New Approaches to the Calculation of Round and Annular Plate Bending, as Well as Estimation of Their Lowest Natural Frequencies

Abstract

The transverse movements of the plates of constant thickness are assumed to be small. In this case the plates are bent by moments applied at the edge with constant intensity. For the first time, a theory of the pure bending of round and annular plates by moments of constant intensity applied to their circular boundaries. Pure bending is understood as a stress-strain state in which shears in the plates are completely absent. Within the framework of the accepted hypotheses, the normal radial deformations of the plates are determined. Based on the continuity equation in the axisymmetric case, it was established that the normal radial and circumferential strains coincide. Using Hooke's law, the normal stresses acting in the plates are determined. Based on the equilibrium equations, the moments necessary for bending the plates to a given curvature are calculated. A differential equation is obtained for determining small lateral displacements of plates under the action of moments of constant intensity applied to the edge of the plate. The solution of this equation is obtained in elementary form for the case of articulating plates around the perimeter. To proceed for solving the problems of plate bending by a transverse normal load, a method for determining equivalent moments from the acting axisymmetric load, both for round and ring plates is proposed. To satisfy the equilibrium conditions for the plates under consideration of the action of a transverse load, it is assumed that the magnitude of the vertical reaction on the supports along the perimeter is uniformly distributed and equal to the integral value of the normal load divided by the length of the perimeter. As an example, the problems of bending plates under their own weight with articulated support are solved. Within the framework of the proposed theory, solutions to the problems of bending round and annular plates located on the Winkler base are demonstrated. For the first time, a technique has been proposed for determining the lowest natural frequency of both round and annular plates in the framework of the proposed theory of pure bending. A technique is also proposed for taking into account the influence of the Winkler base under the plates on the lower natural frequency.