Post-buckling optimization of two-dimensional functionally graded porous beams

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.