Analytical Investigation of Interaction Between Flexible Circular Plate with Transversely Isotropic Half-space

Document Type : پژوهشی

Authors

• ghasem gorji ¹
• morteza asgandari ²
• aziz ardeshir ¹

¹ Babol Noshirvani University of Technology

² University of Tehran

10.22067/civil.v27i1.50062

Abstract

A transversely isotropic half-space with axis of material symmetry perpendicular to the free surface supports a flexible circular plate. The contact area of the plate and the half-space is considered to be both frictionless and unbonded (tensionless). The foundation is affected by a vertical static axisymmetric load. Detailed analysis of the interaction of these two systems is the target of this paper. With the use of ring load Green’s functions for both the plate and the continuum half-space, dual integral equations accompanied with some inequalities are obtained to model the complex boundary value problem. With the incorporation of the ring-shape finite element method, where its size gradually varies, we are capable of capturing both regular and singular solution smoothly. The validity of the combination of the analytical and numerical method is proved with comparing the results of this paper with a number of benchmark cases of both linear and nonlinear interaction of circular plate and half-space. Some new illustrations are presented to portray the aspect of the anisotropy of the half-space.

Keywords

• Transversely isotropic half-space
• Flexible circular plate
• Axial axisymmetric
• Dual integral equations
• Ring-element method

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References

1. رحیمیان، محمد و اسکندری قادی، مرتضی (1379)، "تئوری ارتجاعی." انتشارات دانشگاه تهران.

2. اردشیر بهرستاقی، عزیزالله و اسکندری قادی، مرتضی (2011)، "تحلیل نیم‌فضای ایزوتروپ جانبی تحت اثر صفحه صلب دایره‌ای با استفاده از توابع گرین بار حلقوی." نشریه علوم کاربردی و محاسباتی در مکانیک، 22 (1)، صفحات.

3. Celep, Z. (1988). “Rectangular plates resting on tensionless elastic foundation.” J. Eng. Mech., 114(2), doi: 10.1061/(ASCE)0733-9399-114:12(2083), pp. 2083-2092.

4. Celep, Z. (1988). “Circular plate on tensionless Winkler foundation.” J. Eng. Mech., 114(10), doi: 10.1061/(ASCE)0733-9399-114:10(1723), pp. 1723-1739.

5. Celep, Z., Malaika, A., and Abu-Hussein, M. (1989). “Forced vibrations of a beam on a tensionless foundation.” Journal of Sound and Vibration, Vol. 128, Issue. 2, pp. 235-246.

6. Celep, Z. (1990). “In-plane vibrations of circular rings on a tensionless foundation.” Journal of Sound and Vibration, Vol. 143, Issue. 3, pp. 461-471.

7. Celep, Z., and Demir, F. (2005). “Circular rigid beam on a tensionless two-parameter elastic foundation.” ZAMM- Journal of Applied Mathematics and Mechanics, Vol. 85, Issue. 6,pp. 431-439.

8. Celep, Z., and Demir, F. (2007). “Symmetrically loaded beam on a two-parameter tensionless foundation.” Structural Engineering and Mechanics, Vol. 27, No. 5, pp. 555-574.

9. Celep, Z., and Guler, K. (2007). “Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation.” Journal of Sound and Vibration, Vol. 301, Issues. 3-5, pp. 495-509.

10. Ding, H., and Chen, W., and Zhang, L. (2006). “Elasticity of transversely isotropic materials.” Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

11. Elliott, H. A. (1948). “Three dimensional stress distribution in hexagonal aeolotropic crystals.” Mathematical Proceedings of the Cambridge Philosophical Society, 44, pp. 522-533.

12. Eskandari-Ghadi, M. (2005). “A complete solutions of the wave equations for transversely isotropic media.” J. of Elasticity, 81, pp. 1-19.

13. Eskandari-Ghadi, M., Pak, R. Y. S., and Ardeshir-Behrestaghi, A. (2008). “Transversely isotropic elastodynamic solution of a finite layer on an infinite subgrade under surface loads.” Soil Dynamics and Earthquake Engineering 8, pp. 986-1003.

