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

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 e...

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Bibliographic Details
Published in:Mathematics (Basel) 2020-05, Vol.8 (5), p.653
Main Authors: Li, Xue, Sun, Jun-Yi, Zhao, Zhi-Hang, Li, Shou-Zhen, He, Xiao-Ting
Format: Article
Language:English
Subjects:
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
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8050653