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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 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.
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https://hal.archives-ouvertes.fr/hal-00502994
Contributor : Francesco Dell'Isola <>
Submitted on : Friday, July 16, 2010 - 12:11:26 PM
Last modification on : Thursday, July 29, 2010 - 6:50:36 PM
Long-term archiving on: : Thursday, December 1, 2016 - 2:08:47 PM

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  • HAL Id : hal-00502994, version 1

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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⟩

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