14. Eskandari-Ghadi, M., Fallahi, M., and Ardeshir-Behrestaghi, A. (2010). “Forced vertical vibration of Rigid circular disc on a transversely isotropic half-space.” Journal of engineering mechanics © ASCE/ July, pp. 913-922.

15. Eskandari-Ghadi, M., Ardeshir-Behrestaghi, A. (2010). “Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half space.” Soil dynamics and earthquake engineering 30, doi: 10.1016/j, pp. 547-560.

16. Eskandari-Ghadi, M., Mirzapour, A., and Ardeshir-Behrestaghi, A. (2010). “Rocking vibration of a rigid circular disc buried in an a transversely isotropic full space.” Numerical and analytical method in geomechanics, doi: 10.1002/nag.976, pp. 547-560.

17. Eskandari-Ghadi, M., Pak, R. Y. S., and Ardeshir-Behrestaghi, A. (2011). “A hybrid analytical-numerical method for vertical concentric multi-annular punch contact with a transversely isotropic elastic half-space.” Journal of applaied mechanics, J_ID: JAM DOI: 10.1115/1.4005546.

18. Eubanks, R. A., and Sternberg, E. (1954). “On the axisymmetric problem of elasticity theory for a medium with transverse isotropy.” Journal of rational mechanics and analysis, pp. 89-101.

19. Gladwell, G.M.L., and Iyer, K.P.R. (1974). “Unbonded contact between a circular plate and an elastic half-space.” Journal of Elasticity, 4, pp. 115-130.

20. Gurtin, M.E. (1972). “The linear theory of elasticity.” In: S. Flugge (ed.), Handbuch der Physik, Vol. Via/2, Mechanics of Solids II, ed C. Truesdell. Springer, Berlin Heidelberg New York, pp. 1-295.

21. Hu, H. C. (1953). “On the three dimensional problems of the theory of elasticity of a transversely isotropic body.” Sci. Sinica, Vol. 2, pp. 145-151.

22. Lekhnitskii, S. G. (1981). “Theory of anisotropic elastic bodies.” Holden-Day publishing Co., San Fransisko, Calif.

23. Luco, J. E., and Mita, A. (1987). “Response of a circular foundation on a uniform half-space to elastic waves.” Earthquake Engineering and Structural Dynamics, Vol. 15, pp. 105-118.

24. Michell, J. H. (1900). “The stress in an aelotropic elastic solid with an infinite plane boundary.” Proceeding of the London mathematical society, Vol. 32, pp. 247-258.

25. Nowacki, W. (1954). “The stress function in three dimensional problems concerning an elastic body characterized by transversely isotropy.” Bull. Acad. Polon. Sci., Vol. 2, pp. 21-25.

26. Pak, R. Y. S., Simmons, B. M., and Ashlock, J. C. (2008). “Tensionless contact of a flexible plate and annulus with a smooth half-space under axisymmetric loads by integral equations.” International Journal of mechanical Sciences 50, pp. 1004–1011.

27. Pan, Y. C., and Chou, T. W. (1979). “Green's functions solutions for semi-infinite transversely isotropic materials.” Int, J. Eng. Sci, 17(5), pp. 545-551.

28. Sneddon, I. N. (1951). “Fourier transforms.” McGraw Hill, New York, N. Y.

29. Szilard, R. (1974). “Theory and analysis of plates, Classical and Numerical Methods.” Prentice- Hall, Inc., Englawood Cliffs, New Jersey.

30. Timoshenko, S.P.. and Woinowski-Kreiger, S. (1959). “Theory of plates and shells.” 2nd ed. New York: McGraw Hill.

31. Wang, M. Z., and Wang, W. (1995). “Completeness and nonuniqueness of general solutions of transversely isotropic elasticity.” International Journal of Solids and Structures, 32 (374), pp. 501-513.

32. Weitsman, Y. (1969). “ON the unbounded contact between plates and an elastic half space.” Journal of applied Mechanics, ASME; 36, pp. 198-202.

33. Weitsman, Y. (1971). “Onset of separation between a beam and a tensionless elastic foundation under a moving load.” International Journal of Mechanical Sciences, 13 (8), pp. 707-711.

34. Weitsman, Y. (1972). “A tensionless contact between a beam and an elastic half-space.” International Journal of Engineering Science, 10 (1) pp. 73)ER.