Optimal material tailoring of functionally graded porous beams for buckling and free vibration behaviors

In this paper, assuming porosity varies only along thickness direction, its optimal distributions in functionally graded porous (FGP) beams are tailored. Two multi-objective optimization problems are defined. In the first one, critical buckling load and mass are optimized simultaneously while in the second one, we concentrate on simultaneous optimization of mass and fundamental frequency. Employing Timoshenko beam theory, we present governing equations for a FGP beam. For the solution, we use Ritz method and propose appropriate trial functions according to the boundary conditions (Hinged-Hinged, Clamped-Clamped, Clamped-Hinged and Clamped-Free). Since the porosity distribution along thickness is unknown, we assume an arbitrary number of trial points with unknown porosities. Employing cubic polynomial spline for porosity distribution, optimization problem then reduces to determination of porosity at trial points. The results of the optimization problem with genetic algorithm are compared with available (non-optimal) results in the literature which demonstrate significant improvement, especially in vibration analysis. The results show that most FGP beams have optimal behavior when porosity at edges is minimum while is maximum at beam center. Pareto optimal solutions indicate that, sharp decreasing of the mass has a slight decline in critical buckling load or fundamental frequency when they have large values.