Variational Basis of Nonlinear Flexibility Methods for Structural Analysis of Frames

There have been a number of contributions to the literature on a class of structural analysis methods referred to as nonlinear flexibility methods. These methods appear to perform very well compared to classical stiffness approaches for problems with constitutive nonlinearities. Although most of these methods appeal to variational principles, the exact variational basis of these methods has not been entirely clear. Some of them even seem not to be variationally consistent. We show in this paper that, because the equations of equilibrium and kinematics are directly integrable, a nonlinear flexibility method (in the spirit of those presented in the literature) can be derived without appeal to variational principles. The method does not involve interpolation of the displacement field and the accuracy of the method is limited only by the numerical scheme used to perform element integrals. There is no need for $h$ refinement to improve accuracy. Further, we show that this nonlinear flexibility method is essentially identical, with some subtle algorithmic differences, to a two-field (Hellinger-Reissner) variational principle when the stress interpolation is exact (which is possible for this class of problems). We demonstrate the utility of the nonlinear flexibility method by applying it to a problem involving cyclic inelastic loading wherein the strain fields evolve into functions that are difficult to capture through interpolation.