Saint-Venant's principle, named after

Adhémar Jean Claude Barré de Saint-Venant Adhémar Jean Claude Barré de Saint-Venant (23 August 1797 – 6 January 1886) was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as ...

, a

French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...

elasticity theorist, may be expressed as follows: The original statement was published in French by Saint-Venant in 1855. Although this informal statement of the principle is well known among structural and mechanical engineers, more recent mathematical literature gives a rigorous interpretation in the context of partial differential equations. An early such interpretation was made by

Richard von Mises Richard Edler von Mises (; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordo ...

in 1945.R. von Mises
On Saint-Venant's Principle
Bull. AMS, 51, 555–562, 1945. The Saint-Venant's principle allows elasticians to replace complicated stress distributions or weak boundary conditions with ones that are easier to solve, as long as that boundary is geometrically short. Quite analogous to the

electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...

, where the product of the

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

and

electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

due to the ''i''-th moment of the load (with 0th being the net charge, 1st the

dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...

, 2nd the quadrupole) decays as

1/r^

over space, Saint-Venant's principle states that high order moment of mechanical load (moment with order higher than

torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...

) 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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...

behavior of the

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...

by a point-load.

References

Elasticity (physics) Principles {{math-physics-stub

See also

References