Saint-Venant's theorem

From Wikipedia, the free encyclopedia

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= \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}

Saint-Venant 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 < 4ρ2A

is given by Makai[1]

[edit] References

  1. ^ 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
  2. ^ A J-C Barre de Saint-Venant, Memoire sur la torsion des prismes, Memoires presentes par divers savants a l'Acaddmie des Sciences, 14 (1856), pp. 233--560.
  3. ^ G. Polya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267 277.
  4. ^ G. Polya and G. Szego, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).