The Vlasov Theory of the Variational Asymptotic Beam Sectional Analysis

Abstract

The variational-asymptotic method has been established as an ideal means for splitting the geometrically-nonlinear, three-dimensional elasticity problem into a linear, twodimensional cross-sectional analysis and a nonlinear one-dimensional beam analysis. The zeroth-order analysis is a so-called classical theory, with four one-dimensional generalized strain measures for extension, torsion, and bending in two directions. Alternatively, certain problems call for development of refined theories. For beams undergoing shortwavelength motion it is appropriate to include two one-dimensional transverse shear measures, making a generalized Timoshenko theory. For strips and open-section beams, it is generally accepted that a one-dimensional variable allowing for approximate treatment of end effects is needed, such as the Vlasov correction. Recently, it was shown that the Vlasov correction for thin-walled open sections to the beam theory based on the St. Venant principle can be rigorously derived using the variational-asymptotic method. This has provided a solid foundation for a consistent extension of Vlasov beam theory to beams with general geometric and material cross-sectional properties. The computer program VABS, a general-purpose, finite-element based beam cross-sectional analysis under development for over a decade, now contains a Vlasov correction, a comprehensive validation of which has not yet been published. This methodology is summarized herein and is employed to solve several benchmark problems. Not only are beam results presented, but also the recovered three-dimensional stress distributions are compared with results from a commercial three-dimensional finite element code. The present work focuses on the issues concerning the use of the Vlasov correction in the context of the accuracy of the resulting beam theory. The systematic comparison with three-dimensional finite element analysis results helps to quantitatively demonstrate both the advantages of the Vlasov correction and its limitations. In addition, the relevance of indirect indications about the importance of the Vlasov effect is discussed.