Saint Venant’s Principle: A Biharmonic Eigenvalue Problem

Saint Venant’s principle is formulated as a biharmonic eigenvalue problem for the symmetrical truncated wedge x ≥ 0, |y| ≤ y_b (y_b = 1 + x tan ω) which is stress-free along the lateral edges, and is loaded by self-equilibrating shear and normal tractions τ_k^o(y), σ_k^o(y) along edge x = 0. It is found that for the wedge angle 2ω = 0 the law of decay is of the type e^−α_kτ; for 2ω ≠ 0 the law of decay is of type r^−μk; μ_k is minimum for 2ω = π. The indexes k attached to τ^o, σ^o 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.”