Mechanics of Composite Materials

, Volume 55, Issue 2, pp 241–258 | Cite as

Development of a 2D Isoparametric Finite-Element Model Based on Reddy’s Third-Order Theory for the Bending Behavior Analysis of Composite Laminated Plates

  • K. BelkaidEmail author

The objective of this work is to propose a simple efficient finite element, based on Reddy’s third-order shear deformation theory, for analyzing the bending behavior of isotropic and composite laminated plates. It is a 2D isoparametric quadrilateral C0 four-node element with seven degrees of freedom at each node, three translation components, two rotations, and two higher-order rotational degrees. A selective numerical integration technique is introduced for the numerical formulation in order to improve results and to alleviate the locking problem. The formulation is able to take into account the parabolic distribution of transverse shear across the plate thickness in the bending behavior of isotropic and laminated composite plates, both thin and thick, without a need for correction factors. The performance and reliability of the formulation proposed are confirmed by comparing the author’s results with those obtained using the 3D elasticity theory, analytical solutions, and other advanced finite-element models.


third-order shear deformation theory laminated composite plates finite element bending behavior 


  1. 1.
    D. Gay and S. V. Hoa, Composite Materials: Design and Applications, CRC press (2007).Google Scholar
  2. 2.
    J. N. Reddy, “An evaluation of equivalent-single-layer and layerwise theories of composite laminates,” Compos. Struct., 25, No. 1, 21-35 (1993).CrossRefGoogle Scholar
  3. 3.
    G. Kirchhoff, “Über die Schwingungen einer kreisförmigen elastischen Scheibe,” Annalen der Physik, 157, No. 10, 258-264 (1850).CrossRefGoogle Scholar
  4. 4.
    A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press (2013).Google Scholar
  5. 5.
    E. Reissner, “The effect of transverse shear deformations on the bending of elastic plates,” J. Appl. Mech., 12, No. A69-A77 (1945).Google Scholar
  6. 6.
    J. M. Whitney, “The effect of transverse shear deformation on the bending of laminated plates,” J. Compos. Mater., 3, No. 3, 534-547 (1969).CrossRefGoogle Scholar
  7. 7.
    N. Pagano, “Exact solutions for composite laminates in cylindrical bending,” J. Compos. Mater., 3, No. 3, 398-411 (1969).CrossRefGoogle Scholar
  8. 8.
    J. Whitney, “Shear correction factors for orthotropic laminates under static load,” J. Appl. Mech., 40, No. 1, 302-304 (1973).CrossRefGoogle Scholar
  9. 9.
    Y. Stavsky and R. Loewy, “On vibrations of heterogeneous orthotropic cylindrical shells,” J. Sound and Vibration, 15, No. 2, 235-256 (1971).CrossRefGoogle Scholar
  10. 10.
    J. N. Reddy and D. Robbins, “Theories and computational models for composite laminates,” Appl. Mech. Rev., 47, No. 6, 147-169 (1994).CrossRefGoogle Scholar
  11. 11.
    Y. Ghugal and R. Shimpi, “A review of refined shear deformation theories of isotropic and anisotropic laminated plates,” J. Reinforced Plastics and Compos., 21, No. 9, 775-813 (2002).CrossRefGoogle Scholar
  12. 12.
    J. N. Reddy, “A simple higher-order theory for laminated composite plates,” J. Appl. Mech., 51, No. 4, 745-752 (1984(a)).Google Scholar
  13. 13.
    J. N. Reddy, “A refined nonlinear theory of plates with transverse shear deformation,” Int. J. Solids and Struct., 20, No. 9, 881-896 (1984(c)).Google Scholar
  14. 14.
    J. L. Batozand and M. B. Tahar, “Evaluation of a new quadrilateral thin plate bending element,” Int. J. Numerical Methods in Eng., 18, No. 11, 1655-1677 (1982).CrossRefGoogle Scholar
  15. 15.
    K. Belkaid, A. Tati, and R. Boumaraf, “A simple finite element with five degrees of freedom based on Reddy’s third-order shear deformation theory,” Mech. Compos. Mater., 52, No. 2, 257-270 (2016).CrossRefGoogle Scholar
  16. 16.
    K. Belkaid, and A. Tati, “Analysis of laminated composite plates bending using a new simple finite element based on Reddy’s third order theory,” Revue Des Composites Et Des Materiaux Avances, 25, No. 1, 89-106 (2015).Google Scholar
  17. 17.
    Y. Zhang and C. Yang, “Recent developments in finite element analysis for laminated composite plates,” Compos. Struct., 88, No. 1, 147-157 (2009).CrossRefGoogle Scholar
