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

Francesco Dell'Isola (1, 2) , Giuseppe C. Ruta (1)

1 Dipartimento di Ingegneria Strutturale e Geotecnica
2 Laboratorio Strutture e Materiali Intelligenti - Fondazione Tullio Levi-Civita

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 Dε whose thickness we scale by a parameter ε. For each ε we build a coordinate mapping from a fixed plane domain D onto Dε. The shear stress field in Dε can be rapresented by a Prandtl-like stress flow function [5, 6]. This is naturally done in torsion (torsion, [1]), while in flexure (flexure inégable, [1]) we face a gauge choice where 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 problem 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 attemp 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.

Domains: Mechanics [physics.med-ph]

Francesco Dell'Isola, Giuseppe C. Ruta. Outlooks in Saint-Venant theory III. Torsion and flexure in sections of variable thickness by formal expansions. Archives of Mechanics, 1997, pp.23. ⟨hal-00502994⟩