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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.

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify.

In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large.

In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented.

In this paper, we analytically dealt with the usually so-called prestressed annular membrane problem, that is, the problem of axisymmetric deformation of the annular membrane with an initial in-plane tensile stress, in which the prestressed annular membrane is peripherally fixed, internally connected with a rigid circular plate, and loaded by a shaft at the center of this rigid circular plate. The prestress effect, that is, the influence of the initial stress in the undeformed membrane on the axisymmetric deformation of the membrane, was taken into account in this study by establishing the boundary condition with initial stress, while in the existing work by establishing the physical equation with initial stress. By creating an integral expression of elementary function, the governing equation of a second-order differential equation was reduced to a first-order differential equation with an undetermined integral constant. According to the three preconditions that the undetermined integral constant is less than, equal to, or greater than zero, the resulting first-order differential equation was further divided into three cases to solve, such that each case can be solved by creating a new integral expression of elementary function. Finally, a characteristic equation for determining the three preconditions was deduced in order to make the three preconditions correspond to the situation in practice. The solution presented here could be called the extended annular membrane solution since it can be regressed into the classic annular membrane solution when the initial stress is equal to zero.