniform porosity distributions is more (58% higher post buckling function), at higher values of the mass. Optimum distributions mostly have the higher values of porosity at center line of the beam and minimum values at outer line. Analysis of variance is also provided to create a better understanding about design points contributions on the post-buckling path.

在本研究中，尝试提出二维（2D）FGP梁的后屈曲控制方程，并提出适当的优化程序以实现最佳的后屈曲行为和质量。为此，考虑采用Timoshenko梁理论，Von-Karman非线性关系，虚功原理和广义微分正交方法来推导和求解未知的二维孔隙度分布的控制方程式和相关的边界条件（Hinged-Hinged）。文中的论文对提出的方法进行了验证。然后提出了包括定义孔隙度分布（插值），后屈曲函数和田口方法的优化程序，以优化后屈曲路径并最小化2D-FGP梁的质量。结果表明，通过优化孔隙率分布可以实现很大的改进。对于相同的质量，最佳点的屈曲后路径更接近所需路径（致密结构）。在较高的质量值下，均匀和不均匀的孔隙率分布之间的差异更大（屈曲后函数高58％）。最佳分布通常在梁的中心线处具有较高的孔隙率值，而在外线处具有最小值。还提供了方差分析，以更好地了解屈曲后路径上的设计点贡献。在更高的质量值。最佳分布通常在梁的中心线处具有较高的孔隙率值，而在外线处具有最小值。还提供了方差分析，以更好地了解屈曲后路径上的设计点贡献。在更高的质量值。最佳分布通常在梁的中心线处具有较高的孔隙率值，而在外线处具有最小值。还提供了方差分析，以更好地了解屈曲后路径上的设计点贡献。