Approximating paraboloids with axisymmetric pressurized membranes

Abstract

The pressurization of initially flat and curved axisymmetric membranes to, respectively, approximate and exact parabolic shapes is discussed and the impact of uniform radial edge support displacements on the achieved accuracy is studied. It is shown that the non-parabolic shape to which initially flat membranes inflate better approximates a paraboloid if pretension is applied. For dish shapes and relatively low pressurization levels of current interest, however, the associated stresses render this means to improve performance impractical. The design of initially non-flat membranes to assume a parabolic shape when properly pressurized is discussed. It is seen that radial edge support displacements generally alter the achieved shapes only in the vicinity of the perimeter while the dish center undergoes an axial translation without its shape affected. The size of the dish center of unaltered shape is shown to be generally independent from the magnitude of the support perturbation. However, these trends disappear for shallow dishes (high f/D ratios) and high pressurization levels where the edge effects spread to the dish center. The mechanics of the observed phenomena are discussed. Wrinkling is accounted for. The presented results have been produced with a program which is able to determine the initial geometry of a membrane to achieve any specified pressurized axisymmetric shape. The results are applicable via scaling to any dish diameter.