The wedge subjected to tractions: a paradox re-examined

Abstract

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2α satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for α not equal but very close to 235-3, the classical solution can still be very large as α approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as α approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2α=π and 2π. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r ^-λ and ln r are replaced by R ^-gl and ln R where R=r/r ₀is the dimensionless radial distance and r ₀ is an arbitrary constant having the dimension of length.