Discrete-Layer Model for Prediction of Free Edge Stresses in Laminated Composite Plates

  • Received : 2010.10.25
  • Accepted : 2010.12.07
  • Published : 2010.12.31


The discrete-layer model is proposed to analyze the edge-effect problem of laminates under extension and flexure. Based on three-dimensional elasticity theory, the displacement fields of each layer in a laminate have been treated discretely in terms of three displacement components across the thickness. The displacement fields at bottom and top surfaces within a layer are approximated by two-dimensional shape functions. Then two surfaces are connected by one-dimensional high order shape functions. Thus the p-convergent refinement on approximated one- and two-dimensional shape functions can be implemented independently of each other. The quality of present model is mostly determined by polynomial degrees of shape functions for given displacement fields. For nodal modes with physical meaning, the linear Lagrangian polynomials are considered. Additional modes without physical meaning, which are created by increasing nodeless degrees of shape functions, are derived from integrals of Legendre polynomials which have an orthogonality property. Also, it is assumed that mapping functions are linear in the light of shape of laminated plates. The results obtained by this proposed model are compared with those available in literatures. Especially, three-dimensional out-of-plane stresses in the interior and near the free edges are evaluated and convergence performance of the present model is established with the stress results.


Supported by : Korea Research Foundation (KRF)


