Membrane Dish Analysis: A Summary of Structural and Optical Analysis Capabilities

The governing fourth-order equations are presented for the shell of revolution under axisymmetric load, and an asymptotic solution for edge bending is obtained. The asymptotic solution is based on the assumption that the thickness of the shell is very small relative to the shell's radii of curvature. In order to incorporate the effect of high pressurization on edge bending, the geometricnonlinearity associated with the change in meridional slope is included in the governing equations. The total solution, which is the sum of the membrane (particular) and edge bending (complementary) solutions, is given for the case of uniform pressure load, and the steps required in order to evaluate the solution are explicitly described. As an example, the paraboloidal shell under uniformpressure load is discussed, and the results are compared with a previous ANSYS finite element analysis by Murphy (1987).