The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.