Outlooks in Saint-Venant theory III. Torsion and flexure in sections of variable thickness by formal expansions

Abstract

We study the Saint-Venant shear stress fields [1] arising in a family of sections we call Bredt-like [2, 3, 4], i.e. in a set of plane regions D_ε whose thickness we scale by a parameter ε. For each ε we build a coordinate mapping from a fixed plane domain D onto D_ε. The shear stress field in D_ε can be represented by a Prandtl-like stress flow function [5, 6]. This is naturally done in torsion (torsion, [1]), while in flexure (flexion inégale, [1]) we face a gauge choice whose physical interpretation is uncertain [6]. We then consider the Helmholtz operator in a fixed system of coordinates in D and represent the shear stress field in a basis field which is not the covariant basis associated to any coordinate system. Formal ε-power series expansions for the shear stress field, the warping, the resultant force and torque and the shear shape factors tensor lead to hierarchies of perturbation problems for their coefficients. We obtain all the technical formulae at the lowest iteration steps and their generalization at higher steps - i.e., for thicker sections. No attempt is made to apply the methods proposed in [16] to estimate the distance between the generalized formulae we provide and the true solutions for the Saint-Venant shear stress problem.