DOI QR코드

DOI QR Code

Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations

  • Akgoz, Bekir (Akdeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics) ;
  • Civalek, Omer (Akdeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics)
  • Received : 2011.02.01
  • Accepted : 2011.06.22
  • Published : 2011.09.25

Abstract

In the present manuscript, geometrically nonlinear free vibration analysis of thin laminated plates resting on non-linear elastic foundations is investigated. Winkler-Pasternak type foundation model is used. Governing equations of motions are obtained using the von Karman type nonlinear theory. The method of discrete singular convolution is used to obtain the discretised equations of motion of plates. The effects of plate geometry, boundary conditions, material properties and foundation parameters on nonlinear vibration behavior of plates are presented.

Keywords

Nonlinear dynamics;free vibration;discrete singular convolution;laminated plates;non-linear foundation

References

  1. Baltacolu, A.K. (2009), "Nonlinear static and dynamic analyses of laminated composite plates and shells restingon nonlinear elastic foundations", M.Sc. Seminar, Graduate school of natural and applied sciences, Akdeniz University, (in Turkish).
  2. Bhaskar, A. and Dumir, P.C. (1988), "Nonlinear vibration of orthotropic thin rectangular plates on elastic foundations", J. Sound Vib., 125(1), 1-11. https://doi.org/10.1016/0022-460X(88)90410-5
  3. Chen, C.S. and Fung, C.P. (2004), "Large-amplitude vibration of an initially stressed plate on elastic foundations", Comp. Struct., 82(9-10), 689-701. https://doi.org/10.1016/j.compstruc.2004.02.018
  4. Chien, R.D. and Chen, C.S. (2005), "Nonlinear vibration of laminated plates on a nonlinear elastic foundation", Compos. Struct., 70(1), 90-99. https://doi.org/10.1016/j.compstruct.2004.08.015
  5. Chien, R.D. and Chen, C.S. (2006), "Nonlinear vibration of laminated plates on an elastic foundation", Thin-Walled Struct., 44(8), 852-860. https://doi.org/10.1016/j.tws.2006.08.016
  6. Civalek, O. (2004), "Geometrically non-linear static and dynamic analysis of plates and shells resting on elastic foundation by the method of polynomial differential quadrature (PDQ)", PhD. Thesis, Fyrat University, (in Turkish), Elazig.
  7. Civalek, O. (2005), "Geometrically Nonlinear Dynamic Analysis of Doubly Curved Isotropic Shells Resting on Elastic Foundation by a Combination of HDQ-FD Methods", Int. J. Pressure Vessels and Piping, 82(6), 470-479. https://doi.org/10.1016/j.ijpvp.2004.12.003
  8. Civalek, O. (2006), "Harmonic Differential Quadrature-Finite differences Coupled Approaches for Geometrically Nonlinear Static and Dynamic Analysis of Rectangular Plates on Elastic Foundation", J. Sound Vib., 294(4-5), 966-980. https://doi.org/10.1016/j.jsv.2005.12.041
  9. Civalek, O. (2007a), "Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods", Appl. Math. Model. 31(3), 606-624. https://doi.org/10.1016/j.apm.2005.11.023
  10. Civalek, O. (2007b), "Linear vibration analysis of isotropic conical shells by discrete singular convolution (DSC)", Struct. Eng. Mech., 25, 127-130. https://doi.org/10.12989/sem.2007.25.1.127
  11. Dai, K.Y., Liu, G.R., Lim, K.M. and Chen, X.L. (2004), "A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates", J. Sound Vib., 269(3-5), 633-652. https://doi.org/10.1016/S0022-460X(03)00089-0
  12. Dumir, P.C. (1988), "Nonlinear dynamic response of isotropic thin rectangular plates on elastic foundations", Acta Mechanica, 71(1-4), 233-244. https://doi.org/10.1007/BF01173950
