Calculation Method for Axisymmetric Bending of Circular and Annular Plates on a Changeable Elastic Bed. Part 1. Analytical Relations

Circular and ring plates on a continuous changeable elastic bed are considered. It is assumed that the bed response is described by the Winkler model with a changeable bedding coefficient. An analytical method for axisymmetric bending calculations of plates loaded by a transverse continuously distributed replaceable load is proposed. The method is based on the solution of the corresponding differential bending equation with exchangeable coefficients found by direct integration. The exact solution is expressed through dimensionless fundamental function given in the form of power series of dimensionless parameters with exchangeable coefficients. Recurrence integral relations were obtained to determine the exchangeable coefficients of the series. As a result, formulas for deflections and internal forces, which fully characterize the plate's stress-strain state, in the case where the given load and bed factor are arbitrary continuous functions, are obtained in analytical form. The method can be applied for any given limit conditions on the plate contours. In fact, calculations for axisymmetric bending of plates are reduced to determination of unknown integration constants from the given limit conditions and numerical realization of the obtained solutions. The derived general formulas are adaped for the practical case with the bedding factor and load given by polynomials. It is shown that, in this case, the dimensionless fundamental functions can be represented by power series. To calculate the coefficients of the power series, the corresponding recurrence relations are derived. Compared with approximate methods, the proposed analytical method allows one to obtain a more realistic bending behavior, being a lucrative alternative to the application of approximate methods when solving this class of problems.