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.^[1] 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, ${\displaystyle \rho }$ the radius of its greatest inscribed circle, the torsional rigidity P of D is defined by

${\displaystyle P=4\sup _{f}{\frac {\left(\iint \limits _{D}f\,dx\,dy\right)^{2}}{\iint \limits _{D}{f_{x}}^{2}+{f_{y}}^{2}\,dx\,dy}}.}$

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^[2] conjectured in 1856 that of all domains D of equal area A the circular one has the greatest torsional rigidity, that is

${\displaystyle P\leq P_{\text{circle}}\leq {\frac {A^{2}}{2\pi }}.}$

A rigorous proof of this inequality was not given until 1948 by Pólya.^[3] Another proof was given by Davenport and reported in.^[4] A more general proof and an estimate

${\displaystyle P<4\rho ^{2}A}$

is given by Makai.^[1]