Elastic, Large Deflections of Annular Membranes

In studying various problems arising in the analysis of radially symmetric membranes, existing theories are investigated under the boundary conditions (1) simple continuous circular membrane, uniformly loaded and (2) pressure loaded membrane additionally loaded by an opposing force centrally applied through a smaller disc rigidly cemented to the membrane. Equilibrium equations of a higher degree of non-linearity than those usually encountered in the Foppl-von Karman theory are studied; these reduce to the latter as a special case. Using the method of power series, exact solutions are given to both systems of equations for the case of a uniformly loaded clamped membrane and curves of displacement versus pressure are included for comparison. These are compared with previously derived empirical relationships and experimental data. Two of the principal difficulties associated with the power series approach are avoided by using an iterative technique suitable for a digital computer. For the annular membrane, a numerical approach is developed whereby the two point boundary value problem is replaced by an initial value problem and a Runge Kutta numerical integration process operating directly on the simplified equations of equilibrium. The results of this theory area are compared with those of Iberall.