A problem of unilateral contact of two hanging chains

The unilateral contact of two closely hanging chains (heavy inextensible strings) under the action of gravity and small horizontal loading is considered. Each chain has one end fixed and the other free. The chains have different lengths; the linear densities of the chains may be variable. The horizontal loading slightly deflects the chains from the vertical line, so that the system is geometrically linear. The basic problem is to find the shapes of the chains. This problem is reduced to the problem of finding the density of the forces of interaction between the chains. The accurate formulation of the latter contact problem is propounded. The unknown density of the contact forces is assumed to be the sum of piecewise continuous (and one-sided continuous) function and delta functions describing the concentrated forces. The uniqueness of the solution of the problem is proved. The analytical solution is constructed in some special cases. The following contact patterns are found out: contact at one point, contact along the full length of the shorter chain, contact along a part of the shorter chain, contact at a point and along a part of the shorter chain.