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

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

2009-12-14

arxiv.org/abs/0912.2622v4

Physics

Mathematical Physics

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Scientific paper

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.

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