Saint-Venant's principle

Remarks

As stated, this is a heuristic principle used by engineers, not a theorem: the words "small", "large" and "negligible" are not made precise. Mathematical versions exist for particular geometries. Toupin proved such a version for a prismatic cylinder loaded on one end by self-equilibrated tractions. The strain energy stored in the part of the cylinder beyond distance $s$ from the loaded end decays exponentially in $s$. The rate depends on the cross-section and the elastic constants. The principle justifies replacing complicated end loads on a beam by their resultant force and moment, as in beam theory and in the Saint-Venant torsion and bending solutions. The self-equilibrium hypothesis is necessary: a single point force on a half-space produces stresses that decay only like the inverse square of the distance, as the Boussinesq problem shows.

Going deeper

For a student

In practice the principle lets one solve a simple problem, with idealised end loads, and trust the solution away from the ends. The region where the details matter has a size comparable to the loaded region, for example one or two depths of a beam. Its rigorous forms are decay estimates for solutions of the elasticity equations with zero resultant data, close in spirit to Phragmén–Lindelöf type estimates for elliptic equations.