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3-D Free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM)

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Abstract

In this paper, the free vibration of functionally graded annular plates on elastic foundations, based on the three-dimensional theory of elasticity, using the differential quadrature method for different boundary conditions including simply supported–clamped, clamped–clamped and free–clamped ends is investigated. The foundation is described by the Pasternak or two-parameter model. A semi-analytical approach composed of differential quadrature method (DQM) and series solution are adopted to solve the equations of motions. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The new results can be used as benchmark solutions for future researches.

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References

  1. Finot M., Suresh S.: Small and large deformation of thick and thin-film multilayers: effect of layer geometry, plasticity and compositional gradients. J. Mech. Phys. Solids 44, 683–721 (1996)

    Article  Google Scholar 

  2. Prakash T., Ganapathi M.: Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method. Compos. B. Eng. 37, 642–649 (2006)

    Article  Google Scholar 

  3. Efraim E., Eisenberger M.: Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. J. Sound Vib. 299, 720–738 (2007)

    Article  Google Scholar 

  4. Nie G.J., Zhong Z.: Semi-analytical solution for three-dimensional vibration of functionally graded circular plates. Comput. Methods Appl. Mech. Eng. 196, 4901–4910 (2007)

    Article  MATH  Google Scholar 

  5. Dong C.Y.: Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev–Ritz method. Mater. Des. 29, 1518–1525 (2008)

    Article  Google Scholar 

  6. Gupta U.S., Lal R., Jain S.K.: Effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness. J. Sound Vib. 139, 503–513 (1990)

    Article  Google Scholar 

  7. Laura P.A.A., Gutierrez R.H.: Free vibrations of a solid circular plate of linearly varying thickness and attached to Winkler foundation. J. Sound Vib. 144, 149–161 (1991)

    Article  Google Scholar 

  8. Gupta U.S., Lal R., Sagar R.: Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic Mindlin circular plates. Indian. J. Pure Appl. Math. 25, 1317–1326 (1994)

    MATH  Google Scholar 

  9. Xiang Y., Wang C.M., Kitipornchai S.: Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations. Int. J. Mech. Sci. 36, 311–316 (1994)

    Article  MATH  Google Scholar 

  10. Ju F., Lee H.P.K.H.: Free vibration of plates with stepped variations in thickness on non-homogeneous elastic foundations. J. Sound Vib. 183, 533–545 (1995)

    Article  MATH  Google Scholar 

  11. Xiang Y., Kitipornchai S., Liew K.M.: Buckling and vibration of thick laminates on Pasternak foundations. J. Eng. Mech. ASCE 122, 54–63 (1996)

    Article  Google Scholar 

  12. Matsunaga H.: Vibration and stability of, thick plates on elastic foundations. J. Eng. Mech. ASCE 126, 27–34 (2000)

    Article  Google Scholar 

  13. Gupta U.S., Ansari A.H.: Effect of elastic foundation on axisymmetric vibrations of polar orthotropic linearly tapered circular plates. J. Sound Vib. 254, 411–426 (2002)

    Article  Google Scholar 

  14. Malekzadeh P., Karami G.: Vibration of non-uniform thick plates on elastic foundation by differential quadrature method. Eng. Struct. 26, 1473–1482 (2004)

    Article  Google Scholar 

  15. Zhou D., Cheung Y.K., Lo S.H., Au F.T.K.: Three-dimensional vibration analysis of rectangular thick plates on Pasternak foundation. Int. J. Numer. Methods Eng. 59, 1313–1334 (2004)

    Article  MATH  Google Scholar 

  16. Zhou D., Lo S.H., Au F.T.K., Cheung Y.K.: Three-dimensional free vibration of thick circular plates on Pasternak foundation. J. Sound Vib. 292, 726–741 (2006)

    Article  Google Scholar 

  17. Gupta U.S., Ansari A.H., Sharma S.: Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation. J. Sound Vib. 297, 457–476 (2006)

    Article  Google Scholar 

  18. Hosseini Hashemi S.H., Rokni Damavandi Taher H., Omidi M.: 3-D free vibration analysis of annular plates on Pasternak elastic foundation via p-Ritz method. J. Sound Vib 311, 1114–1140 (2008)

    Article  Google Scholar 

  19. Hosseini-Hashemi Sh., Omidi M., Rokni Damavandi Taher H.: The validity range of CPT and Mindlin plate theory in comparison with 3-D vibrational analysis of circular plates on the elastic foundation. Eur. J. Mech. A Solids 28, 289–304 (2009)

    Article  MATH  Google Scholar 

  20. Ming-Hung Hsu.: Vibration analysis of orthotropic rectangular plates on elastic foundations. Compos. Struct. 92, 844–852 (2010)

    Article  Google Scholar 

  21. Malekzadeh P., Afsari A., Zahedinejad P., Bahadori R.: Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation. Appl. Math. Model. 34, 776–790 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Malekzadeh P.: Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations. Compos. Struct. 89, 367–373 (2009)

    Article  Google Scholar 

  23. Amini, M.H., Soleimani, M., Rastgoo, A.: Three-dimensional free vibration analysis of functionally graded material plates resting on an elastic foundation. Smart Mater. Struct. 180, 85015 (9 pp) (2009)

    Google Scholar 

  24. Yas, M.H., Sobhani Aragh, B.: Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation. Int. J. Eng. Sci. (2010)

  25. Hosseini-Hashemi Sh., Akhavan H., Rokni Damavandi Taher H., Daemi N., Alibeigloo A.: Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation. Mater. Des. 31, 1871–1880 (2010)

    Article  Google Scholar 

  26. Hosseini-Hashemi Sh., Rokni Damavandi Taher H., Akhavan H.: Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations. Compos. Struct. 92, 1734–1743 (2010)

    Article  Google Scholar 

  27. Chen W.Q., Lv C.F., Bian Z.G.: Elasticity solution for free vibration of laminated beams. Compos. Struct. 62, 75–82 (2003)

    Article  Google Scholar 

  28. Shu C., Richards B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stoaks equations. Int. J. Numer. Methods Fluids 15, 8–791 (1992)

    Article  Google Scholar 

  29. Nie G.J., Zhong Z.: Dynamic analysis of multi-directional functionally graded annular plates. Appl. Math. Model. 34, 608–616 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Bert C.W., Malik M.: Differential quadrature method in computational mechanics: A review. Appl. Mech. Rev. 49, 1–27 (1996)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Razi University, Kermanshah, Iran

    M. H. Yas & V. Tahouneh

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  1. M. H. Yas
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  2. V. Tahouneh
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Correspondence to V. Tahouneh.

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Yas, M.H., Tahouneh, V. 3-D Free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM). Acta Mech 223, 43–62 (2012). https://doi.org/10.1007/s00707-011-0543-6

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  • DOI: https://doi.org/10.1007/s00707-011-0543-6

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