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Isogeometric Analysis of Laminated Plates under Free Vibration

  • Lee, Sang Jin (ADOPT Research Group, Department of Architectural Engineering, Gyeongsang National University)
  • Received : 2013.10.26
  • Accepted : 2014.05.25
  • Published : 2014.09.30

Abstract

A plate element is developed by using isogeometric approach in order to determine natural frequencies of laminated composite plates. Reissner-Mindlin (RM) assumptions is adopted to consider the shear deformation and rotatory inertia effect. All terms required in isogeometric element formulation are consistently derived by using Non-uniform rational B-spline surface (NURBS) definition. Gauss quadrature rule is used to form the element stiffness matrix and separately Lobatto quadrature rule is used to calculate element mass matrix. The capability of the present plate element is demonstrated by using numerical examples. From numerical tests, the present isogeometric element produces reliable numerical results for both thin and thick plate situations.

Keywords

References

  1. Akhras, G. and Li, W. (2005) Static and free vibration analysis of composite plates using spline finite strips with higher order shear deformation. Composite Part B: engineering, vol. 36, pp.496-503. https://doi.org/10.1016/j.compositesb.2005.03.001
  2. Cottrell, J.A., Bazilevs, Y. and Hughes, T.J.R. (2009). Isogeometric Analysis: Towards Integration of CAD and FEA. Wiley.
  3. De Boor, C. (1978) A Practical Guide to Splines. Springer.
  4. Ferreira, A.J.M., Roque, C.M.C., Neves, A.M.A., Jorge, R.M.N., Soares, C.M.M. and Liew, K.M. (2011) Buckling and vibration analysis of isotropic and laminated plates by radial basis functions. Composite Part B: engineering, vol. 42, pp. 592-606. https://doi.org/10.1016/j.compositesb.2010.08.001
  5. Haldar, S. and Sengupta, D. (2003) Free vibration analysis of composite right angle triangular plate using a shear flexible element. J. of Reinforced Plastics and Composites, vol. 22 no. 3, pp.229-255. https://doi.org/10.1177/0731684403022003018
  6. Hughes, T.J.R., Cottrell, J. A. and Bazilevs, Y. (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng., vol. 194, no. 39-41, pp. 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008
  7. Hughes, T.J.R. and Evans J.A. (2010). Isogeometric analysis. ICES Report 10-18, The Institute of Computational Engineering and Science, University of Texas Austin.
  8. Kant, T and Swaminathan, K. (2001) Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher order refined theory. Composite Structures, vol.53, pp.73-85. https://doi.org/10.1016/S0263-8223(00)00180-X
  9. Lee, S.J. and Kim, H.R. (2012) Vibration and buckling of thick plates using isogeometric approach. Architectural Research, vol. 15, pp. 35-42. https://doi.org/10.5659/AIKAR.2013.15.1.35
  10. Lee, S.J. and Kim, H.R. (2012) Analysis of plates using isogeometric approach based on Reissner-Mindlin Theory. J. AIK, vol. 28, no. 9, pp.75-82.
  11. Lee, S.J. and Park, K.S. (2011) Free vibration analysis of elastic bars using isogeometric approach, Architectural Research, vol. 13, pp. 41-47. https://doi.org/10.5659/AIKAR.2011.13.3.41
  12. Lee, S.J. and Park, K.S. (2013) Vibrations of Timoshenko beams with isogeometric approach. Applied Mathematical Modelling. vol. 37, no. 22, pp.9174-9190. https://doi.org/10.1016/j.apm.2013.04.034
  13. Liew, K.M., Huang, Y.Q. and Reddy, J.N. (2003) Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method. Computer Methods in Applied Mechanics and Engineering, vol. 192, pp.2203-22. https://doi.org/10.1016/S0045-7825(03)00238-X
  14. Mindlin, R.D. (1951) Influence of rotatory inertia and shear on flexural vibration of isotropic, elastic plates. J. of Applied Mechanics, 18, pp. 31-38.
  15. Nguyen, N., Mai-Duy, N., Karunasena, W., Tran-Cong, T. (2011) Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral at shell element with in-plane rotations. Composite and Structures, vol. 89, pp.612-625. https://doi.org/10.1016/j.compstruc.2011.01.005
  16. Nguyen-Van, H., Mai-Duy, N., Tran-Cong, T. (2008) Free vibration analysis of laminated plate/shell structures based on FSDT with a stabilized nodal-integrated quadrilateral element. J. of Sound and Vibration, vol. 313, pp.205-223. https://doi.org/10.1016/j.jsv.2007.11.043
  17. Reddy, J.N. (1997) Mechanics of laminated composite plate. CRC Press.
  18. Reddy, J.N. and Phan, N.D. (1985) Stability and vibration of isotropic and laminated plates according to higher order shear deformation theory. J. Sound and Vibration, vol. 98, pp.157-170. https://doi.org/10.1016/0022-460X(85)90383-9
  19. Reissner, E. (1945) The effect of transverse shear deformation on the bending of elastic plates. J. of Applied Mechanics, 67, pp. 69-77.
  20. Thai, C. H., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T-H., Nguyen-Thoi, T. and Rabczuk, T. (2012) Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach. International Journal for Numerical Methods in Engineering vol. 91, pp.571-603 https://doi.org/10.1002/nme.4282
  21. Whitney, J.M. and Pagano, N.J. (1970) Shear deformation in heterogeneous anisotropic plates. ASME J. Appl. Mech. vol.37, no.4, pp.1031-1306. https://doi.org/10.1115/1.3408654