Singularities of contact problems for systems of strings and beams with weakly restrained elements

In known contact problems for strings and beams, contact elastic elements are restrained in such a way that each of them can be in the equilibrium for the arbitrary applied load. However, other types of restraint are possible where the above-mentioned condition does not hold for one of the elements; this element can be called weakly restrained. At the same time, the system as a whole can be in the equilibrium for the wide set of external loads. The systems of this type have some singularities in the contact problem statement, in the proof of the uniqueness of solution and in constructing the analytical solution. The singularities are as follows: the special condition for the weakly restrained element equilibrium, the additional unknown parameter describing the indefinite part of the weakly restrained element displacement, the necessity to prove the uniqueness of not only the contact forces but also this parameter, the expansion of the set of allowable contact loads, the exclusion of zero external loads. These singularities are considered for two following examples: 1) two strings; the first one has the free edges; 2) two beams with the gap between them; the first one has the hinted bearing and the free edge. The modification of the usual plan of consideration of contact problems for strings and beams is proposed. This modification keeps the main ideas of the usual plan and gives the opportunity to take the weakly restrained elements into account. For each example, the uniqueness of the solution of the contact problem is proved and the analytical solution is built.