On non-axisymmetric buckling modes of inhomogeneous circular plates

Authors

  • Svetlana M. Bauer St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
  • Eva B. Voronkova St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2021.201

Abstract

Unsymmetrical buckling of nonuniform circular plates with elastically restrained edge and subjected to normal pressure is studied in this paper. The asymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value at which waves in the circumferential direction can appear. The effect of material heterogeneity and boundary on the buckling load is examined. For a plate with elastically restrained edge, the buckling pressure and mode number increase with a rise of spring stiffness. Increasing of the elasticity modulus to the plate edge leads to increasing of the buckling pressure, but the mode number does not change. If the translational flexibility coefficient is small, decreasing of the elasticity modulus to the shell (plate) edge leads to sufficient lowering of the buckling pressure.

Keywords:

circular plate, buckling, heterogeneity

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References

Литература

1. Adachi J. Stresses and buckling in thin domes under internal pressure. Tech. Rep. MS68-01, U.S. Army Materials and Mechanics Research Center, Watertown (1968).

2. Bushnell D. Buckling of shells-pitfall for designers. AIAA Journal 19, 1183–1226 (2015). https://doi.org/10.2514/3.60058

3. Панов Д.Ю., Феодосьев В.И. О равновесии и потере устойчивости пологих оболочек при больших прогибах. Прикладная математика и механика 12, 389–406 (1948).

4. Cheo L. S., Reiss E.L. Unsymmetric wrinkling of circular plates. Quarterly of Applied Mathematics 31 (1), 75–91 (1973). https://doi.org/10.1090/qam/99710

5. Feodos’ev V.I. On a method of solution of the nonlinear problems of stability of deformable systems. Journal of Applied Mathematics and Mechanics 27 (2), 392–404 (1963). https://doi.org/10.1016/0021-8928(63)90008-X

6. Морозов Н.Ф. К вопросу о существовании несимметричного решения в задаче о больших прогибах круглой пластинки, загруженной симметричной нагрузкой. Известия высших учебных заведений. Математика 2, 126–129 (1961).

7. Piechocki W. On the nonlinear theory of thin elastic spherical shells: Nonlinear partial differential equations solutions in theory of thin elastic spherical shells subjected to temperature fields and external loading. Archiwum mechaniki stosowanej 21 (1), 81–102 (1969).

8. Coman C.D., Bassom A.P. Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap. International Journal of Non-Linear Mechanics 81, 8–18 (2016). https://doi.org/10.1016/j.ijnonlinmec.2015.12.004

9. Bauer S.M., Voronkova E.B. Unsymmetrical wrinkling of nonuniform annular plates and spherical caps under internal pressure. In:Altenbach H., Chroscielewski J., Eremeyev V., Wisniewski K. (eds) Recent Developments in the Theory of Shells. Advanced Structured Materials, vol. 110, 79–89. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17747-8_6

10. Бауэр С.М., Воронкова Е.Б. Влияние условий закрепления на появление несимметричных форм равновесия у круглых пластин под действием нормального давления. Журнал Белорусского государственного университета. Математика. Информатика 1, 38–46 (2020). https://doi.org/10.33581/2520-6508-2020-1-38-46

11. Bauer S.M., Voronkova E.B. On Buckling Behavior of Inhomogeneous Shallow Spherical Caps with Elastically Restrained Edge. In:Altenbach H., Chinchaladze N., Kienzler R., MullerW. (eds) Analysis of Shells, Plates, and Beams. Advanced Structured Materials, vol. 134, 65–74. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-47491-1_4

References

1. Adachi J. Stresses and buckling in thin domes under internal pressure. Tech. Rep. MS68–01, U.S. Army Materials and Mechanics Research Center, Watertown (1968).

2. Bushnell D. Buckling of shells-pitfall for designers. AIAA Journal 19, 1183–1226 (2015). https://doi.org/10.2514/3.60058

3. Panov D.Y., Feodosiev V.I. On the equilibrium and loss of stability of shallow shells in the case of large displacement. Journal of Applied Mathematics and Mechanics 12, 389–406 (1948). (In Russian)

4. Cheo L. S., Reiss E.L. Unsymmetric wrinkling of circular plates. Quarterly of Applied Mathematics 31 (1), 75–91 (1973). https://doi.org/10.1090/qam/99710

5. Feodos’ev V.I. On a method of solution of the nonlinear problems of stability of deformable systems. Journal of Applied Mathematics and Mechanics 27 (2), 392–404 (1963). https://doi.org/10.1016/0021-8928(63)90008-X

6. Morozov N.F. On the existence of a non-symmetric solution in the problem of large deflections of a circular plate with a symmetric load. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 2, 126–129 (1961). (In Russian)

7. Piechocki W. On the nonlinear theory of thin elastic spherical shells: Nonlinear partial differential equations solutions in theory of thin elastic spherical shells subjected to temperature fields and external loading. Archiwum mechaniki stosowanej 21 (1), 81–102 (1969).

8. Coman C.D., Bassom A.P. Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap. International Journal of Non-Linear Mechanics 81, 8–18 (2016). https://doi.org/10.1016/j.ijnonlinmec.2015.12.004

9. Bauer S.M., Voronkova E.B. Unsymmetrical wrinkling of nonuniform annular plates and spherical caps under internal pressure. In: Altenbach H., Chroscielewski J., Eremeyev V., Wisniewski K. (eds) Recent Developments in the Theory of Shells. Advanced Structured Materials, vol. 110, 79–89. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17747-8_6

10. Bauer S.M., Voronkova E.B. Influence of boundary constraints on the appearance of asymmetrical equilibrium states in circular plates under normal pressure. Journal of the Belarusian State University. Mathematics and Informatics 1, 38–46 (2020). https://doi.org/10.33581/2520-6508-2020-1-38-46 (In Russian)

11. Bauer S.M., Voronkova E.B. On Buckling Behavior of Inhomogeneous Shallow Spherical Caps with Elastically Restrained Edge. In: Altenbach H., Chinchaladze N., Kienzler R., Muller W. (eds) Analysis of Shells, Plates, and Beams. Advanced Structured Materials, vol. 134, 65–74. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-47491-1_4

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Published

2021-07-21

How to Cite

Bauer, S. M., & Voronkova, E. B. (2021). On non-axisymmetric buckling modes of inhomogeneous circular plates. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(2), 204–211. https://doi.org/10.21638/spbu01.2021.201

Issue

Vol. 8 No. 2 (2021)

Section

In memoriam of P. E. Tovstik

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.

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