On Kelvin's formula for torsion of thin cylinders

As it is well known, there is in general no closed-form solution for either the shear stress or the longitudinal displacement in the torsion of a Saint-Venant cylinder. For technical purposes, approximate formulae for cylinders with thin-walled sections are at disposal. In the case of open sections (i. e., sections which are topologically equivalent to a rectangle) the (dominant component of the) shear stress turns out to be an affine function (with zero mean) of the coordinate along the thickness. Kelvin & Tait pointed out that this formula is not sufficient to determine the torque resisted by a cylinder with rectangular sections. Indeed, the dominant component of the stress is equivalent only to one half of the applied torque; the other half is carried by a secondary component of the shear stress near the edges. This seems paradoxal, as the secondary component of the stress is negligible in magnitude with respect to the dominant, as was also proved by means of formal expansions. Popov helped to clarify the matter, showing that the secondary component of the shear stress becomes relevant near the 'short' edges of the section and that any open section behaves like the rectangular one. In this paper a simple geometrical description of an open section of a Saint-Venant cylinder of arbitrary (but regular) varying thickness is introduced. By means of simple differential geometry techniques and of the fields equations of Saint-Venant problem. Kelvin's and Popov's formulae are immediately obtained and generalized in a straightforward way.