  18. 18.
    T. Kant, D. Owen, and O. Zienkiewicz, “A refined higher-order C0 plate bending element,” Computers & Struct., 15, No. 2, 177-183 (1982).CrossRefGoogle Scholar
  19. 19.
    B. Pandya and T. Kant, “Flexural analysis of laminated composites using refined higher-order C0 plate bending elements,” Computer Methods in Appl. Mech. Eng., 66, No. 2, 173-198 (1988).CrossRefGoogle Scholar
  20. 20.
    B. Pandya and T. Kant, “A refined higher-order generally orthotropic C0 plate bending element,” Computers & Struct., 28, No. 2, 119-133 (1988).CrossRefGoogle Scholar
  21. 21.
    B. Pandya and T. Kant, “Higher-order shear deformable theories for flexure of sandwich plates-finite element evaluations,” Int. J. Solids and Struct., 24, No. 12, 1267-1286 (1988).CrossRefGoogle Scholar
  22. 22.
    T. Kant and J. Kommineni, “C0 finite element geometrically non-linear analysis of fibre reinforced composite and sandwich laminates based on a higher-order theory,” Computers & Struct., 45, No. 3, 511-520 (1992).CrossRefGoogle Scholar
  23. 23.
    T. Kant and B. Manjunatha, “An unsymmetric FRC laminate C° finite element model with 12 degrees of freedom per node,” Engineering Computations, 5, No. 4, 300-308 (1988).CrossRefGoogle Scholar
  24. 24.
    O. Polit and M. Touratier, “A new laminated triangular finite element assuring interface continuity for displacements and stresses,” Compos. Struct., 38, No. 1, 37-44 (1997).CrossRefGoogle Scholar
  25. 25.
    O. Polit and M. Touratier, “High-order triangular sandwich plate finite element for linear and non-linear analyses,” Computer Methods in Appl. Mech. Eng., 185, No. 2, 305-324 (2000).CrossRefGoogle Scholar
  26. 26.
    O. Polit and M. Touratier, “A multilayered/sandwich triangular finite element applied to linear and non-linear analyses,” Compos. Struct., 58, No. 1, 121-128 (2002).CrossRefGoogle Scholar
  27. 27.
    M. Touratier, “An efficient standard plate theory,” Int. J. Eng. Sci., 29, No. 8, 901-916 (1991).CrossRefGoogle Scholar
  28. 28.
    J. J. Engblom and O. O. Ochoa, “Through-the-thickness stress predictions for laminated plates of advanced composite materials,” Int. J. Numerical Methods in Eng., 21, No. 10, 1759-1776 (1985).CrossRefGoogle Scholar
  29. 29.
    S. Goswami and W. Becker, “A new rectangular finite element formulation based on higher order displacement theory for thick and thin composite and sandwich plates,” World J. of Mech., 3, No. 03, 194 (2013).CrossRefGoogle Scholar
  30. 30.
    P. Bose and J. Reddy, “Analysis of composite plates using various plate theories Part 1: Formulation and analytical solutions,” Struct. Eng. Mech., 6, No. 6, 583-612 (1998).CrossRefGoogle Scholar
  31. 31.
    N. Putcha and J. Reddy, “Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory,” J. Sound and Vibration, 104, No. 2, 285-300 (1986).CrossRefGoogle Scholar
  32. 32.
    C.-P. Wu and H.-C. Kuo, “An interlaminar stress mixed finite element method for the analysis of thick laminated composite plates,” Compos. Struct., 24, No. 1, 29-42 (1993).CrossRefGoogle Scholar
  33. 33.
    W. Chih-Ping and Y. Chung-Bing, “Interlaminar stress mixed finite element analysis of unsymmetrically laminated composite plates,” Computers & Struct., 49, No. 3, 411-419 (1993).CrossRefGoogle Scholar
  34. 34.
    N. Phan and J. Reddy, “Analysis of laminated composite plates using a higher-order shear deformation theory,” Int. J. Numerical Methods in Eng., 21, No. 12, 2201-2219 (1985).CrossRefGoogle Scholar
  35. 35.
    T. Kant and B Pandya, “A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates,” Compos. Struct., 9, No. 3, 215-246 (1988).CrossRefGoogle Scholar
  36. 36.
    R. Averill and J. Reddy, “An assessment of four-noded plate finite elements based on a generalized third-order theory,” Int. J. Numerical Methods in Eng., 33, No. 8, 1553-1572 (1992).CrossRefGoogle Scholar
  37. 37.
    J. Ren and E. Hinton, “The finite element analysis of homogeneous and laminated composite plates using a simple higher order theory,” Communications in Applied Numerical Methods, 2, No. 2, 217-228 (1986).CrossRefGoogle Scholar
  38. 38.
    A. H. Sheikh and A. Chakrabarti, “A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates,” Finite Elements in Analysis and Design, 39, No. 9, 883-903 (2003).CrossRefGoogle Scholar
