The use of Castigliano's theorem in Coulomb friction problems

This paper presents a systematic way of analyzing the statics of elastic beams with Coulomb friction by means of Castigliano's theorem. For this purpose, the handling of the equilibrium conditions in Castigliano's theorem is unified first. It is shown that all equations for the unknown reaction forces and the displacements of force application points can be derived from an augmented energy function in which the force and moment balance equations are introduced via Lagrange multipliers. As a consequence, the case differentiation between stick and slip in Coulomb's friction laws can be described in a natural way, where the set of equations determining the unknown reaction forces of an elastic body with several frictional supports is always obtained from the same function for any stick-slip sequence. To demonstrate the benefit of this approach, multiple statically indeterminate, semi-circular beams are discussed.