DOI QR코드

DOI QR Code

Analytical Asymptotic Solutions for Rectangular Laminated Composite Plates

  • Received : 2011.02.11
  • Accepted : 2011.06.21
  • Published : 2011.06.30

Abstract

An analytical solution for rectangular laminated composite plates was obtained via a formal asymptotic method. From threedimensional static equilibrium equations, the microscopic one-dimensional and macroscopic two-dimensional equations were systematically derived by scaling of the thickness coordinate with respect to the characteristic length of the plate. The onedimensional through-the-thickness analysis was performed by applying a standard finite element method. The derived twodimensional plate equations, which take the form of recursive equations, were solved under sinusoidal loading with simplysupported boundary conditions. To demonstrate the validity and accuracy of the present method, various types of composite plates were studied, such as cross-ply, anti-symmetric angle-ply and sandwich plates. The results obtained were compared to those of the classical laminated plate theory, the first-order shear deformation theory and the three-dimensional elasticity. In the present analysis, the characteristic length of each composite was dependent upon the layup configurations, which affected the convergence rate of the method. The results shown herein are promising that it can serve as an efficient tool for the analysis and design of laminated composite plates.

Keywords

Laminated composite;Sandwich plate;asymptotic analysis;Formal asymptotic method-based plate analysis;Analytical solution

References

  1. Berdichevskii, V. L. (1979). Variational-asymptotic method of constructing a theory of shells. Journal of Applied Mathematics and Mechanics, 43, 711-736. https://doi.org/10.1016/0021-8928(79)90157-6
  2. Berdichevskii, V. L. (1979). Variational-asymptotic method of constructing a theory of shells. Journal of Applied Mathematics and Mechanics, 43, 711-736. https://doi.org/10.1016/0021-8928(79)90157-6
  3. Buannic, N. and Cartraud, P. (2001). Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method. International Journal of Solids and Structures, 38, 7139-7161. https://doi.org/10.1016/S0020-7683(00)00422-4
  4. Carrera, E. (2003). Historical review of Zig-Zag theories for multilayered plates and shells. Applied Mechanics Reviews, 56, 287-308. https://doi.org/10.1115/1.1557614
  5. Cho, M. and Kim, J. H. (1996a). Postprocess method using displacement field of higher order laminated composite plate theory. AIAA Journal, 34, 362-368. https://doi.org/10.2514/3.13072
  6. Cho, M. and Kim, J. S. (1996b). Four-noded finite element post-process method using a displacement field of higherorder laminated composite plate theory. Computers and Structures, 61, 283-290. https://doi.org/10.1016/0045-7949(96)00043-0
  7. Cho, M. and Kim, J. S. (1997). Improved mindlin plate stress analysis for laminated composites in finite element method. AIAA Journal, 35, 587-590. https://doi.org/10.2514/2.145
  8. Dauge, M. and Gruais, I. (1996). Asymptotics of arbitrary order for a thin elastic clamped plate, I. Optimal error estimates. Asymptotic Analysis, 13, 167-197.
  9. Duva, J. M. and Simmonds, J. G. (1992). Influence of two-dimensional end effects on the natural frequencies of cantilevered beams weak in shear. Journal of Applied Mechanics, Transactions ASME, 59, 230-232. https://doi.org/10.1115/1.2899441
  10. Jones, R. M. (1975). Mechanics of Composite Materials. Washington, DC: Scripta Book Co.
  11. Kapania, R. K. and Raciti, S. (1989). Recent advances in analysis of laminated beams and plates. Part I. Shear effects and buckling. AIAA Journal, 27(7), 923-934. https://doi.org/10.2514/3.10202
  12. Kim, J. S. (2009). An asymptotic analysis of anisotropic heterogeneous plates with consideration of end effects. Journal of Mechanics of Materials and Structures, 4, 1535- 1553.
  13. Kim, J. S., Cho, M., and Smith, E. C. (2008). An asymptotic analysis of composite beams with kinematically corrected end effects. International Journal of Solids and Structures, 45, 1954-1977. https://doi.org/10.1016/j.ijsolstr.2007.11.005
  14. Kim, J. S. and Wang, K. W. (2010). Vibration analysis of composite beams with end effects via the formal asymptotic method. Journal of Vibration and Acoustics, Transactions of the ASME, 132, 0410031-0410038.
  15. Niordson, F. I. (1979). An asymptotic theory for vibrating plates. International Journal of Solids and Structures, 15, 167- 181. https://doi.org/10.1016/0020-7683(79)90020-9
  16. Noor, A. K. and Burton, W. S. (1989). Assessment of shear deformation theories for multilayered composite plates. Applied Mechanics Reviews, 42, 1-13. https://doi.org/10.1115/1.3152418
  17. Novotny, B. (1970). On the asymptotic integration of the three-dimensional non-linear equations of thin elastic shells and plates. International Journal of Solids and Structures, 6, 433-451. https://doi.org/10.1016/0020-7683(70)90095-8
  18. Pagano, N. J. (1970). Exact solutions for rectangular bidirectional composites and sandwich plates. Journal of Composite Materials, 4, 20-34.
  19. Park, I. J., Jung, S. N., Kim, D. H., and Yun, C. Y. (2009). General purpose cross-section analysis program for composite rotor blades. International Journal of Aeronautical and Space Sciences, 10, 77-85. https://doi.org/10.5139/IJASS.2009.10.2.077
  20. Reddy, J. N. (2004). Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. 2nd ed. Boca Raton: CRC Press.
  21. Reddy, J. N., Robbins, D. H., and Jr. (1994). Theories and computational models for composite laminates. Applied Mechanics Reviews, 47, 147-169. https://doi.org/10.1115/1.3111076
  22. Savoia, M. and Reddy, J. N. (1992). A variational approach to three-dimensional elasticity solutions of laminated composite plates. Journal of Applied Mechanics, 59, S166-S175. https://doi.org/10.1115/1.2899483
  23. Wang, Y. M. and Tarn, J. Q. (1994). A three-dimensional analysis of anisotropic inhomogeneous and laminated plates. International Journal of Solids and Structures, 31, 497-515. https://doi.org/10.1016/0020-7683(94)90089-2
  24. Yu, W. (2005). Mathematical construction of a Reissner- Mindlin plate theory for composite laminates. International Journal of Solids and Structures, 42, 6680-6699. https://doi.org/10.1016/j.ijsolstr.2005.02.049
  25. Yu, W., Hodges, D. H., and Volovoi, V. V. (2002). Asymptotic construction of Reissner-like composite plate theory with accurate strain recovery. International Journal of Solids and Structures, 39, 5185-5203. https://doi.org/10.1016/S0020-7683(02)00410-9