Optimal Layouts of Stiffeners for Plates in Bending - Topology Optimization Approach

This paper applies optimum structural topological design methodologies to the stiffening of plates in bending. The purpose of the design is to position a given amount of material, in the form of orthogonal stiffeners, minimizing the compliance of a given plate. Cheng and Olhoff have shown that if left unchecked, this formulation includes dense sets of infinitely thin stiffeners - an optimal yet obviously impractical solution. In order to circumvent the obstacle, the present formulation assumes a plate meshed by an orthogonal lattice with rectangular elements allowing the stiffeners to be positioned only along the lines of the mesh. The basic problem is to add uniform beams, of generally variable stiffness, in an optimal fashion along the gridlines, having a constant amount of material at hand. A beam stiffness parameter, usually the width or the height, is taken as the design variable xi of the problem, having i=1,..,l+t, assuming the lattice has l longitudinal and t transversal lines. In this form, we have stated a standard topological design problem that calls for the minimization of the compliance, subjected to a constraint on the amount of material, which can be solved by classical optimization techniques, not unlike the erstwhile topology optimization for 2D membrane problems, using density design variables. It is agreed that when limiting the positions of the stiffeners to prescribed gridlines, one does not necessarily obtain the best possible layouts. However, the discretization of the problem allows for using topological optimization methodologies that produce noteworthy results. In particular, we treat the problem of getting 0/1 solutions by using rigidity interpolation and/or a norm constraint. Herein, the issue of the convexity of the problem is addressed. The optimal designs obtained exhibit interesting features. Influence of the plate-to-beam stiffness ratio is discussed. The technique is illustrated by clamped rectangular plates under different lateral loading conditions. Below is a typical example of a result produced by the presented formulation.

Figure 1

Optimal layout of orthogonal stiffeners for hydrostatic loading