Advertisement

Finite Element Analysis of Composite Laminates

  • O. O. Ochoa
  • J. N. Reddy
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 7)

Abstract

The partial differential equations governing composite laminates (see Section 2.4) of arbitrary geometries and boundary conditions cannot be solved in closed form. Analytical solutions of plate theories are available (see Reddy [1–5]) mostly for rectangular plates with all edges simply supported (i.e., the Navier solutions) or with two opposite edges simply supported and the remaining edges having arbitrary boundary conditions (i.e., the Levy solutions). The Rayleigh-Ritz and Galerkin methods can also be used to determine approximate analytical solutions, but they too are limited to simple geometries because of the difficulty in constructing the approximation functions for complicated geometries. The use of numerical methods facilitates the solution of these equations for problems of practical importance. Among the numerical methods available for the solution of differential equations defined over arbitrary domains, the finite element method is the most effective method. A brief introduction to the finite element method is presented in Section 3.2.

Keywords

Finite Element Analysis Composite Laminate Plate Theory Laminate Plate Shear Deformation Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Reddy, J. N., Mechanics of Laminated Composite Structures: Theory and Analysis, Lecture Notes, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, October 1988, also see Chapters 14 and 15 in: Finite Element Analysis for Engineering Design, J. N. Reddy, C. S. Krishna Moorthy, and K. N. Seetharamu (eds.), Vol. 37, Springer-Verlag, Berlin (1988).CrossRefGoogle Scholar
  2. 2.
    Reddy, J. N., Energy and Variational Methods in Applied Mechanics, John Wiley, New York (1984).zbMATHGoogle Scholar
  3. 3.
    Khdeir, A. A., Reddy, J. N., and Librescu, L., “Levy Type solutions for Symmetrically Laminated Rectangular Plates Using First-Order Shear Deformation Theory,” Journal of Applied Mechanics, 54, pp. 640–642 (1987).Google Scholar
  4. 4.
    Reddy, J. N. and Khdeir, A. A., “Buckling and Vibration of Laminated Composite Plates Using Various Plate Theories,” AIAA Journal, 27 (12), pp. 1808–1817 (1989).MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Nosier, A. and Reddy, J. N., “On Vibration and Buckling of Symmetric Laminated Plates According to Shear Deformation Theories,” Acta Mechanica, in press.Google Scholar
  6. 6.
    Reddy, J. N., An Introduction to the Finite Element Method, first edition, McGraw-Hill, New York (1984); second edition, in press (1993).Google Scholar
  7. 7.
    Reddy, J. N., Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York (1986); reprinted by Krieger Publishing Co., Melbourne, FL (1991).zbMATHGoogle Scholar
  8. 8.
    Reddy, J. N. and Chao, W. C, “A Comparison of Closed-Form and Finite Element Solutions of Thick Laminated Anisotropic Rectangular Plates,” Nuclear Engineering and Design, 64, pp. 153–167 (1981).CrossRefGoogle Scholar
  9. 9.
    Reddy, J. N., “Simple Finite Elements with Relaxed Continuity for Nonlinear Analysis of Plates,” Proceedings of the 3rd International Conference on Finite Element Methods, Australia, pp. 265–281 (July 1979).Google Scholar
  10. 10.
    Reddy, J. N., “A Penalty Plate-Bending Element for the Analysis of Laminated Anisotropic Composite Plates,” International Journal for Numerical Methods in Engineering, 15, pp. 1187–1206 (1980).ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Reddy, J. N., “Analysis of Layered Composite Plates Accounting for Large Deflections and Transverse Shear Strains,” in Recent Advances in Nonlinear Computational Mechanics, E. Hinton, D. R. J. Owen, and C. Taylor (Eds.), pp. 155–202 (1982).Google Scholar
  12. 12.
    Reddy, J. N., “Geometrically Nonlinear Transient Analysis of Laminated Composite Plates,” AIAA Journal, 21 (4), pp. 621–629 (1983).ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Reddy, J. N., “On Refined Computational Models of Composite Laminates,” International Journal for Numerical Methods in Engineering, 27, pp. 361–382 (1989).MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Hughes, T. J. R., The Finite Element Method, Prentice-Hall, Engelwood Cliffs, NJ (1986).Google Scholar
  15. 15.
    Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, Engelwood Cliffs, NJ (1982).Google Scholar
  16. 16.
