On Saint-Venant's Principle in Elasticity

Abstract

Publisher Summary The Saint-Venant's Principle is considered from the angle of its present practical interest in Structural Mechanics, especially for composite structures. The interior large wavelength effect must be separated from the edge or extremity effects with a small wavelength to be computed. In a more precise way, it is a theorem that expresses conditions ensuring localization of displacements and stresses. This point of view is built on certain characteristic properties of the solutions and not on the properties of zero resultant-moment loadings. The diameter O of the beam or the thickness 2h of the plate are not considered as small parameters. The approach has nothing to do with asymptotic methods. Beams and plates are studied as far as possible from a unitary point of view. The problem is restricted to the so-called Saint-Venant Problem for which the non-zero loadings and displacements are only prescribed on the edges of the structure. This chapter presents several major properties for the solutions, localized or not; they are inferred from the particularities of the geometry. The localization concepts are stated precisely. The corresponding solutions decrease exponentially as a function of the distance to the edges such that the decaying length is 0(O) for beams and 0(h) for plates. The Saint-Venant Principle that characterizes such solutions expresses the orthogonality to the interior large wavelength solutions.