Large Deflection Analysis of Axially Symmetric Deformation of Prestressed Circular Membranes under Uniform Lateral Loads

In this study, the problem of axisymmetric deformation of peripherally fixed and uniformly laterally loaded circular membranes with arbitrary initial stress is solved analytically. This problem could be called the generalized Föppl–Hencky membrane problem as the case where the initial stress in the membrane is equal to zero is the well-known Föppl–Hencky membrane problem. The problem can be mathematically modeled only in terms of radial coordinate owing to its axial symmetry, and in the present work, it is reformulated by considering an arbitrary initial stress (tensile, compressive, or zero) and by simultaneously improving the out-of-plane equilibrium equation and geometric equation, while the formulation was previously considered to fail to improve the geometric equation. The power-series method is used to solve the reformulated boundary value problem, and a new and more refined analytic solution of the problem is presented. This solution is actually observed to be able to regress into the well-known Hencky solution of zero initial stress, allowing the considered initial stress to be zero. Moreover, the numerical example conducted shows that the obtained power-series solutions for stress and deflection converge very well, and have higher computational accuracy in comparison with the existing solutions.