Saint–Venant's theorem Information & Saint–Venant's theorem Links at HealthHaven.com

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 , $ρ$ the radius and $σ$ the area of its greatest inscribed circle, the torsional rigidity P of D is defined by

$P= 4\,\mathrm{sup}_f \frac{\left( \int\int\limits_D f\, dx\, dy\right)^2}{\int\int\limits_D f_x^2+f_y^2\, dx\, dy}.$

here the suppremum is taken over all the continuously differentiable functions vanishing on the boundary of D. The existence of this suppremum 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

$P \le P_{\mathrm{circle}} \le \frac{A^2}{2 \pi}$

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

$P< 4 \rho^2 A\$

is given by Makai.^[1]

[edit] Notes

^ ^a ^b E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966doi:10.1007/BF01894885
^ A J-C Barre de Saint-Venant, Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560.
^ G. Polya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267 277.
^ G. Polya and G. Szego, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).

[Makai-0] E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966doi:10.1007/BF01894885

[1] A J-C Barre de Saint-Venant, Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560.

[2] G. Polya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267 277.

[3] G. Polya and G. Szego, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).

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