Coupled systems mechanics

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A parametric study on the free vibration of a functionally graded material circular plate with non-uniform thickness resting on a variable Pasternak foundation by differential quadrature method

  • Abdelbaki, Bassem M. (Mathematics and Physics Department, Faculty of Engineering, Fayoum University) ;
  • Ahmed, Mohamed E. Sayed (Mathematics and Physics Department, Faculty of Engineering, Fayoum University) ;
  • Al Kaisy, Ahmed M. (Mathematics and Physics Department, Faculty of Engineering, Fayoum University)
  • Received : 2022.04.23
  • Accepted : 2022.07.11
  • Published : 2022.08.25
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Abstract

This paper presents a parametric study on the free vibration analysis of a functionally graded material (FGM) circular plate with non-uniform thickness resting on a variable Pasternak elastic foundation. The mechanical properties of the material vary in the transverse direction through the thickness of the plate according to the power-law distribution to represent the constituent components. The equation of motion of the circular plate has been carried out based on the classical plate theory (CPT), and the differential quadrature method (DQM) is employed to solve the governing equations as a semi-analytical method. The grid points are chosen based on Chebyshev-Gauss-Lobatto distribution to achieve acceptable convergence and better accuracy. The influence of geometric parameters, variable elastic foundation, and functionally graded variation for clamped and simply supported boundary conditions on the first three natural frequencies are investigated. Comparisons of results with similar studies in the literature have been presented and two-dimensional mode shapes for particular plates have been plotted to illustrate the effect of variable thickness profile.

