A contact problem of bending of two beams with the internal joint

Abstract

The joint bending of two Bernoulli–Euler’s beams is considered. Each beam has one end fixed and the other free. The beams have the different lengths and thicknesses. The long beam is loaded by the concentrated force. This beam is composite as it includes the internal joint. There is the frictionless unilateral contact between the beams. The elastic lines of the beams are to be found. This problem is reduced to finding of the density of forces of interaction between the beams and the constant that describes the unknown term in the displacement of the unrestrained part of the composite beam. The mathematical formulation of this contact problem is propounded. The density is assumed to be the sum of piecewise continuous function and delta-functions describing the concentrated forces. The uniqueness of the solution of the problem is proved and the analytical solution is constructed. Two possible contact patterns are found out. The former is contact at one point at the end of the short beam. The latter is contact at the same point and at one more point located at the unrestrained part of the composite beam. The coordinate of this point is the root of the cubic equation. The obtained analytical solution is used for the optimization of the structure. The optimization problem is to find the beams thicknesses that minimize the maximum stress for the given loading, beams lengths and the overall deflection. This problem is solved numerically for some values of the given parameters. The hypothesis of the equal-stressed optimum structure is set up on the basis of the numerical results. This hypothesis enables to construct the analytical solution of the optimization problem.