Large Deflection Analysis of Peripherally Fixed Circular Membranes Subjected to Liquid Weight Loading: A Refined Design Theory of Membrane Deflection-Based Rain Gauges

Abstract

The anticipated use of elastic membranes for deflection-based rain gauges has provided an impetus for this paper to revisit the large deflection problem of a peripherally fixed circular membrane subjected to liquid weight loading, a statics problem when the fluid–structure interaction of membrane and liquid reaches static equilibrium. The closed-form solution of this statics problem of fluid–structure interaction is necessary for the design of such membrane deflection-based rain gauges, while the existing closed-form solution, due to the use of the small rotation angle assumption of the membrane, cannot meet the design requirements for computational accuracy. In this paper, the problem under consideration is reformulated by giving up the small rotation angle assumption, which gives rise to a new and somewhat intractable nonlinear integro-differential equation of the governing out-of-plane equilibrium. The power series method has played an irreplaceable role in analytically solving membrane equations involving both integral and differential operations, and a new and more refined closed-form solution without the small rotation angle assumption is finally presented. Numerical examples conducted show that the new and more refined closed-form solution presented has satisfactory convergence, and the effect of giving up the small rotation angle assumption is also investigated numerically. The application of the closed-form solution presented in designing such membrane deflection-based rain gauges is illustrated, and the reliability of the new and more refined closed-form solution presented was confirmed by conducting a confirmatory experiment.

An Improved Mathematical Theory for Designing Membrane Deflection-Based Rain Gauges

“…Such a fluid-solid coupling problem presents serious analytical difficulties, which involves the analytical solution to differential-integral governing equations. The mathematical theory for designing such rain gauges, which is presented in our previous study [13], is observed to be inaccurate. It is suitable only for the design of nonlinear (rather than linear) rain gauges, because the ranges of the nearly-linear (seemingly following a linear-dependent relationship) segments in the nonlinear input-output relationships of the rain gauges, which it can provide, are too small or too narrow.…”

“…Many thin films are able to exhibit large elastic deflections when subjected to external loads [1][2][3][4][5][6], which makes it possible to develop devices or instruments based on film deflection [7][8][9][10][11][12]. In our previous study [13], to overcome the shortcomings of the existing rain gauges, a new type of membrane deflection-based rain gauge was proposed, which involves a liquid-structure interaction of a peripherally fixed circular membrane under liquid weight loading. Such a fluid-solid coupling problem presents serious analytical difficulties, which involves the analytical solution to differential-integral governing equations.…”

“…In general, due to various factors, such as calibration errors, the measurements of precipitation may be underestimated by 5% to 40% [32]. In our previous study [13], we proposed developing a new type of rain gauge using the elastic deflection of conductive membranes, namely the membrane elastic deflection-based rain gauge. As opposed to tipping-bucket or weighing rain gauges, this new type of rain gauge does not require performing the calibration for volumetric metering of rainwater, thus avoiding the measurement errors of precipitation caused by this.…”

An Exact In-Plane Equilibrium Equation for Transversely Loaded Large Deflection Membranes and Its Application to the Föppl-Hencky Membrane Problem

“…The "membrane" is a term in mechanics which refers to a laterally loaded thin plate whose edge displacement is always completely constrained (fixed) during deflection. Therefore, "thin plate bending problems" [1-5] differ from "membrane deflecting problems" [6][7][8][9][10], mainly in that the thin plates under lateral loading undergo both tensile stress and compressive stress in bending problems (i.e., undergoing the so-called bending stresses), and undergo only tensile stress in deflecting problems due to the always fully fixed edge displacements (as if the bending stiffness of the plates vanishes). In general, the large deflection problems of laterally loaded thin plates or membranes are solved analytically by establishing the governing equations such as geometric equations and equilibrium equations, in which the accuracies of these governing equations depend usually on whether the deflection effect is properly taken into account, which further affects the accuracies of the solutions of these large deflection problems.…”

Improved Power Series Solution of Transversely Loaded Hollow Annular Membranes: Simultaneous Modification of Out-of-Plane Equilibrium Equation and Radial Geometric Equation

“…Essential to these technical applications is the ability to solve these problems of a large deflection of the membranes under lateral loading analytically and accurately. However, these large deflections often give rise to strong geometric nonlinearity, and these nonlinear differential equations often present analytical difficulties [22][23][24]. In the existing literature, annular membrane problems are less in evidence in comparison with circular membrane problems [25][26][27][28][29], but are often more difficult to deal with analytically due to the boundary conditions at the inner and outer edges which need to be satisfied simultaneously [30][31][32].…”