Beam Dynamics Using a Generalized Beam Theory Based on the Solution of a Reference Beam Problem

Abstract

Mechanical properties of slender, prismatic structures are typically analyzed based on classical beam mechanics (Timoshenko’s shear force bending, Vlasov’s theory of warping torsion, …). There it is assumed that the cross-section remains rigid in its projection plane and in-plane distortional deformations of the cross-section are neglected. Such a model is predictive in case of static gradually distributed loading, and solid cross-sections, however, in case of thin-walled crosssections and dynamic loading severe deviations might occur. Therefore, a generalized beam theory is proposed, where warping fields and accompanied distortional fields of the cross-section are axially distributed each based on one generalized degree of freedom. The evaluation of pairs ofwarping and distortional fields in ascending order of importance is performed using a specific reference beam problem (RBP), where three-dimensional elasticity theory is applied in connection with semi-analytical finite elements (SAFE). Convergence of the resulting formulation is ensured by increasing the number of contributing pairs of warping and distortional fields. The resulting formulation yields significantly better results compared to classical beam mechanics especially in the dynamic regime.