Uniqueness Theorem for Nonlinear Plane Frames

A plane structural framework is considered subject to loading by concentrated or distributed forces or concentrated moments, or prescribed distributed curvature, displacement or rotation (absolute or relative) at the joints. All prescribed quantities are in the plane of the frame. The deformations are assumed to be small, and deformations caused by shear and normal forces are neglected. The moment is assumed to be a monotonically increasing function (with a finite number of discontinuities admitted) of the curvature of each infinitesimal beam element. This frame problem, typical for example of a plane concrete frame subject to miscellaneous loading, support settlements, etc., is presumed to admit a solution; it is shown that such a solution is unique. This theorem of uniqueness is a useful adjunct with certain non-incremental methods of analysis in the literature. A simple continuous beam example illustrates uniqueness in a way that appeals to engineering intuition.