# Saint-Venant's theorem 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. 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 and {displaystyle sigma } the area of its greatest inscribed circle, the torsional rigidity P of D is defined by {displaystyle P=4sup _{f}{frac {left(iint limits _{D}f,dx,dyright)^{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 conjectured in 1856 that of all domains D of equal area A the circular one has the greatest torsional rigidity, that is {displaystyle Pleq P_{text{circle}}leq {frac {A^{2}}{2pi }}.} A rigorous proof of this inequality was not given until 1948 by Pólya. Another proof was given by Davenport and reported in. A more general proof and an estimate {displaystyle P<4rho ^{2}A} is given by Makai. Notes ^ Jump up to: a b E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966 doi:10.1007/BF01894885 ^ A J-C Barre de Saint-Venant,popularly known as संत वनंत Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560. ^ G. Pólya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267, 277. ^ G. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951). Categories: Elasticity (physics)Calculus of variationsInequalitiesPhysics theorems

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