Augmented beam elements using unit deflection shapes of the cross section

Abstract

Wave behavior of line-shaped structures can be analyzed with the help of beam theories if the structures have a simple cross section (beams, rods, etc.), and if the frequency or wavelength is within a range where the theories are still applicable. If the frequencies and wavelengths exceed these limits, the modeling of line- or beam-shaped structures requires volume elements, which results in considerably higher computation time and modeling effort. The goal of this work is to keep the beam-like modeling of structures, but enhance the solution space such that arbitrary deflection shapes of the cross section can be covered. New degrees of freedom are introduced which correspond to the contribution of a unit deflection shape at each node. The unit deflection shapes are defined and computed on the basis of a 2D finite element mesh of the cross section: warp shapes for primary and secondary torsion and shear forces, eigenmodes for the problem of a plate in membrane and a plate in bending action, derived warp shapes for in-plane shapes, shapes which cover the lateral strains and—most importantly—eigenmodes of the infinite waveguide structure. By this transformation of unknowns, the number of system degrees of freedom can be reduced considerably. Different sets of unit deflection shapes are compared according to their efficiency.