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# Saint-Venant's principle: Wikis

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# Encyclopedia

Updated live from Wikipedia, last check: May 22, 2013 05:34 UTC (47 seconds ago)

Saint-Venant's principle, named after the French elasticity theorist Jean Claude Barré de Saint-Venant can be stated as saying that[1]

"... the strains that can be produced in a body by the application, to a small part of its surface, of a system of forces statically equivalent to zero force and zero couple, are of negligible magnitude at distances which are large compared with the linear dimensions of the part." A.E.H. Love

The original statement was published in French by Saint-Venant in 1855[2]. Although this informal statement of the principle is well known among mechanical engineers, more recent mathematical literature gives a rigorous interpretation in the context of partial differential equations. An early such intepratation was made by von Mises in 1945[3]

The Saint-Venant's principle allows elasticians to replace complicated stress distributions or weak boundary conditions into ones that are easier to solve, as long as that boundary is geometrically short. Quite analogous to the electrostatics, where the electric field due to the i-th momentum of the load ( with 0th being the net charge, 1st the dipole, 2nd the quadrupole) decays as $\,\frac{1}{r^{i+2}}$ over space, Saint-Venant's principle states that high order momentum of mechanical load ( momentum with order higher than torque) decays so fast that they never need to be considered for regions far from the short boundary. Therefore, the Saint-Venant's principle can be regarded as a statement on the asymptotic behavior of the Green's function by a point-load.

## References

1. ^ A.E.H. Love, "A treatise on the mathematical theory of elasticity" Cambridge University Press, 1927. (Dover reprint ISBN 0486601749)
2. ^ A. J. C. B. Saint-Venant, 1855, Memoire sur la Torsion des Prismes, Mem. Divers Savants, 14, pp. 233-560
3. ^ R. von Mises, On Saint-Venant's Principle. , Bull. AMS, 51, 555-562, 1945