Some Solutions for the Large Deflections of Uniformly Loaded Circular Membranes

SummaryInconsistent citations in the literature and questions about convergence prompt reexamination ofHencky's classic solution for the large deflections of a clamped, circular isotropic membrane under uni-form pressure. This classic solution is observed actually to be for uniform lateral loading because theradial component of the pressure acting on the deformed membrane is neglected. An algebraic error inHencky's solution is corrected, additional terms are retained in the power series to assess convergence,and results are obtained for two additional values of Poisson' s ratio.To evaluate the importance of the neglected radial component of the applied pressure, the problemis reformulated with this component included and is solved, with escalating algebraic complexity, by asimilar power-series approach. The two solutions agree quite closely for lightly loaded membranes anddiverge slowly as the load intensifies. Differences in maximum stresses and deflections are substantialonly when stresses are very high. The more nearly spherical deflection shape of the membrane undertrue pressure loading suggests that a near-parabolic membrane reflector designed on the basis of themore convenient Hencky theory would not perform as well as expected.In addition, both theories are found to yield closed-form, nonuniform membrane-thickness distribu-tions that produce parabolic middle-surface deflections under loading. Both distributions require thatthe circular boundary expand radially in amounts that depend on material and loading parameters.IntroductionConcepts for orbiting inflatable reflectors are of interest primarily because of their relative mechan-ical simplicity, high area-to-mass ratio, and compact packaging characteristics. Essential to the designand fabrication of inflatable reflectors is the ability to predict the reflector shape upon inflation. Whenneither a deep reflector nor extreme surface precision is required, an attractively simple configuration isan initially flat and unstressed circular elastic membrane attached at its perimeter to a stiff ring and sub-jected to differential pressure. While uniform loading applied to a constant-thickness membrane will notproduce the exact paraboloidal reflector shape that is desired in numerous applications, the shape differ-ence may be small enough for many purposes.This paper, because of inconsistent citations in the literature (e.g., ref. 1) and questions about con-vergence, reexamines Hencky's original analysis (ref. 2) of the large deflections of a clamped, circularmembrane under uniform pressure. Hencky's power-series approach is again employed, an algebraicerror is corrected, more terms are retained to assess convergence, and results are generated for two addi-tional values of Poisson's ratio.Also, because Hencky's problem actually involves uniform lateral loading (i.e., the radial compo-nent of pressure on the deformed membrane is neglected), the boundary value problem for true uniform-pressure loading is formulated and solved. Results for lateral deflections and membrane stresses fromboth Hencky's solution (corrected) and the uniform-pressure solution are presented in tabular andgraphical form, and comparisons are made between the two solutions.In addition, although neither problem solution yields an exactly paraboloidal deflection shape,nonuniform membrane-thickness distributions that yield such shapes can be found for both loadingconditions. These distributions are derived in the appendix.Symbols