  13. Dumir, P.C. and Bhaskar, A. (1988), "Nonlinear static analysis of rectangular plates on elastic foundations by the orthogonal point collocation method", Comp. Meth. Appl. Mech. Eng., 67(1), 111-124. https://doi.org/10.1016/0045-7825(88)90071-0
  14. Ganapathi, M., Varadan, T.K. and Sarma, B.S. (1991), "Nonlinear flexural vibrations of laminated orthotropic plates", Comp. Struct., 39(6), 685-688. https://doi.org/10.1016/0045-7949(91)90211-4
  15. Kant, T., Arora, C.P. and Varaiya, J.H. (1992), "Finite element transient analysis of composite and sandwich plates based on a refined theory and mode superposition method", Compos. Struct., 22(2), 109-120. https://doi.org/10.1016/0263-8223(92)90071-J
  16. Kant, T. and Kommineni, J.R. (1994), "Large amplitude free vibration analysis of cross-ply composite and sandwich laminates with a refined theory and $C^0$ finite elements", Comp. Struct., 51(1), 123-134.
  17. Kant, T. and Mallikarjuna, B.N. (1989), "Transient dynamic of composite sandwich plates using 4-, 8-, 9- noded isoparametric quadrilateral elements", Finite Elements in Anal. Design, 5(4), 307-318. https://doi.org/10.1016/0168-874X(89)90010-3
  18. Kant, T., Ravichandran, R.V., Pandya, B.N. and Mallikarjuna, B.N. (1988), "Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order theory", Compos. Struct., 9(4), 319-342. https://doi.org/10.1016/0263-8223(88)90051-7
  19. Kant, T., Varaiya, J.H. and Arora, C.P. (1990), "Finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration schemes", Comp. Struct., 36(3), 401-420. https://doi.org/10.1016/0045-7949(90)90279-B
  20. Lai, S.K. and Xiang, Y. (2009), "DSC analysis for buckling and vibration of rectangular plates with elastically restrained edges and linearly varying in-plane loading", Int. J. Struct. Stab. Dynamics, 9(3), 511-531. https://doi.org/10.1142/S0219455409003119
  21. Liew, K.M., Han, J.-B. and Xiao, Z.M. (1996), "Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility", Int. J. Solids Struct., 33(18), 2647-2658. https://doi.org/10.1016/0020-7683(95)00174-3
  22. Liew, K.M. and Huang, Y.Q. (2003), "Bending and buckling analysis of thick symmetric rectangular laminates using the moving least-squares differential quadrature method", Int. J. Mech. Sci., 45(1), 95-114. https://doi.org/10.1016/S0020-7403(03)00037-7
  23. Liew, K.M., Huang, Y.Q. and Reddy, J.N. (2003), "Vibration analysis of symmetrically laminated plates based FSDT using the moving least squares differential quadrature method", Comput. Meth. Appl. Mech. Eng., 192(19), 2203-2222. https://doi.org/10.1016/S0045-7825(03)00238-X
  24. Liew, K.M., Wang, J., Tan, M.J. and Rajendran, S. (2004), "Nonlinear analysis of laminated composite plates using the mesh-free kp-Ritz method based on FSDT", Comp. Meth. Appl. Mech. Eng., 193(45-47), 4763-4779. https://doi.org/10.1016/j.cma.2004.03.013
  25. Liew, K.M., Zhang, J.Z., Ng, T.Y. and Meguid, S.A. (2003), "Three-dimensional modeling of elastic bonding in composite laminates using layerwise differential quadrature", Int. J. Solids Struct., 40(7), 1745-1764. https://doi.org/10.1016/S0020-7683(02)00666-2
  26. Liu, G.R. and Achenbach, J.D. (1994), "A strip element method for stress analysis of anisotropic linearly elastic solid", ASME J. Appl. Mech., 61(2), 270-277. https://doi.org/10.1115/1.2901440
  27. Liu, G.R. and Chen, X.L. (2001), "A mesh free method for static and free vibration analysis of thin plates of arbitrary shape", J. Sound Vib., 241(5), 839-855. https://doi.org/10.1006/jsvi.2000.3330
  28. Liu, G.R. and Chen, X.L. (2002), "Bucking of symmetrically laminated composite plates using the element-free Galerkin method", Int. J. Struct. Stab. Dynamics, 2(3), 281-294. https://doi.org/10.1142/S0219455402000634
