Some Remarks on the Wedge Paradox and Saint-Venant’s Principle

Abstract

The solutions for the wedge loaded by a concentrated couple at its vertex, known as the Carothers-Sternberg-Koiter problem, are examined critically. The cause of the divergence are terms which are fundamental self-equilibrated solutions with unbounded energy arising from the eigenvalues of the wedge and with exponent depending on the angle. The “paradox” occurs at the angle at which the term of the (angle-independent exponent) r−2 stress singularity is superseded by the next eigenvalue. These terms are conjectured to be the weight functions of the wedge. The breaking down of St. Venant’s principle for the wedge in connection with these terms is also demonstrated. As for the definition of a concentrated couple, it can only exist if the limiting solutions are independent of the path of loading (stable node); for wedge angles bigger than a half-space, they depend on the path of loading, and the loadings that give path independence are characterized.