Abstract A new theorem for elastic structures obeying Hooke’s law is proved and enunciated as follows: If the strain energy can be expressed in terms of given external loads and the positions of points constrained to have no displacement, then the expression will have a maximum value when the positions are such as make the constraining forces zero. The theorem is generalized to allow a wider class of constraint, such as the attachment of a second, rigid structure to the elastic structure under analysis, in such a way that the two have certain displacement components in common. The principle has been applied by E. H. Mansfield to certain problems in the bending of plates.