A couple contact loading at the unilateral contact of beams

The contact problem for the structure consisting of two beams is considered. The beams have the different lengths and the different variable thicknesses. One end of the shorter beam is clamped coinciding with the hinged end of the longer beam. The other ends of the beams are free. The given loading is applied to the longer beam. The beams undergo the weak joint bending with the unilateral (receding) contact. There is no friction between the beams. The bending of each beam is described by Bernoulli–Euler model. The contact problem is to find the contact loading, i.e. the forces of interaction of beams. This problem has a number of well-known characteristic features. Some of them inhere in the contact problems for the beam structures on the whole. The others inhere in the structures containing the beam that cannot be in the equilibrium for the arbitrary loading. Besides, this problem has the novel peculiarity consisting in the appearance of the concentrated couple in the contact loading. The non-negativity of the contact loading, as the necessary condition of the unilateral contact, is not spoiled because the concentrated couple is at the end of the beams and its “negative part” is located outside the beams and does not belong to the contact loading. The mathematical formulation of the contact problem is propounded. The uniqueness of the solution of this problem is proved. The analytical solution is constructed in some special cases. The relation between the problem under consideration and the well-known contact problem for two cantilever beams is established.