  1. Ahn, J.S., Woo, K.S., Basu, P.K., Park, J.H. (2009) Subparametric Element Based on Partial-Linear Layerwise Theory for the Analysis of Orthotropic Laminate Composites, Journal of the Computational Structural Engineering Institute of Korea, 22(2), pp.189-196..
  2. Ahn, J.S., Basu, P.K., Woo, K.S. (2010) Analysis of Cracked Aluminum Plates with One-Sided Patch Repair Using $\rho$-Convergent Layered Model, Finite Elements in Analysis and Design, 46(5), pp.438-448. https://doi.org/10.1016/j.finel.2010.01.008
  3. Babuska I., Szabo, B.A., Katz, I.N. (1981) The $\rho$-Version of the Finite Element Method, Journal of the Society for Industrial and Applied Mathematics Series B: Numerical Analysis, 18(3), pp.515-545.
  4. Cho, M., Kim, H.S. (2000) Iterative Free-Edge Stress Analysis of Composite Laminates under Extension, Bending, Twisting and Thermal Loading, International Journal of Solids and Structures, 37, pp.435-459. https://doi.org/10.1016/S0020-7683(99)00014-1
  5. Duster, A, Rank, E. (2001) The $\rho$-Version of the Finite Element Method Compared to an Adaptive h-Version for the Deformation Theory of Plasticity, Computer Methods in Applied Mechanics and Engineering, 190(15-17), pp.1925-1935. https://doi.org/10.1016/S0045-7825(00)00215-2
  6. Ghosh, D.K., Basu, P.K. (1998) A Parallel Programming Environment for Adaptive $\rho$-Version Finite Element Analysis, Advances in Engineering Software, 29(3-6), pp.227-240. https://doi.org/10.1016/S0965-9978(97)00069-0
  7. Hong, C.H., Woo, K.S., Shin, Y.S. (1996) $\rho$-Version Finite Element Model of Stiffened Plates by Hierarchic C0-Element, Journal of the Computational Structural Engineering Institute of Korea, 8(1), pp.33-45.
  8. Hsu, P.W., Herakovich, C.T. (1977) Edge Effects in Angle-Ply Composite Laminates, Journal of Composite Materials, 11(4), pp.422-428. https://doi.org/10.1177/002199837701100405
  9. Nguyen, V.T., Caron, J.F. (2006) A New Finite Element for Free Edge Effect Analysis in Laminated Composites, Computers & Structures, 84(22-23), pp.1538-1546. https://doi.org/10.1016/j.compstruc.2006.01.038
  10. Nguyen, V.T., Caron, J.F. (2009) Finite Element Analysis of Free-Edge Stresses in Composite Laminates under Mechanical and Thermal Loading, Composites Science and Technology, 69(1), pp.40-49. https://doi.org/10.1016/j.compscitech.2007.10.055
  11. Pagano, N.J. (1974) On the Calculation of Interlaminar Normal Stress in Composite Laminate, Journal of Composite Materials, 8(1), pp.65-81. https://doi.org/10.1177/002199837400800106
  12. Pagano, N.J. (1978) Stress Fields in Composite Laminates, International Journal of Solids and Structures, 14(5), pp.385-400. https://doi.org/10.1016/0020-7683(78)90020-3
  13. Pipes, R.B., Pagano, N.J. (1970) Interlaminar Stresses in Composite Laminates under Uniform Axial Extension, Journal of Composite Materials, 4(4), pp.538-548.
  14. Puppo, A.H., Evensen, H.A. (1970) Interlaminar Shear in Laminated Composites under Generalized Plane Stress, Journal of Composite Materials, 4(2), pp.204-220. https://doi.org/10.1177/002199837000400206
  15. Ramesh, S.S., Wang, C.M., Reddy, J.N., Ang, K.K. (2008) Computation of Stress Resultants in Plate Bending Problems Using Higher-Order Triangular Elements, Engineering Structures, 30(10), pp.2687-2706. https://doi.org/10.1016/j.engstruct.2008.03.003
  16. Ramesh, S.S., Wang, C.M., Reddy, J.N., Ang, K.K. (2009) A Higher-Order Plate Element for Accurate Prediction of Interlaminar Stresses in Laminated Composite Plates, Composite Structures, 91(3), pp.337-357. https://doi.org/10.1016/j.compstruct.2009.06.001
  17. Reddy, J.N. (2004) Mechanics of Laminated Composites Plates and Shells; Theory and Analysis, Second Edition, CRC Press, p.831.
  18. Robbins Jr., D.H., Reddy, J.N. (1996) Variable Kinematic Modeling of Laminated Composite Plates, International Journal for Numerical Methods in Engineering, 39(13), pp.2283-2317. https://doi.org/10.1002/(SICI)1097-0207(19960715)39:13<2283::AID-NME956>3.0.CO;2-M
  19. Rybicki, E.F. (1971) Approximate Three-Dimensional Solutions for Symmetric Laminates under Inplane Loading, Journal of Composite Materials, 5(3), pp.354-360. https://doi.org/10.1177/002199837100500305
  20. Tahani, M., Nosier, A. (2003) Three-Dimensional Interlaminar Stress Analysis at Free Edges of General Cross-Ply Composite Laminates, Materials & Design, 24(2), pp.121-130. https://doi.org/10.1016/S0261-3069(02)00107-3
  21. Wang, A.S.D., Crossman, F.W. (1977) Some New Results on Edge Effect in Symmetric Composite Laminates, Journal of Composite Materials, 11(1), pp.92-106. https://doi.org/10.1177/002199837701100110
  22. Wang, S.S., Choi, I. (1982a) Boundary-Layer Effects in Composite Laminates: Part 1-Free-Edge Stress Singularities, Journal of Applied Mechanics, 49(3), pp.541-548. https://doi.org/10.1115/1.3162514
  23. Wang, S.S, Choi, I. (1982b) Boundary-layer Effects in Composite Laminates: Part 2-Free-edge Stress Solutions and Basic Characteristics, Journal of Applied Mechanics, 49(3), pp.549-560. https://doi.org/10.1115/1.3162521
  24. Woo, K.S., Basu, P.K. (1989) Analysis of Singular Cylindrical Shells by $\rho$-Version of FEM, International Journal of Solids and Structures, 25(2), pp.151-165. https://doi.org/10.1016/0020-7683(89)90004-8
  25. Woo, K.S., Hong, C.H., Basu, P.K. (2003) Materially and Geometrically Nonlinear Analysis of Laminated Anisotropic Plates by $\rho$-Version of FEM, Computers & Structures, 81(16), pp.1653-1662. https://doi.org/10.1016/S0045-7949(03)00151-2
  26. Woo, K.S., Jo, J.H., Lee, S.J. (2007) The Selective $\rho$-Distribution for Adaptive Refinement of L-Shaped Plates Subjected to Bending, Journal of the Computational Structural Engineering Institute of Korea, 20(5), pp.533-542 .
  27. Zienkiewicz, O.C., Gago, J.P.D.S.R., Kelly, D.W. (1983) The Hierarchical Concept in Finite Element Analysis, Computers & Structures, 16(1-4), pp.53-65. https://doi.org/10.1016/0045-7949(83)90147-5