On shear and torsion factors in the theory of linearly elastic rods

TL;DR

This paper establishes lower bounds for shear and torsion factors in linearly elastic rods using a new criterion based on energy equivalence, confirming that shear factors exceed one and torsion factors exceed one except for circular cross sections.

Contribution

It introduces a simple, new method to derive bounds for shear and torsion factors, providing proofs that are more straightforward than traditional approaches.

Findings

Shear factor is always greater than one for any cross section.

Torsion factor exceeds one except for circular or annular cross sections.

The new criterion links stored-energy densities in rod theory and Saint-Venant cylinder solutions.

Abstract

Lower bounds for the factors entering the standard notions of shear and torsion stiffness for a linearly elastic rod are established in a new and simple way. The proofs are based on the following criterion to identify the stiffness parameters entering rod theory: the rod's stored-energy density per unit length expressed in terms of force and moment resultants should equal the stored-energy density per unit length expressed in terms of stress components of a Saint-Venant cylinder subject to either flexure or torsion, according to the case. It is shown that the shear factor is always greater than one, whatever the cross section, a fact that is customarily stated without proof in textbooks of structure mechanics; and that the torsion factor is also greater than one, except when the cross section is a circle or a circular annulus, a fact that is usually proved making use of Saint-Venant's solution in terms of displacement components.