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Using a rod theory formulation, we derive equations of state for a thin elastic membrane subjected to several different boundary conditions--clamped, simply supported, and periodic. The former is applicable to membranes supported on a softer substrate and subjected to uniaxial compression. We show that a wider family of quasistatic equilibrium shapes exist beyond the previously obtained analytical solutions. In the latter case of periodic membranes, we were able to derive exact solutions in terms of elliptic functions. These equilibria are verified by considering a fluid-structure interaction problem of a periodic, length-preserving bilipid membrane modeled by the Helfrich energy immersed in a viscous fluid. Starting from an arbitrary shape, the membrane dynamics to equilibrium are simulated using a boundary integral method. [ABSTRACT FROM AUTHOR]