Mechanics of Composite Materials

, Volume 49, Issue 6, pp 629–640 | Cite as

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

  • K. Nedri
  • N. El Meiche
  • A. Tounsi


The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.


free vibration shear deformation theory of plates laminated composite plate elastic foundation 


  1. 1.
    R. D. Mindlin, “Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates,” J. of Appl. Mechanics, 18, 31-38 (1951).Google Scholar
  2. 2.
    Y. Stavski, “On the theory of symmetrically heterogeneous plates having the same thickness variation of the elastic moduli,” Topics in Applied Mechanics. American Elsevier, N. Y., 105 (1965),Google Scholar
  3. 3.
    P. C Yang, C. H. Norris, and Y. Stavsky, “Elastic wave propagation in heterogeneous plates,” Int. J. of Solids and Structure, 2, 665-684 (1966).Google Scholar
  4. 4.
    S. Srinivas and A. K. Rao, “Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates,” Int. J. of Solids and Structures, 6, 1463-1481 (1970).CrossRefGoogle Scholar
  5. 5.
    J. M. Whitney and C. T. Sun, “A higher-order theory for extensional motion of laminated composites,” J. of Sound and Vibration, 30, 85-97 (1973).CrossRefGoogle Scholar
  6. 6.
    C. W. Bert, “Structure design and analysis: Part I,” in: C. C. Chamis (Ed.), Analysis of Plates. Academic Press, N. Y. (Chapter 4), (1974).Google Scholar
  7. 7.
    J. M. Whitney and N. J. Pagano, “Shear deformation in heterogeneous anisotropic plates,” J. of Appl. Mechanics, 37, 1031-1036 (1970).CrossRefGoogle Scholar
  8. 8.
    R. C. Fortier and J. N. Rossettos, “On the vibration of shear-deformable curved anisotropic composite plates,” J. of Appl. Mechanics, 40, 299-301 (1973).CrossRefGoogle Scholar
  9. 9.
    P. K., Shinha and A. K. Rath, “Vibration and buckling of cross-ply laminated circular cylindrical panels,” Aeronautical Quarterly, 26, 211-218 (1975).Google Scholar
  10. 10.
    C. W. Bert and, T. L. C. Chen, “Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates,” Int. J. of Solids and Structure, 14, 465-473 (1978).CrossRefGoogle Scholar
  11. 11.
    A. K. Noor, “Free vibrations of multilayered composite plates,” AIAA J, 11, 1038-1039 (1973).CrossRefGoogle Scholar
  12. 12.
    J. N. Reddy, “Free vibration of antisymmetric angle-ply laminated plates including transverse shear deformation by the finite element method,” J. of Sound and Vibration, 66 (4), 565-576 (1979).CrossRefGoogle Scholar
  13. 13.
    J. N. Reddy, “A simple higher-order theory for laminated composite plates,” ASME J Appl. Mech., 51, 745-752 (1984).CrossRefGoogle Scholar
  14. 14.
    J. N. Reddy and A. A. Khdeir, “Buckling and vibration of laminated composite plates using various plate theories,” AIAAJ, 27(12), 1808-1817 (1989).CrossRefGoogle Scholar
  15. 15.
    C. A. Shankara and N. G. Iyengar, “A C0 element for the free vibration analysis of laminated composite plates,” J. of Sound and Vibration, 191 (5), 721-738 (1996).CrossRefGoogle Scholar
  16. 16.
    A. A. Khdeir and J. N. Reddy, “Free vibration of laminated composite plates using second-order shear deformation theory,” Compos. Struct., 71, 617-626 (1999).Google Scholar
  17. 17.
    B. N. Singh, D. Yadav, and N. G. R. Iyengar, “Natural frequencies of composite plates with random material properties using higher-order shear deformation theory,” Int. J. of Mechanical Sci., 43, 2193-2214 (2001).CrossRefGoogle Scholar
  18. 18.
    M. Rastgaar, Agaah, M. Mahinfalah, and G. Nakhaie Jazar, “Natural frequencies of laminated composite plates using third-order shear deformation theory,” Composite Structures, 72, 273-279 (2006).CrossRefGoogle Scholar
  19. 19.
    M. Şimşek, “Vibration analysis of a functionally graded beam under a moving mass by using different beam theories,” Compos. Struct., 92, 904-917 (2010).CrossRefGoogle Scholar
  20. 20.
    M. Şimşek, “Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories,” Nuclear Engineering and Design, 240, 697-705 (2010).CrossRefGoogle Scholar
  21. 21.
    H. Matsunaga, “Stress analysis of functionally graded plates subjected to thermal and mechanical loadings,” Compos. Struct., 87, 344-357 (2009).CrossRefGoogle Scholar
  22. 22.
    J. L. Mantari, A. S. Oktem, and C. Guedes Soares, “A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates,” Int. J. of Solids and Structures, 49, 43-53 (2012).CrossRefGoogle Scholar
  23. 23.
    J. N. Reddy, Mechanics of Laminated Composite Plate: Theory and Analysis, N. Y.: CRC Press, 1997.Google Scholar
  24. 24.
    M. Karama, K. S. Afaq, and S. Mistou, “Mechanical behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity,” Int. J. Solids and Structures, 40, 1525-1546 (2003).CrossRefGoogle Scholar
  25. 25.
    M. Touratier, “An efficient standard plate theory,” Int. J. Eng. Sci., 29, 901-916 (1991).CrossRefGoogle Scholar
  26. 26.
    H. Ait Atmane, A. Tounsi, I. Mechab, and E. A. Adda Bedia, “Free vibration analysis of functionally graded plates resting on Winkler-Pasternak elastic foundations using a new shear deformation theory,” Int. J. Mech. Mater. Des., 6 (2), 113-121 (2010).CrossRefGoogle Scholar
  27. 27.
    H. Akhavan, Sh. Hosseini Hashemi, H. Rokni Damavandi Taher, A. Alibeigloo, and Sh. Vahabi, “Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation. Part II: Frequency analysis,” Computational Materials Sci., 44, 951-961 (2009).CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Laboratoire des Matériaux et HydrologieUniversité de Sidi Bel AbbesSidi Bel AbbesAlgérie
  2. 2.Université de Sidi Bel Abbes, Faculté des Sciences de l’Ingénieur, Département de Génie CivilSidi Bel AbbesAlgérie

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