Mathematical Problems in Engineering

Volume 2018, Article ID 5989010, 9 pages

https://doi.org/10.1155/2018/5989010

## Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

^{1}School of Civil Engineering, Chongqing University, Chongqing 400045, China^{2}Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Ministry of Education, Chongqing 400045, China

Correspondence should be addressed to Xiao-ting He; nc.ude.uqc@gnitoaixeh

Received 4 January 2018; Accepted 26 March 2018; Published 8 May 2018

Academic Editor: Fabrizio Greco

Copyright © 2018 Teng-fei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of axisymmetric deformation of a peripherally fixed and uniformly loaded circular membrane under deflection restrictions (by a frictionless horizontal rigid plate) was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and a closed-form solution of this problem was presented for the first time. The numerical analysis shows that the closed-form solution presented here has higher calculation accuracy than the existing approximate solution.

#### 1. Introduction

Elastic membrane structures and components are widely used in many fields [1–7]. The large deflection phenomena of membrane problem usually give rise to nonlinear differential equations [8–11]. These nonlinear equations generally present serious analytical difficulties when applied to boundary-value problems. Due to these somewhat intractable nonlinear equations, the analytical solutions of membrane problems are available in a few cases, but in practice they are often found to be necessary.

Hencky [12] originally dealt with the problem of axisymmetric deformation of the circular membrane fixed at the outer edge under the action of a uniformly distributed transverse loads and presented the power series solution of the problem, as shown in Figure 1, where is the uniformly distributed transverse loads. A calculation error in [12] was corrected by Chien [13] and Alekseev [14], respectively. This problem and its solution are usually called well-known Hencky problem and well-known Hencky solution for short, which are often referred to or cited in a number of related studies [15–22]. However, if we use a frictionless horizontal rigid plate to restrict the deflection of the membrane in the well-known Hencky problem, as shown in Figure 2, then such a problem will probably become somewhat complicated, where is the radial coordinate, is the transversal displacement, is the radius of the membrane contacting with the frictionless rigid plate, and is the gap between the frictionless rigid plate and the initially flat membrane. So, Xu and Liechti had to use the following four assumptions to deal with this problem [23]: the membrane has negligible flexural rigidity and only membrane stresses are considered; the slope angle of membrane is so small that the condition could approximately hold, that is, the so-called small-rotation-angle assumption; a constant radial stress is assumed, that is, the radial stress in the membrane under loads has nothing to do with the radial coordinate; the contact between the membrane and the rigid plate is frictionless. Obviously, assumption above seems to be too harsh, and, in fact, it can be given up, as will be seen later.