Journal Article

A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation

Zhao, Zhi-Hang, Li, Shou-Zhen, Sun, Jun-Yi, Li, Xue, He, Xiao-Ting

2020

Overview

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.

• Publisher: MDPI AG
• Subject: Axial strain / axisymmetric deformation / Boundary conditions / circular membrane / Closed form solutions / Deflection / equilibrium equation / Equilibrium equations / Exact solutions / large deflection / Mathematics / Membranes / Nonlinear differential equations / Power series / power series method / Stress concentration