Mixed beam formulation with cross-section warping for dynamic analysis of thin-walled structures

This paper presents the formulation of a three-dimensional beam finite element (FE) that accounts for cross-section warping and dynamic inertia effects. The model is the extension of an existing mixed formulation, originally developed for the static analysis of thin-walled beams, to the case of dynamic loading conditions. Four independent fields are considered to derive the element governing equations, i.e. material rigid displacements, strains and stresses and an additional displacement field, describing the out-of-plane warping displacement of the beam cross-sections. The latter is independently interpolated in the element volume by including additional degrees of freedom (DOF) to the nodal translations and rotations classically considered in beam formulations. To obtain a consistent form of the element mass matrix, the cross-section displacement shape functions are computed, relating the generalized cross-section displacement fields to the element nodal variables. In mixed FE formulations, these are not assigned a priori, as in displacement-based approaches, but are derived on the basis of material stiffness and element geometry, together with compatibility conditions. Thus, the Unit Load method is applied to deduce the expressions of the shape functions consistent with the force-based approach, assuming the simply-supported beam as reference element configuration. As opposed to the original FE model, the additional warping DOFs are not condensed-out with the definition of the element quantities but are treated as additional global unknowns. This permits a correct description of the inertia effects and ensures continuity of the warping displacement fields between adjacent FEs. Correlation studies are presented to validate the proposed model and investigate the effects of cross-section warping on the dynamic behavior of thin-walled structures. For selected specimens, the studies compare solutions obtained adopting the proposed beam element with those resulting from shell or brick FE models. Modal decompositions and time-history analyses are conducted, assuming both linear elastic and nonlinear constitutive behavior for the latter.