From Sokolnikoff, I. S. (1941). Mathematical theory of elasticity. Brown University.
It is obvious from the formulation of the fundamental boundary value problems of the theory of Elasticity that the exact solution of these problems is likely to present formidable mathematical difficulties because of the complicated form of the boundary conditions. Frequently it is possible to obtain a solution of the problem if the boundary conditions are slightly modified, and it is worth noting that in the technological applications of the theory of Elasticity one can only approximate the mathematical formulation of the boundary conditions, so that the mathematical solution of the problem represents only an approximation to the actual situation.
In 1855 B. de Saint Venant expressed a principle which agrees well with the applications of the theory of Elasticity to practical problems. The essence of the principle can be stated as follows:
If some distribution of forces acting on a portion of the surface of a body is replaced by a different distribution of forces, acting on the same portion of the body, then the effects of the two different distributions on the parts of the body sufficiently far removed from the region of application of the forces are essentially the same, provided that the two distributions of forces are statically equivalent.
The phrase “statically equivalent” means that the two distributions of forces have the same resultant force and the same resultant moment.
To illustrate the meaning of the principle, consider a long beam one end of which is fixed in a rigid wall, while the other is acted upon by a distribution of forces which gives rise to a resultant force F and a couple of moment M.
Now there are infinitely many distributions of forces that may act on the end of the beam and that will have the same resultant F and the same resultant moment M.
The principle of Saint Venant asserts that while the distributions of stresses and strains near the region of application may differ greatly, the eccentricities of the local distribution will have no appreciable effect on the state of stress far enough from the points of application, so long as the systems of applied forces are statically equivalent.
This principle is of great usefulness in practical applications since it permits one to alter the boundary conditions and thus simplify the problem.
One would suspect from the generality of the statement of the Principle that the latter is not easy to justify in all cases on purely mathematical grounds. In specific instances one can calculate the distribution of stresses produced by various statically equivalent systems of forces, and in problems on beams, for example, it is reasonable to assume that the local eccentricities are not felt at distances which are about ten times the greatest linear dimension of the area over which the forces are distributed.
We shall make some use of this principle in the next chapters.