A mixed problem of plate bending for doubly connected domains with partially unknown boundaries in the presence of cyclic symmetry

Abstract

This paper addresses the problem of plate bending for a doubly connected body with outer and inner boundaries in the form of regular polygons with a common center and parallel sides. The neighborhoods of the vertices of the inner boundary are equal full-strength smooth arcs symmetric about the rays coming from the vertices to the center, but have unknown positions. Rigid bars are attached to the linear parts of the boundary. The plate bends by the moments applied to the middle point bars. The unknown arcs are free from external stresses. The same problem of plate bending is considered for a regular hexagon weakened by a full-strength hole. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body), the plate deflection and unknown parts of its boundary are determined under the condition that the tangential normal moment on that plate takes a constant value. Numerical analyses are also performed and the corresponding graphs are constructed.