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A New Solution to Főppl-Hencky Membrane Equation

Published online by Cambridge University Press:  12 January 2017

Z. X. Yang ,
J. Y. Sun ,
G. M. Ran  and
X. T. He
Z. X. Yang
Affiliation:
School of Civil EngineeringChongqing UniversityChongqing, China Key Laboratory of New Technology for Construction of Cities in Mountain AreaChongqing UniversityMinistry of EducationChongqing, China
J. Y. Sun*
Affiliation:
School of Civil EngineeringChongqing UniversityChongqing, China Key Laboratory of New Technology for Construction of Cities in Mountain AreaChongqing UniversityMinistry of EducationChongqing, China
G. M. Ran
Affiliation:
School of Civil EngineeringChongqing UniversityChongqing, China Key Laboratory of New Technology for Construction of Cities in Mountain AreaChongqing UniversityMinistry of EducationChongqing, China
X. T. He
Affiliation:
School of Civil EngineeringChongqing UniversityChongqing, China Key Laboratory of New Technology for Construction of Cities in Mountain AreaChongqing UniversityMinistry of EducationChongqing, China
*
*Corresponding author (sunjunyi@cqu.edu.cn)
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Abstract

In this note, Föppl-Hencky membrane equation in the case of axisymmetric deformation was derived, and its power series solution was presented by using the displacement-based solution method. The result shows that both the displacement-based solution method and the stress-based solution method are effective for the solution to Föppl-Hencky equation. But in comparison with the latter, the former makes the solving process some more concise. In addition, some issues concerned were also discussed.

Keywords

Föppl-Hencky membraneAxisymmetric deformationPower series solutionDisplacement
Type
Technical Note
Information
Journal of Mechanics , Volume 33 , Issue 4 , August 2017 , pp. N7 - N11
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2016 

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References

1. Föppl, A., Vorlesungen über technische,” Mechanik, 5, pp. 132–144 (1907).Google Scholar
2. von Kármán, T., Festigkeitsprobleme im maschinenbau,” Encyklopäie der Mathematischen Wissenschaften, 4, pp. 348–351 (1910).Google Scholar
3. Van Gorder, R. A., Analytical method for the construction of solutions to the Föppl-von Kármán equations governing deflections of a thin flat plate,” International Journal of Non-Linear Mechanics, 47, pp. 1–6 (2012).CrossRefGoogle Scholar
4. Coman, C. D., On the compatibility relation for the Föppl–von Kármán plate equations,” Applied Mathematics Letters, 25, pp. 2407–2410 (2012).CrossRefGoogle Scholar
5. Yeh, K. Y., Zheng, X. J. and Zhou, Y. H., An analytical formula of the exact solution to von Kármán's equations of a circular plate under a concentrated load,” International Journal of Non-Linear Mechanics, 24, pp. 551–560 (1989).Google Scholar
6. Hencky, H., Über den Spannungszustand in kreisrunden Platten mit verschwindender Biegungssteifigkeit,” Zeitschrift Für Mathematik und Physik, 63, pp. 311–317 (1915).Google Scholar
7. Hencky, H., Die Berechnung dünner rechteckiger Platten mit verschwindender Biegungsteifigkeit,” Zeitschrift für Angewandte Mathematik und Mechanik, 1, pp. 81–88 (1921).Google Scholar
8. Chien, W. Z., Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection,” The Science Reports of National Tsinghua University, 5, pp. 193–208 (1948).Google Scholar
9. Alekseev, S. A., Elastic circular membranes under the uniformly distributed loads,” Engineering Corpus, 14, pp. 196–198 (1953) (in Russian).Google Scholar
10. Campbell, J. D., On the theory of initially tensioned circular membranes subjected to uniform pressure,” Quarterly Journal of Mechanics and Applied Mathematics, 9, pp. 84–93 (1956).Google Scholar
11. Alekseev, S. A., Elastic annular membranes with a stiff centre under the concentrated force,” Engineering Corpus, 10, pp. 71–80 (1951) (in Russian).Google Scholar
12. Sun, J. Y., Hu, J. L., He, X. T. and Zheng, Z. L., A theoretical study of a clamped punch-loaded blister configuration: The quantitative relation of load and deflection,” International Journal of Mechanical Sciences, 52, pp. 928–936 (2010).Google Scholar
13. Chien, W. Z., Wang, Z. Z., Xu, Y. G. and Chen, S. L., The symmetrical deformation of circular membrane under the action of uniformly distributed loads in its portion,” Applied Mathematics and Mechanics (English Edition), 2, pp. 653–668 (1981).Google Scholar
14. Chen, S. L. and Zheng, Z. L., Large deformation of circular membrane under the concentrated force,” Applied Mathematics and Mechanics (English Edition), 24, pp. 28–31 (2003).Google Scholar
15. Jin, C. R., Large deflection of circular membrane under concentrated force,” Applied Mathematics and Mechanics (English Edition), 29, pp. 889–896 (2008).CrossRefGoogle Scholar
16. Sun, J. Y., Hu, J. L., He, X. T., Zheng, Z. L. and Geng, H. H., A theoretical study of thin film delamination using clamped punch-loaded blister test: Energy release rate and closed-form solution,” Journal of Adhesion Science and Technology, 25, pp. 2063–2080 (2011).CrossRefGoogle Scholar
17. Fichter, W. B., Some solutions for the large deflections of uniformly loaded circular membranes,” NASA Technical paper No. 3658 (1997).Google Scholar
18. Lim, T. C., Large deflection of circular auxetic membranes under uniform load,” Journal of Engineering Materials and Technology, 138, 041011 (2016).CrossRefGoogle Scholar
19. Plaut, R. H., Linearly elastic annular and circular membranes under radial, transverse, and torsional loading. Part I: large unwrinkled axisymmetric deformations,” Acta Mechanica, 202, pp. 79–99 (2008).Google Scholar
20. Sun, J. Y., Rong, Y., He, X. T., Gao, X. W. and Zheng, Z. L., Power series solution of circular membrane under uniformly distributed loads: investigation into Hencky transformation,” Structural Engineering and Mechanics, 45, pp. 631–641 (2013).CrossRefGoogle Scholar
21. Tseng, W. D. and Tarn, J. Q., Exact elasticity solution for axisymmetric deformation of circular plates,” Journal of Mechanics, 31, pp. 617–629 (2015).Google Scholar
22. He, X. T., Cao, L., Li, Z. Y., Hu, X. J. and Sun, J. Y., Nonlinear large deflection problems of beams with gradient: A biparametric perturbation method,” Applied Mathematics and Computation, 219, pp. 7493–7513 (2013).Google Scholar
23. He, X. T., Cao, L., Sun, J. Y. and Zheng, Z. L., Application of a biparametric perturbation method to large-deflection circular plate problems with a bimodular effect under combined loads,” Journal of Mathematical Analysis and Applications, 420, pp. 48–65 (2014).CrossRefGoogle Scholar