Optimal Grillage Layout for Given Natural Frequency

The problem of minimizing the volume of structural material of an elastic grillage subject to a prescribed value or the fundamental frequency of free transverse vibrations, is considered. The domain and the boundary conditions are given, and the grillage is equipped with a specified, distributed non-structural mass, which is large in comparison with the structural mass. It is the objective to determine the optimal layout of the grillage, and to optimize the beams, which are of given depth and varying width. It is shown that the optimal grillage layout is the same as that found earlier for three different types of design problems, namely plastic design for given collapse load, elastic design for given permissible stress, and elastic design for given compliance. An an example, a simply supported, square grillage with uniformly distributed mass loading is optimized, and it is shown that the amount of structural material of the optimal grillage is significantly smaller than that to be used for a uniform plate with the same fundamental natural frequency of transverse vibrations.