Saint-Venant's Theorem

In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle.E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966 It is named after the French mathematician Adhémar Jean Claude Barré de Saint-Venant.

Given a simply connected domain ''D'' in the plane with area ''A'', $\rho$ the radius and $\sigma$ the area of its greatest inscribed circle, the torsional rigidity ''P''
of ''D'' is defined by

:$P= 4\sup_f \frac.$

Here the supremum is taken over all the continuously differentiable functions vanishing on the boundary of ''D''. The existence of this supremum is a consequence of Poincaré inequality.

Saint-Venant conjectured in 1856 that
of all domains ''D'' of equal area ''A'' the circular one has the greatest torsional rigidity, that is

:$P \le P_ \le \frac{2 \pi}.$

A rigorous proof of this inequality was not given until 1948 by Pólya. Another proof was given by Davenport and reported in.G. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951). A more general proof and an estimate
:$P< 4 \rho^2 A$

is given by Makai.

Notes

Elasticity (physics)
Calculus of variations
Inequalities
Physics theorems