Theory of Anisotropic Thin-Walled Beams

TLDR

Prior to this work, no analytical theories for beams with closed cross sections consistently included shell bending strain measures. The study presents an asymptotically correct, linear theory for thin‑walled prismatic beams made of generally anisotropic materials. Using intrinsic small parameters, the authors develop a unified framework for all thin‑walled beams, employing four classical one‑dimensional variables and deriving analytical formulas for the fully populated 4×4 cross‑sectional stiffness matrix and strain field. The theory shows that corrections from shell bending strain measures are significant in some cases, that cross sections are not rigid in their own plane, that Vlasov’s correction is negligible for closed sections but asymptotically correct formulas are provided for open sections, extending prior I‑beam results. S0021-8936(00)03003-8.

Abstract

Asymptotically correct, linear theory is presented for thin-walled prismatic beams made of generally anisotropic materials. Consistent use of small parameters that are intrinsic to the problem permits a natural description of all thin-walled beams within a common framework, regardless of whether cross-sectional geometry is open, closed, or strip-like. Four “classical” one-dimensional variables associated with extension, twist, and bending in two orthogonal directions are employed. Analytical formulas are obtained for the resulting 4×4 cross-sectional stiffness matrix (which, in general, is fully populated and includes all elastic couplings) as well as for the strain field. Prior to this work no analytical theories for beams with closed cross sections were able to consistently include shell bending strain measures. Corrections stemming from those measures are shown to be important for certain cases. Contrary to widespread belief, it is demonstrated that for such “classical” theories, a cross section is not rigid in its own plane. Vlasov’s correction is shown to be unimportant for closed sections, while for open cross sections asymptotically correct formulas for this effect are provided. The latter result is an extension to a general contour of a result for I-beams previously published by the authors. [S0021-8936(00)03003-8]