    Averill, R. C. and Reddy, J. N., “On the Behavior of Plate Elements Based on the First-Order Shear Deformation Theory,” Engineering Computations, 7 (1), pp. 57–74 (1990).CrossRefGoogle Scholar
  17. 17.
    Zienkiewicz, O. C, Taylor, R. L. and Too, J. M., “Reduced Integration Technique in General Analysis of Plates and Shells,” International Journal for Numerical Methods in Engineering, 3, pp. 275–290 (1971).ADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Pawsey, S. F. and Clough, R. W., “Improved Numerical Integration of Thick Shell Finite Elements,” International Journal for Numerical Methods in Engineering, 3, pp. 575–586 (1971).ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Hughes, T. J. R., Cohen, M., and Haroun, M., “Reduced and Selective Integration Techniques in the Finite Eement Analysis of Plates,” Nuclear Engineering and Design, 46, pp. 203–222 (1978).CrossRefGoogle Scholar
  20. 20.
    Malkus, D. S., and Hughes, T. J. R., “Mixed Finite Element Methods-Reduced and Selective Integration Techniques: A Unification of Concepts,” Computer Methods in Applied Mechanics and Engineering, 15, pp. 63–81 (1978).ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Lee, S. W., and Pian, T. H. H., “Improvement of Plate and Shell Finite Elements by Mixed Formulations,” AIAA Journal, 16, pp. 29–34 (1978).ADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Spilker, R. L., and Munir, N. I., “The Hybrid-Stress Model for Thin Plates,” International Journal for Numerical Methods in Engineering, 15, pp. 1239–1260 (1980).MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Park, K. C, and Stanley, G. M., “A Curved Shell Element Based on Assumed Natural-Coordinate Strains,” Journal of Applied Mechanics, 53, pp. 278–290 (1986).ADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Bathe, K. J., Dvorkin, E. N., and Ho, L. W., “On Discrete-Kirchhoff and Isoparametric Shell Elements for Nonlinear Analysis — An Assessment,” Computers and Structures, 16, pp. 89–98 (1983).zbMATHCrossRefGoogle Scholar
  25. 25.
    Zienkiewicz, O. C., and Hinton, E., “Reduced Integration, Function Smoothing and Non-Conformity in Finite Element Analysis (with special reference to thick plates),” Journal of the Franklin Instaure, 302 (5 & 6), pp. 443–461 (1976).zbMATHCrossRefGoogle Scholar
  26. 26.
    Pugh, E. D. L., Hinton, E., and Zienkiewicz, O. C, “A Study of Quadrilateral Plate Bending Elements with ‘Reduced’ Integration,” International Journal for Numerical Methods in Engineering, 12, pp. 1059–1079 (1978).ADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Phan, N. D., and Reddy, J. N., “Analysis of Laminated Composite Plates Using a Higher-Order Shear Deformation Theory,” International Journal for Numerical Methods in Engineering, 12, pp. 2201–2219 (1985).ADSCrossRefGoogle Scholar
  28. 28.
    Ren, J. G., and Hinton, E., “The Finite Element Analysis of Homogeneous and Laminated Composite Plates Using a Simple Higher Order Theory,” Communications in Applied Numerical Methods, 2 (2), pp. 217–228 (1986).zbMATHCrossRefGoogle Scholar
  29. 29.
    Ren, J. G., “Bending of Simply-Supported, Antisymmetrically Laminated Rectangular Plate Under Transverse Loading,” Composites Science and Technology, 28 (3), pp. 231–243 (1987).CrossRefGoogle Scholar
  30. 30.
    Putcha, N. S., and Reddy, J. N., “A Mixed Finite Element for the Analysis of Laminated Plates,” in Advances in Aerospace Structures, Materials and Dynamics, U. Yuceoglu, et al. (Eds.), ASME AD-06, pp. 31–39 (1983).Google Scholar
  31. 31.
    Putcha, N. S., and Reddy, J. N., “A Refined Mixed Shear Flexible Finite Element for the Nonlinear Analysis of Laminated Plates,” Computers and Structures, 22 (2), pp. 529–538 (1986).zbMATHCrossRefGoogle Scholar
  32. 32.
    Putcha, N. S., and Reddy, J. N., “Stability and Natural Vibration Analysis of Laminated Plates by Using a Mixed Element Based on a Refined Plate Theory,” Journal of Sound and Vibration, 104 (2), pp. 285–300 (1986).ADSCrossRefGoogle Scholar
  33. 33.
    Reddy, J. N., “On Mixed Finite-Element Formulations of a Higher-Order Theory of Composite Laminates,” Finite Element Methods for Plate and Shell Structures, T. J. R. Hughes and E. Hinton (Eds.), Pineridge Press, U.K., pp. 31–57, (1986).Google Scholar
  34. 34.
    Averill, R. C, and Reddy, J. N., “An Assessment of Four-Noded Plate Elements Based on a Generalized Third-Order Theory,” International Journal for Numerical Methods in Engineering, 33, pp. 1553–1572 (1992).ADSzbMATHCrossRefGoogle Scholar
  35. 35.
