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, right-circular 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 cross-sectional stiffness matrix and stressstrain-displacement recovery relations, are obtainable as closed-form analytical expressions. These results, coupled with the VAM-based beam analysis being devoid of ad hoc assumptions, culminate in what is expected to be of significance when formulating a general oblique cross-sectional analysis for beams with anisotropic material and initial curvature/twist, the detailed treatment of which will be alluded to in a later paper.In beam theory, a natural choice of the reference crosssectional plane is the one normal to the reference line. However, using a cross section that is not perpendicular to the beam reference line is of interest in several cases, notable of which are swept wings (see Fig. 1). Relaxing the necessity of a section to be perpendicular to the reference line could also be convenient for users. An oblique cross-sectional analysis is therefore a dimensional reduction that uses a cross section that is not constrained to be perpendicular to the reference line. A review of literature reveals that studies pertaining to this problem are limited. A previous work on oblique cross-sectional analysis [12] was only able to correctly construct the classical model in VABS. Borri et al.[13] mention the possibility of "a cross section which could be tilted with respect to the reference line" and though it may be that Borri and Journal of Applied Mechanics

The dimensional reduction approach for 2D non-prismatic beam modelling: A solution based on Hellinger–Reissner principle

“…Starting from this model, Hodges et al (2010) provide ways to recover stresses and strains, and they have shown the excellent accuracy of the model. More recently, Rajagopal et al (2012) have used the variational-asymptotic method for the analysis of initially curved isotropic strips, whereas Rajagopal and Hodges (2014) have used the same approach to perform oblique cross-section analysis. In both papers, the proposed approaches lead to accurate and promising results, although the generalization to non-prismatic beams does not seem to be available.…”

Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

“…The adopted geometrical description implies that cross-sections are non-orthogonal to the beam's neutral axis. Instead, orthogonality is preserved in the Cartesian coordinate system, noting similar methodology is adopted in related works (Balduzzi et al 2016;Gimena et al 2008;Rajagopal and Hodges 2014;Balduzzi et al 2017a, c, d;Mercuri et al 2020;Vilar et al 2021a). Consequently, the cross-sectional layer area A (k) and total cross-sectional area A are given by The domain of each layer Ω (k) is then defined by…”

“… 2016 ; Gimena et al. 2008 ; Rajagopal and Hodges 2014 ; Balduzzi et al. 2017a , c , d ; Mercuri et al.…”

Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix

“…Such a modeling choice is substantially different from standard modeling approach used for curved beams where the cross-section is assumed to be perpendicular to the centerline [2, Chapter IV], [33]. However, as discussed in [54], such an exotic choice may have advantages in several situations. Certainly, it allows to bypass the need of parametric definition of the beam centerline and the use of additional local coordinate systems, with the associated coordinate transformations and derivative chains.…”