Asymptotic Approach to Oblique Cross Sectional Analysis of Beams

Abstract

Structural and aeroelastic analyses using beam theories by default choose a cross section that is perpendicular to the reference line. In several cases, such as swept wings with high AR, a beam theory that allows for the choice of a cross section that is oblique to the reference line may be more convenient. This work uses the variational asymptotic method (VAM) to develop such a beam theory. The problems addressed are the planar deformation of a strip and the full 3D deformation of a solid, prismatic, rightcircular cylinder, both made of homogeneous, isotropic material. The motivation for choosing these problems is primarily the existence of 3D elasticity solutions, which comprise a complete validation set for all possible deformations and which are shown to be accurately captured by the current analysis. A secondary motivation was that the development and final results of the beam theory, i.e., the crosssectional stiffness matrix and stressstraindisplacement recovery relations, are obtainable as closedform analytical expressions. These results, coupled with the VAMbased beam analysis being devoid of ad hoc assumptions, culminate in what is expected to be of significance when formulating a general oblique crosssectional analysis for beams with anisotropic material and initial curvature/twist, the detailed treatment of which will be alluded to in a later paper.