  39. 39.
    I.-W. Liu, “An element for static, vibration and buckling analysis of thick laminated plates,” Computers & Struct., 59, No. 6, 1051-1058 (1996).CrossRefGoogle Scholar
  40. 40.
    A. Nayak, S. Moy, and R. Shenoi, “Free vibration analysis of composite sandwich plates based on Reddy’s higher-order theory,” Composites: Part B, Eng., 33, No. 7, 505-519 (2002).Google Scholar
  41. 41.
    M. Rezaiee-Pajand, F. Shahabian, and F. Tavakoli, “A new higher-order triangular plate bending element for the analysis of laminated composite and sandwich plates,” Struct. Eng. and Mech., 43, No. 2, 253-271 (2012).CrossRefGoogle Scholar
  42. 42.
    S. J. Lee and H. R. Kim, “FE analysis of laminated composite plates using a higher order shear deformation theory with assumed strains,” Latin Am. J. of Solids and Struct., 10, No. 3, 523-547 (2013).CrossRefGoogle Scholar
  43. 43.
    J. Whitney and N. Pagano, “Shear deformation in heterogeneous anisotropic plates,” J. Appl. Mech., 37, No. 4, 1031-1036 (1970).Google Scholar
  44. 44.
    J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. Second edition ed, Boca Raton, FL, CRC Press (2004).Google Scholar
  45. 45.
    T. J. Hughes, M. Cohen, and M. Haroun, “Reduced and selective integration techniques in the finite element analysis of plates,” Nuclear Engineering and Design, 46, No. 1, 203-222 (1978).CrossRefGoogle Scholar
  46. 46.
    O. C. Zienkiewicz and Y. K. Cheung. The Finite Element Method for Analysis of Elastic Isotropic and Orthotropic Slabs, in ICE Proceedings, Thomas Telford (1964)Google Scholar
  47. 47.
    T. Kant, “Numerical analysis of thick plates,” Computer Methods in Applied Mech. Eng., 31, No. 1, 1-18 (1982).CrossRefGoogle Scholar
  48. 48.
    A. Dobyns, “Analysis of simply-supported orthotropic plates subject to static and dynamic loads,” AIAA J., 19, No. 5. 642-650 (1981).CrossRefGoogle Scholar
  49. 49.
    N. Pagano and H. J. Hatfield, “Elastic behavior of multilayered bidirectional composites,” AIAA J., 10, No. 7, 931-933 (1972).CrossRefGoogle Scholar
  50. 50.
    X. Wang and G. Shi, “A refined laminated plate theory accounting for the third-order shear deformation and interlaminar transverse stress continuity,” Appl. Math. Modelling, 39, No. 18, 5659-5680 (2015).CrossRefGoogle Scholar
  51. 51.
    K. Lee, W. Lin, and S. Chow, “Bidirectional bending of laminated composite plates using an improved zig-zag model,” Compos. Struct., 28, No. 3< 283-294 (1994).Google Scholar
  52. 52.
    S. Goswami, “A C0 plate bending element with refined shear deformation theory for composite structures,” Compos. Struct., 72, No. 3, 375-382 (2006).CrossRefGoogle Scholar
  53. 53.
    T. Pervez, A. Seibi, and F. Al-Jahwari, “Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory,” Compos. Struct., 71, No. 3-4, 414-422 (2005).CrossRefGoogle Scholar
  54. 54.
    T. Kant, A. B. Gupta, S. S. Pendhari, and Y. M. Desai, “Elasticity solution for cross-ply composite and sandwich laminates,” Compos. Struct., 83, No. 1, 13-24 (2008).CrossRefGoogle Scholar
  55. 55.
    N. Pagano, “Exact solutions for rectangular bidirectional composites and sandwich plates,” J. Compos. Mater., 4, No. 1. 20-34 (1970).CrossRefGoogle Scholar
  56. 56.
    S. S. Ramesh, C. Wang, J. Reddy, and K. Ang, “A higher-order plate element for accurate prediction of interlaminar stresses in laminated composite plates,” Compos. Struct., 91, No. 3, 337-357 (2009).CrossRefGoogle Scholar
  57. 57.
    K. Soldatos, “A transverse shear deformation theory for homogeneous monoclinic plates,” Acta Mechanica, 94, No. 3-4, 195-220 (1992).CrossRefGoogle Scholar
  58. 58.
    A. Chakrabarti and A. Sheikh, “A new triangular element to model inter-laminar shear stress continuous plate theory,” Int. J. Numerical Methods in Eng., 60, No. 7, 1237-1257 (2004).CrossRefGoogle Scholar
  59. 59.
    H. Kabir, “A shear-locking free robust isoparametric three-node triangular finite element for moderately-thick and thin arbitrarily laminated plates,” Computers & Struct., 57, No. 4, 589-597 (1995).CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Research Center in Industrial Technologies CRTICheragaAlgeria

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