    Barbero, E. J., and Reddy, J. N., “Nonlinear Analysis of Composite Laminates Using a Generalized Laminate Plate Theory,” AIAA Journal, 28(11), pp. 1987–1994 (1990).ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Robbins, D. H., and Reddy, J. N., “Modelling of Thick Composites Using a Layer-Wise Laminate Theory,” International Journal for Numerical Methods in Engineering, in press.Google Scholar
  37. 37.
    Barlow, J., “Optimal Stress Location in Finite Element Models,” International Journal for Numerical Methods in Engineering, 10, pp. 243–251 (1976).ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Barlow, J., “More on Optimal Stress Points — Reduced Integration Element Distortions and Eror Estimation,” International Journal for Numerical Methods in Engineering, 28, pp. 1486–1504 (1989).ADSCrossRefGoogle Scholar
  39. 39.
    Zienkiewicz, O. C, and R. L. Taylor, The Finite Element Method, Vols. 1 and 2, McGraw-Hill, New York, 1989 and 1991.Google Scholar
  40. 40.
    Riks, E., “An Incremental Approach to the Solution of Snapping and Buckling Problem,” International Journal of Solids and Structures, 15, pp. 529–551 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Crisfield, M. A., “A Fast Incremental/Iterative Solution Procedure that Handles Snap-Through,” Computers and Structures, 13, pp. 55–62 (1981).zbMATHCrossRefGoogle Scholar
  42. 42.
    Ramm, E., “Strategies for Tracing the Nonlinear Response Near Limit Points,” Nonlinear Finite Element Analysis in Structural Mechanics, W. Wunderlich, M. Stein, and K. J. Bathe (eds.), Springer-Verlag, Berlin, pp. 63–89 (1981).CrossRefGoogle Scholar
  43. 43.
    Bathe, K. J., Ramm, E., and Wilson, E. L., “Finite Element Formulations for Large Deformation Dynamic Analysis,” International Journal for Numerical Methods in Engineering, 9, pp. 353–386 (1975).ADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Chao, W. C, and Reddy, J. N., “Analysis of Laminated Composite Shells Using a Degenerated 3-D Element,” International Journal for Numerical Methods in Engineering, 20, pp. 1991–2007 (1984).ADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Reddy, J. N., and Heyliger, P. R., “A Mixed Updated Lagrangian Formulation for Plane Elastic Problems,” Journal of Composites Technology and Research, 9 (4), pp. 131–140 (1987).CrossRefGoogle Scholar
  46. 46.
    Reddy, J. N., and Roy, S., “Non-Linear Analysis of Adhesively Bonded Joints,” Journal of Non-Linear Mechanics, 23 (2), pp. 97–112 (1988).zbMATHCrossRefGoogle Scholar
  47. 47.
    Liao, C. L., and Reddy, J. N., “Continuum-Based Stiffened Composite Shell Element for Geometrically Nonlinear Analysis,” AIAA Journal, 27 (1), pp. 95–101 (1989).ADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Liao, C. L., and Reddy, J. N., “Analysis of Anisotropic, Stiffened Composite Laminates Using a Continuum-Based Shell Element,” Computers and Structures, 34 (6), pp. 805–815 (1990).zbMATHCrossRefGoogle Scholar
  49. 49.
    Engblom, J. J., and Ochoa, O. O., “Through-the-Thickness Stress Distribution for Laminated Plates of Advanced Composite Materials,” International Journal for Numerical Methods in Engineering, 21, pp. 1759–1776 (1985).ADSzbMATHCrossRefGoogle Scholar
  50. 50.
    Reddy, J. N., “A Note on Symmetry Considerations in the Transient Response of Unsymmetrically Laminated Anisotropic Plates,” International Journal for Numerical Methods in Engineering, 20, pp. 175–194 (1984).ADSzbMATHCrossRefGoogle Scholar
  51. 51.
    Pagano, N. J., “Exact Solutions for Composite Laminates in Cylindrical Bending,” Journal of Composite Materials, 3, pp. 398–411 (1967).ADSCrossRefGoogle Scholar
  52. 52.
    Pagano, N. J., “Eact Solutions for Rectangular Bidirectional Composites and Sandwich Plates,” Journal of Composite Materials, 4, pp. 20–34 (1970).Google Scholar
  53. 53.
    Sun, C. T., and Chin, H., “Analysis of Asymmetric Composite Laminates,” AIAA Journal, 26 (6), pp. 714–718 (1988).ADSCrossRefGoogle Scholar
  54. 54.
    Zaghloul, S. A., and Kennedy, J. B., “Nonlinear Behavior of Symmetrically Laminated Plates,” Journal of Applied Mechanics, 42, pp. 234–236 (1975).ADSCrossRefGoogle Scholar
  55. 55.
    Bathe, K. J., and Ho, L. W., “A Simple Effective Element for Analysis of General Shell Structures,” Computers and Structures, 13, pp. 673–681 (1981).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • O. O. Ochoa
    • 1
  • J. N. Reddy
    • 1
  1. 1.Texas A&M UniversityTexasUSA

Personalised recommendations