Geometric optimization based on the constructal design of perforated thin plates subject to buckling

Abstract

Elastic buckling is an instability phenomenon that can occur if a slender and thin-walled plate is subjected to axial compressive load. It is well known that the presence of holes in structural plate elements is almost inevitable in inspection, maintenance, and service purposes, or to reduce the structural weight. In this paper constructal design was employed to optimize the geometry of thin perforated plates submitted to elastic buckling. Simply supported rectangular perforated plates were analyzed with three different shapes of centered holes: elliptical, rectangular, and diamond. The purpose was to obtain the optimal geometry that maximizes the critical buckling load. The ratio between the height and length of the plate was kept constant, while the ratio between the characteristic dimensions of the holes was optimized for several hole volume fractions (φ). A finite-element model was used to assess the plate buckling load, and the Lanczos method was applied to the solution of the corresponding eigenvalue problem. When φ ≤ 0.20 the optimum geometry is the diamond hole, reaching maximum buckling loads around 80.0,21.5, and 17.4% higher than a plate without perforation and plates with elliptical and rectangular holes, respectively. For intermediate and higher values of φ, the elliptical and rectangular holes, respectively, led to the best performance. The optimal shapes were obtained according to the constructal principle of minimization of distribution of imperfections, showing that the constructal design also can be employed to define the optimized geometries in the mechanics of material problems.