Acta Mechanica Sinica

, Volume 35, Issue 1, pp 174–189 | Cite as

Buckling analysis of functionally graded plates partially resting on elastic foundation using the differential quadrature element method

  • Arash ShahbaztabarEmail author
  • Koosha Arteshyar
Research Paper


We extend the differential quadrature element method (DQEM) to the buckling analysis of uniformly in-plane loaded functionally graded (FG) plates fully or partially resting on the Pasternak model of elastic support. Material properties of the FG plate are graded in the thickness direction and assumed to obey a power law distribution of the volume fraction of the constituents. To set up the global eigenvalue equation, the plate is divided into sub-domains or elements and the generalized differential quadrature procedure is applied to discretize the governing, boundary and compatibility equations. By assembling discrete equations at all nodal points, the weighting coefficient and force matrices are derived. To validate the accuracy of this method, the results are compared with those of the exact solution and the finite element method. At the end, the effects of different variables and local elastic support arrangements on the buckling load factor are investigated.


Differential quadrature element method Pasternak elastic support Critical buckling load Functionally graded plates 


  1. 1.
    Yamanouchi, M., Koizumi, M., Hirai, T., et al.: In: Proceedings of First International Symposium on Functionally Gradient Materials, Sendai, Japan, (1990)Google Scholar
  2. 2.
    Latifi, M., Farhatnia, F., Kadkhodaei, M.: Buckling analysis of rectangular functionally graded plates under various edge conditions using Fourier series expansion. Eur. J. Mech. A Solids. 41, 16–27 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang, L.W., Zhu, P., Liew, K.M.: Thermal buckling of functionally graded plates using a local Kriging meshless method. Compos. Struct. 108, 472–492 (2014)CrossRefGoogle Scholar
  4. 4.
    Yu, T., Yin, S.H., Bui, T.Q., et al.: Buckling isogeometric analysis of functionally graded ceramic-metal plates under combined thermal and mechanical loads. Compos. Struct. 162, 54–69 (2017)CrossRefGoogle Scholar
  5. 5.
    Chen, X.L., Liew, K.M.: Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads. Smart Mater. Struct. 13, 1430–1437 (2004)CrossRefGoogle Scholar
  6. 6.
    Van Do, V.N., Lee, C.H.: Thermal buckling analyses of FGM sandwich plates using the improved radial point interpolation mesh-free method. Compos. Struct. 177, 171–186 (2017)CrossRefGoogle Scholar
  7. 7.
    Kandasamy, R., Dimitri, R., Tornabene, F.: Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments. Compos. Struct. 157, 207–221 (2016)CrossRefGoogle Scholar
  8. 8.
    Parida, S., Mohanti, S.C.: Vibration and stability analysis of functionally graded skew plate using higher order shear deformation theory. Int. J. Appl. Comput. Math. 4, 22 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fallah, A., Aghdam, M.M., Kargarnovin, M.H.: Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method. Arch. Appl. Mech. 83, 177–191 (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kiani, Y.: Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method. Compos. Part B Eng. 105, 176–187 (2016)CrossRefGoogle Scholar
  11. 11.
    Sari, M.S., Ceballes, S., Abdelkefi, A.: Nonlocal buckling analysis of functionally graded nano-plates subjected to biaxial linearly varying forces. Microsyst. Technol. 24(4), 1935–1948 (2018)CrossRefGoogle Scholar
  12. 12.
    Thai, H.T., Kim, S.E.: Closed-form solution for bucking analysis of thick functionally graded plates on elastic foundation. Int. J. Mech. Sci. 75, 34–44 (2013)CrossRefGoogle Scholar
  13. 13.
    Bodaghi, M., Saidi, A.R.: Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Appl. Math. Model. 34, 3659–3673 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kulkarni, K., Singh, B.N., Maiti, D.K.: Analytical solution for bending and buckling analysis of functionally graded plates using inverse trigonometric shear deformation theory. Compos. Struct. 134, 147–157 (2015)CrossRefGoogle Scholar
  15. 15.
    Bellman, R.E., Casti, J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34, 235–238 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bert, C.W., Wang, X., Striz, A.G.: Differential quadrature for static and free vibration analyses of anisotropic plates. Int. J. Solids Struct. 30(13), 1737–1744 (1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Strokes equations. Int. J. Numer. Methods Fluids 15, 791–798 (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Bert, C.W., Jang, S.K., Striz, A.G.: Two new approximate methods for analyzing free vibration of structural components. AIAA J. 26, 612–618 (1988)CrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, X., Bert, C.W.: A new approach in applying differential quadrature to static and free vibration analyses of beams and plates. J. Sound Vib. 162(3), 566–572 (1993)CrossRefzbMATHGoogle Scholar
  20. 20.
    Shu, C., Du, H.: A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates. Int. J. Solids Struct. 34(7), 837–846 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    Striz, A.G., Chen, W., Bert, C.W.: Static analysis of structures by the quadrature element method (QEM). Int. J. Solids Struct. 31(20), 2807–2818 (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, X., Tan, M., Zhou, Y.: Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method. Thin-Walled. Struct. 41, 15–29 (2003)CrossRefGoogle Scholar
  23. 23.
    Chen, C.N.: A generalized differential quadrature element method. Comput. Methods Appl. Mech. Eng. 188, 553–566 (2000)CrossRefzbMATHGoogle Scholar
  24. 24.
    Han, J.B., Liew, K.M.: Static analysis of Mindlin plates: the differential quadrature element method (DQEM). Comput. Methods Appl. Mech. Eng. 177, 51–75 (1999)CrossRefzbMATHGoogle Scholar
  25. 25.
    Liu, F.L., Liew, K.M.: Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method. J. Sound Vib. 225(5), 915–934 (1999)CrossRefzbMATHGoogle Scholar
  26. 26.
    Karami, G., Malekzadeh, P.: A new differential quadrature methodology for beam analysis and the associated differential quadrature element method. Comput. Methods Appl. Mech. Eng. 191, 3509–3526 (2002)CrossRefzbMATHGoogle Scholar
  27. 27.
    Nobakhti, S., Aghdam, M.M.: Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips. Eur. J. Mech. A/Solids 30, 442–448 (2011)CrossRefzbMATHGoogle Scholar
  28. 28.
    Jahromi, H.N., Aghdam, M.M., Fallah, A.: Free vibration analysis of Mindlin plates partially resting on Pasternak foundation. Int. J. Mech. Sci. 75, 1–7 (2013)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Ocean EngineeringAmirKabir University of TechnologyTehranIran
  2. 2.Department of Mechanical EngineeringIslamic Azad UniversityKarajIran

Personalised recommendations