Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

Authors

DOI:

https://doi.org/10.20535/2521-1943.2021.5.2.218595

Keywords:

infinite cylindrical shell, concentrated radial force, Galerkin method, Navier method, accuracy, number of terms, short and long solution

Abstract

Solution for cylindrical shell under concentrated force is a fundamental problem which allow to consider many other cases of loading and geometries. Existing solutions were based on simplified assumptions, and the ranges of accuracy of them still remains unknown. The common idea is the expansion of them into Fourier series with respect to circumferential coordinate. This reduces the problem to 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relation of 8 constant of integrations with boundary conditions are still beyond the possibilities of analytical treatment. In this paper we apply the decaying exponential functions in Galerkin-like version of weighted residual method to above-mentioned 8th order equation. So, we construct the sets of basic functions each to satisfy boundary conditions as well as axial and circumferential equilibrium equations. The latter gives interdependencies between the coefficients of circumferential and axial displacements with the radial ones. As to radial equilibrium, it is satisfied only approximately by minimizations of residuals. In similar way we developed technique for application of Navier like version of WRM. The results and peculiarities of WRM application are discussed in details for cos2j concentrated loading, which methodologically is the most complicated case, because it embraces the longest distance over the cylinder. The solution for it clearly exhibits two types of behaviors – long-wave and short-wave ones, the analytical technique of treatment of them was developed by first author elsewhere, and here was successfully compared. This example demonstrates the superior accuracy of two semi analytical WRM methods. It was shown that Navier method while being simpler in realization still requires much more (at least by two orders) terms than exponential functions.

References

  1. D. Ren and K.-C. Fu, “Solutions of complete circular cylindrical shell under concentrated loads”, Journal of engineering mechanics, vol. 127, no. 3, pp. 248–253, 2001. doi: 10.1061/(ASCE)0733-9399(2001)127:3(248)
  2. S.W. Yuan, “Thin cylindrical shells subjected to concentrated loads”, Quarterly of Applied Mathematics, vol. 4, no. 1, pp. 13–26, 1946. doi.org/10.1090/qam/16031
  3. L.S.D. Morley, “The thin-walled circular cylinder subjected to concentrated radial loads”, The Quarterly Journal of Mechanics and Applied Mathematics, vol. 13 (1), pp. 24–37, 1960. doi: 10.1093/qjmam/13.1.24
  4. S. Lukasiewicz, Local loads in plates and shells, Warszawa: PWN-Polish Scientific Publishers, 1979.
  5. P.P. Bijlaard, “Stresses from radial loads in cylindrical pressure vessels”, Welding Journal, vol. 33 (12), pp. 615-623, 1954.
  6. K. Mizoguchi, H. Shiota and K. Shirakawa, “Deformation and stress in a cylindrical shell under concentrated loading: 1st report, Radial loading”, Bulletin of JSME, vol. 11 (45), pp. 393–403, 1968. doi.org/10.1299/jsme1958.11.393
  7. V.Z. Vlasov, General theory of shells and its applications in engineering, Washington: National Aeronautics and Space Administration, 1964.
  8. B.V. Nerubailo, “Radial displacement of a long cylindrical shell subjected to radial concentrated forces”, Soviet Applied Mechanics, vol. 10, pp. 1128–1131, 1974. doi: 10.1007/BF00882358
  9. V.P. Ol'shanskii, “Maximal deflection of cylindrical shells under a concentrated force”, Strength of Materials, vol. 22, pp. 1523–1526, 1990. doi: 10.1007/BF00767243
  10. V.P. Shevchenko, “Fundamental-solution methods in stress-concentration problems for thin elastic shells”, International Applied Mechanics, vol. 43, pp. 707–725, 2007. doi: 10.1007/s10778-007-0070-2
  11. C.R. Calladine, Theory of shell structures. Cambridge University Press, 1983. doi: 10.1017/CBO9780511624278
  12. I. Orynyak, A. Bogdan and I. Selivestrova, “The application of long and short cylindrical shell solutions for stress and flexibility determination in a single mitred bend”, Proceedings of the ASME 2016 Pressure Vessels and Piping Conference, 2016. doi: 10.1115/PVP2016-63598
  13. I. Orynyak and A. Oryniak, “Efficient solution for cylindrical shell based on short and long (enhanced Vlasov’s) solutions on example of concentrated radial force”, Proceedings of the ASME 2018 Pressure Vessels and Piping Conference, 2018. doi: 10.1115/PVP2018-85032
  14. A. Oryniak and I. Orynyak, “Application of short and long (enhanced Vlasov's) solutions for cylindrical shell on example of concentrated radial force”, J. Pressure Vessel Technol., vol. 143 (1): 014501, 2021. doi: 10.1115/1.4047828
  15. I.V. Orynyak and Y.P. Bai, “Application of exponential functions in weighted residuals method in structural mechanics. Part 1: axisymmetrical shell problem”, Mechanics and Advanced Technologies, no. 3 (90), pp. 19–28, 2020. doi: 10.20535/2521-1943.2020.0.209618
  16. S.P. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells. 2nd ed. New York, USA: McGraw – Hill, 1959.

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Published

2021-11-09

How to Cite

[1]
I. Orynyak, Y. Bai, and A. . Hryhorenko, “Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces”, Mech. Adv. Technol., vol. 5, no. 2, pp. 165–176, Nov. 2021.

Issue

Vol. 5 No. 2 (2021)

Section

Mechanics

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Copyright (c) 2021 I. Ориняк, Y. Bai

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