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References

  1. Abdelbaki, B., Sayed-Ahmed, M. and Al-Kaisy, A. (2021), "Axisymmetric bending of FGM circular plate with parabolically-varying thickness resting on a non-uniform two-parameter partial foundation by DQM", Adv. Dyn. Syst. Appl., 16(2), 1393-1413.
  2. Abdelbaki, B.M. and Ahmed, M. (2022), "Variation ofsoilsubgrade modulus under circular foundation", J. Al-Azhar Univ. Eng. Sector, 17(63), 549-556. https://doi.org/10.21608/auej.2022.233759
  3. Al Kaisy, A., Esmaeel, R.A. and Nassar, M.M. (2007), "Application of the differential quadrature method to the longitudinal vibration of non-uniform rods", Eng. Mech., 14(5), 303-310.
  4. Arshid, E. and Khorshidvand, A.R. (2018), "Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method", Thin Wall. Struct., 125, 220-233. https://doi.org/10.1016/j.tws.2018.01.007.
  5. Bert, C., Jang, S. and Striz, A. (1987), "New methods for analyzing vibration of structural components", Structures, Structural Dynamics and Materials Conference, Monterey, CA, April.
  6. Bert, C.W., Jang, S.K. and Striz, A.G. (1988), "Two new approximate methods for analyzing free vibration of structural components", AIAA J., 26(5), 612-618. https://doi.org/10.2514/3.9941.
  7. Birman, V. (2011), Plate Structures, Springer Science & Business Media.
  8. Civalek, O. and Ersoy, H. (2009), "Free vibration and bending analysis of circular Mindlin plates using singular convolution method", Commun. Numer. Meth. Eng., 25(8), 907-922. https://doi.org/10.1002/cnm.1138.
  9. Farhatnia, F., Babaei, J. and Foroudastan, R. (2018), "Thermo-Mechanical nonlinear bending analysis of functionally graded thick circular plates resting on Winkler foundation based on sinusoidal shear deformation theory", Arab. J. Sci. Eng., 43(3), 1137-1151. https://doi.org/10.1007/s13369-017-2753-2.
  10. Farhatnia, F., Saadat, R. and Oveissi, S. (2019), "Functionally graded sandwich circular plate of non-uniform varying thickness with homogenous core resting on elastic foundation: Investigation on bending via differential quadrature method", Am. J. Mech. Eng., 7(2), 68-78. https://doi.org/10.12691/ajme-7-2-3
  11. Gupta, U., Ansari, A. and Sharma, S. (2006), "Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation", J. Sound Vib., 297(3-5), 457-476. https://doi.org/10.1016/j.jsv.2006.01.073.
  12. Gupta, U., Lal, R. and Sharma, S. (2006), "Vibration analysis of non-homogeneous circular plate of nonlinear thickness variation by differential quadrature method", J. Sound Vib., 298(4-5), 892-906. https://doi.org/10.1016/j.jsv.2006.05.030.
  13. Hamzehkolaei, S., Malekzadeh, P. and Vaseghi, J. (2011), "Thermal effect on axisymmetric bending of functionally graded circular and annular plates using DQM", Steel Compos. Struct., 11(4), 341-358. https://doi.org/10.12989/scs.2011.11.4.341.
  14. Hosseini-Hashemi, S., Akhavan, H., Taher, H.R.D., Daemi, N. and Alibeigloo, A. (2010), "Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation", Mater. Des., 31(4), 1871-1880. https://doi.org/10.1016/j.matdes.2009.10.060.
  15. Leissa, A.W. (1969), Vibration of Plates, Vol. 160, Scientific and Technical Information Division, National Aeronautics and Space Administration.
  16. Liew, K., Han, J.B. and Xiao, Z. (1997), "Vibration analysis of circular Mindlin plates using the differential quadrature method", J. Sound Vib., 205(5), 617-630. https://doi.org/10.1006/jsvi.1997.1035.
  17. Ma, L. and Wang, T. (2004), "Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory", Int. J. Solid. Struct., 41(1), 85-101. https://doi.org/10.1016/j.ijsolstr.2003.09.008.
  18. Nie, G. and Zhong, Z. (2007), "Axisymmetric bending of two-directional functionally graded circular and annular plates", Acta Mechanica Solida Sinica, 20(4), 289-295. https://doi.org/10.1007/s10338-007-0734-9.
  19. Prakash, T. and Ganapathi, M. (2006), "Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method", Compos. Part B: Eng., 37(7-8), 642-649. https://doi.org/10.1016/j.compositesb.2006.03.005.
  20. Rad, A.B. and Shariyat, M. (2013), "A three-dimensional elasticity solution for two-directional FGM annular plates with non-uniform elastic foundations subjected to normal and shear tractions", Acta Mechanica Solida Sinica, 26(6), 671-690. https://doi.org/10.1016/S0894-9166(14)60010-0.
  21. Reddy, J., Wang, C. and Kitipornchai, S. (1999), "Axisymmetric bending of functionally graded circular and annular plates", Eur. J. Mech.-A/Solid., 18(2), 185-199. https://doi.org/10.1016/S0997-7538(99)80011-4.
  22. Senjanovic, I., Hadzic, N., Vladimir, N. and Cho, D.S. (2014), "Natural vibrations of thick circular plate based on the modified Mindlin theory", Arch. Mech., 66(6), 389-409.
  23. Shariyat, M. and Alipour, M. (2010), "A differential transform approach for modal analysis of variable thickness two-directional FGM circular plates on elastic foundations", Iran. J. Mech. Eng., 11(2), 15-38.
  24. Shariyat, M. and Alipour, M. (2011), "Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations", Arch. Appl. Mech., 81(9), 1289-1306. https://doi.org/10.1007/s00419-010-0484-x.
  25. Sharma, S., Srivastava, S.and Lal, R. (2011), "Free vibration analysis of circular plate of variable thickness resting on Pasternak foundation", J. Int. Acad. Phys. Sci., 15, 1-13.
  26. Shen, H.S. (2016), Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC Press.
  27. Shu, C. (2000), Application of Differential Quadrature Method to Structural and Vibration Analysis, Differential Quadrature and Its Application in Engineering, Springer.
  28. Wu, T., Wang, Y. and Liu, G. (2002), "Free vibration analysis of circular plates using generalized differential quadrature rule", Comput. Meth. Appl. Mech. Eng., 191(46), 5365-5380. https://doi.org/10.1016/S0045-7825(02)00463-2.
  29. Yamanouchi, M., Koizumi, M., Hirai, T. and Shiota, I. (1990), "Proceedings of the first international symposium on functionally gradient materials", Sendai, Japan.
  30. Zheng, L. and Zhong, Z. (2009), "Exact solution for axisymmetric bending of functionally graded circular plate", Tsinghua Sci. Technol., 14, 64-68. https://doi.org/10.1016/S1007-0214(10)70033-X.
  31. Zhou, D., Lo, S., Au, F. and Cheung, Y. (2006), "Three-dimensional free vibration of thick circular plates on Pasternak foundation", J. Sound Vib., 292(3-5), 726-741. https://doi.org/10.1016/j.jsv.2005.08.028.