Parametrization in laminate design for optimal compliance

In this paper we consider the maximal stiffness design of laminated plates subjected to single and multiple loads. The stiffness of the laminates are parametrized in terms of the so-called lamination parameters. These express the relation between the material parameters for the laminate and the laminate lay-up and are given as moments of the trigonometric functions that appear in the well-known rotation formulae for stiffness matrices. These relations are here given in a form suitable for optimization studies. The conditions for the laminate itself to be orthotropic are also given directly in terms of the lamination parameters. The design problem is analyzed by performing a reformulation to an equivalent problem which is local in character and it is shown how this, together with an enlargement of the design space to allow for out of plane chattering designs, leads to a significant simplification of the problem. Thus, the number of variables is reduced to only four for the stiffness problem at hand, even in the general case with coupling stiffnesses and multiple loads. Moreover, in the special case of in-plane loads, the optimal solution for each design element of the plate can be realized as a single rotated ply of material or in special strain situations by two plies. A computational solution procedure for the simplified problem is described and several numerical examples illustrate basic features of the design approach.