Saint Venant’s Principle: A Biharmonic Eigenvalue Problem

Abstract

Abstract Saint Venant’s principle is formulated as a biharmonic eigenvalue problem for the symmetrical truncated wedge x ≥ 0, |y| ≤ yb (yb = 1 + x tan ω) which is stress-free along the lateral edges, and is loaded by self-equilibrating shear and normal tractions τko(y), σko(y) along edge x = 0. It is found that for the wedge angle 2ω = 0 the law of decay is of the type e−αkτ; for 2ω ≠ 0 the law of decay is of type r−μk; μk is minimum for 2ω = π. The indexes k attached to τo, σo indicate that we deal with systems of characteristic tractions which produce characteristic decay rates. Practical implications of the results, as they apply to structural design, are discussed in a separate paper, “Some Aspects of Saint Venant’s Principle.”