Mixed methods and flexibility approaches for nonlinear frame analysis

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The principle of virtual forces has long been an attractive tool for linear matrix structural analysis because it provides a means to compute the exact flexibility matrix for a linear, elastic, non-prismatic Bernoulli–Euler beam element while the classical stiffness method, based on the principle of virtual displacements, does not. Nonlinear flexibility methods have recently appeared in the literature as a means of improving the accuracy of frame analysis when the curvature fields are more complex than the moment fields (e.g., when inelastic response is important). This paper surveys the basic methods of formulating nonlinear Bernoulli–Euler beam elements based on classical and mixed variational principles. We show that the variational structure provided by the Hellinger–Reissner and Hu–Washizu functionals gives a framework that reaps the benefits of the so-called nonlinear flexibility methods while allowing a standard treatment of non-linear problems by Newton's method.