Using a dual variational formulation, criteria for bounds on the minimum cost of rigid, perfectly plastic structures are derived. The results are used to point out a duality which exists between plastic analysis and plastic optimal design. The relations between the theories of Prager, Shield, Drucker, Heyman, and Michell are discussed. The proposed theory is applied to Tresca sandwich plates and fiber-reinforced plates. Identical upper and lower bounds are used to show that the “membrane solution” gives a minimum volume for a certain class of Tresca sandwich-plates.