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

On the Importance of Cross-Sectional Distortions and their Inclusion in an Efficient GBT-Formulation

“…There, a comma in the index refers to partial derivative and c mnkl denotes the elasticity tensor, where we assume one-dimensional elasticity for each stress component. A straight forward derivation detailed in [1] leads to GBT beam element stiffness and mass matrices, which are assembled to the global problem definition of the true beam problem. Physical convergence of the GBT results compared to continuum results is ensured by the number of contributing warping and distortion couples i max , while numerical convergence occurs due to the number of semi-analytical finite elements in the cross-sectional analysis and due to the number of GBT beam elements in the member analysis.…”

“…It is shown in [1,4] that the proposed solution algorithm improves the performance of CBT significantly, especially in cases where thin-walled box type cross-section are considered and torsional loading is applied. Additionally, our GBT formulation enhances the predictive quality in beam problems with unsymmetric cross-sections where Schramm et al [8,9] observed that shear elastic transverse bending load cases do not decouple based on the eigenvector of the bending stiffness tensor.…”

“…A coupling of membrane and bending part is not required (and membrane locking is therefore circumvented by principle), if the line mesh of the semi-analytical finite shell element is straight. Transverse shear locking is circumvented by using cubic shape functions, while the principal ideas of the GBT formulation originated in [1,4] and summarized above remain unchanged. Consider a four noded semi-analytical finite shell element (SAFSE) of constant wall-thickness s and elemental length l e depicted in Fig.…”

“…These effects are also observed by Sapountzakis et al [5,6] where also buckling analyses are considered. In order to circumvent the insufficient accuracy of CBM in thin-walled single or multi-cell box type cross-sections, however, maintaining a moderate number of global degrees of freedom, a Generalized Beam Theory (GBT) can be applied (see [1] and [5] for a comprehensive literature review and a discussion of main contributions in that field). After all, GBT is a two-step algorithm consisting of a cross-sectional analysis defining warping and distortional fields, followed by a member analysis, where those deformation fields are weighed axially along the beam.…”

“…After all, GBT is a two-step algorithm consisting of a cross-sectional analysis defining warping and distortional fields, followed by a member analysis, where those deformation fields are weighed axially along the beam. Kugler et al propose a new form of GBT and the main ingredients are summarized subsequently (for additional details refer to [1,4]):…”