Energy Transformation to Generalized Timoshenko Form by the Variational Asymptotic Beam Sectional Analysis

Abstract

The 2D beam cross-sectional analysis tool VABS finds stiffness constants for the generalized Timoshenko theory by an energy transformation from a 2-order asymptoticallycorrect strain energy. Two inconsistencies in the transformation used in previous versions are identified. These are resolved by re-deriving the energy transformation and formulating new solution procedures based on the revised transformation. An iterative method and a perturbation-based analytical solution are each presented to solve for the stiffness constants from the resulting equations. A comparison between results obtained from the new solutions and those from a previous version of VABS shows that this improved methodology sometimes changes the results measurably. As a form of validation and an internal consistency check, results obtained using curvilinear coordinate axes featuring twist for the 1D analysis of non-rotating, initially twisted, isotropic beams are shown to be equivalent to those obtained using Cartesian coordinates. Similarly, results obtained from a 1D analysis of rotating, initially twisted, isotropic beams using a set of piecewise Cartesian coordinates are shown to be equivalent to those from calculations using a global set of Cartesian coordinates.