Beam-column elements, discretized with elastoplastic fibers at the section level to account for material nonlinearities and equipped with corrotational transformation at the element level to account for geometrical nonlinearities, constitute a powerful tool in the analysis of structural engineering systems. Based on the interpolated fields included in the description of their underlying variational principle, various formulations exist, giving rise to displacement-based, force-based, mixed, and hybrid elements.

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The major goal of all these formulations is to provide elements that accurately capture nonlinear responses without excessively encumbering computational complexity. As a result of the Newton schemes which are predominant in their numerical solutions, the major driver of complexity is the inversion of global or elemental matrices (cubic complexity with the degrees of freedom, under standard inversion methods). The dimensions of these matrices depend on the degrees of freedom of the model and, hence, on the number of unknown variables to be determined (this is greatly associated with the mesh discretization density).

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Based on this remark, there are at least two restrictions we can lift: we can develop efficient numerical schemes beyond Netwon-based methods to reduce the need for costly matrix inversions, or develop formulations that reduce the number of unknown variables involved in inversions without of course compromising accuracy, or, at best, combine both the previous.

In this research, the analysis problem is formulated as a nonlinear programming problem where the saddle point of a discretized variational principle is sought. This renders the problem solvable by the abundance of optimization schemes existing in the literature, also beyond Newton schemes. To reduce the number of unknowns, exact beam kinematics following the Reissner-Simo theory are utilized, which allow us to avoid the inaccuracies introduced by corrotational transformations and reduce the discretization density.

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The performance of the element based on the above principles can be seen below in various benchmark applications. Comparisons with the OpenSees corrotational flexibility elements and Abaqus quadrilateral elements are provided to assess the accuracy of the results. Results include elastic and inelastic cases, with or without shear deformable sections.

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