Optimal design of two-dimensional porosity distribution in shear deformable functionally graded porous beams for stability analysis

In the present study, considering two-dimensional porosity distribution through a functionally graded porous (FGP) beam, its optimal distributions are obtained. A multi-objective optimization problem is defined to maximize critical buckling load and minimize mass of the beam, simultaneously. To this end, Timoshenko beam theory is employed and equilibrium equations for two-dimensional functionally graded porous (2D-FGP) beam are derived. For the solution, we present generalized differential quadrature method (GDQM) and consider two symmetric boundary conditions (Clamped-Clamped and Hinged-Hinged). Solving generalized eigenvalue problem, critical buckling load for 2D-FGP beam is then obtained. During optimization procedure, a cubic polynomial spline interpolating on a finite number of design variables is considered as porosity distribution function. Solving the multi-objective optimization problem using bio-inspired genetic algorithm (NSGA II), leads to a set of optimal porosity distributions known as Pareto optimal solutions. To show the validity of the proposed formulation, we compare results with those reported (1D and 2D porosity distributions) in the literature as well as finite element simulations. We also compare Pareto solutions with optimization result of one dimensional porosity distribution which clearly demonstrates the importance of the presented optimization procedure. In general, optimum porosity distributions are different in each boundary condition. However, in most of the optimum solutions, middle line of the beam is composed of the material with higher values of porosity and outer corners have lower values of porosity. Pareto optimal solutions also indicate that, sharp decreasing of the mass makes a slight decline in critical buckling load when it has large values. The proposed approach can be used for design of porosity distribution in FGP structures.