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A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations

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
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 631; https://doi.org/10.3390/math8040631
Received: 24 March 2020 / Revised: 10 April 2020 / Accepted: 14 April 2020 / Published: 20 April 2020
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. View Full-Text
Keywords: Föppl–Hencky membrane; boundary value problem; power series method; closed-form solution; geometric equation Föppl–Hencky membrane; boundary value problem; power series method; closed-form solution; geometric equation
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Lian, Y.-S.; Sun, J.-Y.; Zhao, Z.-H.; He, X.-T.; Zheng, Z.-L. A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations. Mathematics 2020, 8, 631.

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