Equilibrium problem for a Timoshenko plate contacting by the side and face surfaces

A new model for a Timoshenko plate contacting by the side surface or the edge of the bottom surface (with respect to the chosen coordinate system) with a rigid obstacle of a given configuration is justified. The non-deformable obstacle is defined by a cylindrical surface, the generators of which are perpendicular to the middle plane of the plate, as well as by a part of the plane that is parallel to the middle plane of the plate. A corresponding variational problem is formulated as a minimization of an energy functional over a non-convex set of admissible displacements. The set of admissible displacements is defined taking into account a condition of fixing and a nonpenetration condition. The nonpenetration condition is given as a system of inequalities describing two cases of possible contacts of the plate and the rigid obstacle. Namely, these two cases correspond to different types of contacts by the plate side edge and by the edge of the plate bottom surface. The solvability of the problem is established. In particular case, when contact zones is previously known, an equivalent differential statement is found under the assumption of additional regularity for the solution to the variational problem.