Verification of the Variational-Asymptotic Sectional Analysis for Initially Curved and Twisted Beams

Abstract

The variational–asymptotic method is a powerful mathematical method to construct asymptotically correct models of dimensionally reducible structures. Variational–asymptotic method is used to rigorously split a general three-dimensional nonlinear elasticity problem into a two-dimensional linear cross-sectional analysis and a onedimensional nonlinear beam analysis. This operation results in the replacement of a three-dimensional model with a reduced-order model in terms of an asymptotic series of certain small parameters inherent to the structure. Variational–Asymptotic Beam Sectional Analysis (VABS) is an engineering software package that uses the variational–asymptoticmethod to perform a generalized cross-sectional modeling of any initially twisted and curved anisotropic beam having arbitrary cross-sectional geometry, with transverse shear and Vlasov refinements. Several new cases of verification are presented. The capability of VABS to recover pointwise three-dimensional stressfields is demonstrated for an orthotropic cantilever and verified against known solutions from anisotropic elasticity theory. Next, a form of verification is proposed for beams with either initial twist or curvature by using two different modeling approaches: one based on curvilinear coordinates and the other on Cartesian coordinates for initially twisted beams or piecewise-Cartesian coordinates for initially curved beams. Excellent correlation of results between the two approaches indicates the correct modeling of effects of initial curvature and twist in both cross-sectional and global beam analyses. Further, a numerical verification of results for initially twisted and/or curved beams is provided by comparing values of one-dimensional variables extracted from three-dimensional finite element analyses with results from a one-dimensional, nonlinear beam analysis (using cross-sectional constants fromVABS). Finally, numerical examples are shown to highlight the importance of including curvature measures in the crosssectional modeling by comparing with finite element results.