  29. Manoj, T., Ayyappan, M., Krishnan, K.S. and Rao, B. N. (2000), "Nonlinear vibration analysis of thin laminated rectangular plates on elastic foundations", ZAMM, 80(3), 183-192. https://doi.org/10.1002/(SICI)1521-4001(200003)80:3<183::AID-ZAMM183>3.0.CO;2-P
  30. Nath, Y., Kumar, S. (1995a), "Chebyshev series solution to nonlinear boundary value problems in rectangular domain", Comp. Meth. Appl. Mech. Eng., 125(1-4), 41-52. https://doi.org/10.1016/0045-7825(95)00801-7
  31. Nath, Y., Prithviraju, M. and Mufti, A.A. (2006), "Nonlinear static and dynamics of antisymmetric composite laminated square plates supported on nonlinear elastic subgrade", Commun. Nonlinear Sci. Numer. Simulat., 11(3), 340-354. https://doi.org/10.1016/j.cnsns.2004.11.003
  32. Nath, Y. and Shukla, K.K. (2001), "Non-linear transient analysis of moderately thick laminated composite plates", J. Sound. Vib., 273(3), 509-526.
  33. Nath, Y., Varma, K.K. and Mahrenholtz, D. (1986), "Nonlinear Dynamic response of rectangular plates on linear elastic foundation", Comp. Struct., 24(3), 391-399. https://doi.org/10.1016/0045-7949(86)90316-0
  34. Seckin, A. and Sarygul, A.S. (2009a), "Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: Algorithm and verification", J. Sound Vib. 315(1-2), 197-211.
  35. Seckin, A. and Sarygul, A.S. (2009b), "A novel scheme for the discrete prediction of high-frequency vibration response: Discrete singular convolution-mode superposition approach", J. Sound Vib., 320(4-5), 1004-1022. https://doi.org/10.1016/j.jsv.2008.08.031
  36. Shen, H.-S. (1999), "Large deflection of composite laminated plates under transverse and in-plane loads resting on elastic foundation", Compos. Struct., 45(2), 115-123. https://doi.org/10.1016/S0263-8223(99)00007-0
  37. Shen, H.-S. (2000a), "Nonlinear analysis of composite laminated thin plates subjected to lateral pressure and thermal loading and resting on elastic foundations", Compos. Struct., 49(2), 115-128. https://doi.org/10.1016/S0263-8223(99)00053-7
  38. Shen, H.-S. (2000b), "Nonlinear bending of shear deformable laminated plates under transverse and in-plane loads resting on elastic foundation", Compos. Struct., 50(2), 131-142. https://doi.org/10.1016/S0263-8223(00)00088-X
  39. Shen, H.-S. (2000c), "Nonlinear bending of shear deformable laminated plates under lateral pressure and thermal loading and resting on elastic foundations", J. Strain Analy., 35(2), 93-108. https://doi.org/10.1243/0309324001514053
  40. Shih, Y.-S. and Blotter, P.T. (1993), "Non-linear vibration analysis of arbitrarily laminated thin rectangular plates on elastic foundations", J. Sound Vib., 167(3), 433-459. https://doi.org/10.1006/jsvi.1993.1347
  41. Shukla, K.K. and Nath, Y. (2000), "Non-linear analysis of moderately thick laminated rectangular plates", J. Eng. Mech. ASCE, 26, 831-838.
  42. Shukla, K.K., Nath, Y. and Kreuzer, E. (2005), "Buckling and transient behaviour of layered composite plates under thermomechanical loading", ZAMM Journal, 85(3), 163-175. https://doi.org/10.1002/zamm.200310170
  43. Sofiyev, A.H. (2010), "Buckling analysis of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation", Mech. Research Commun, 37(6), 539-544. https://doi.org/10.1016/j.mechrescom.2010.07.019
  44. Teo, T.M. and Liew, K.M. (2002), "Differential cubature method for analysis of shear deformable rectangular plates on Pasternak foundations", Int. J. Mech. Sci., 44(6), 1179-1194. https://doi.org/10.1016/S0020-7403(02)00034-6
  45. Wang, Y.Y., Lam, K.Y. and Liu, G.R. (2000), "Bending Analysis of Classical Symmetric Laminated Plates by the Strip Element Method", Mech. Compos. Mater. Struct. 7(3), 225-247. https://doi.org/10.1080/10759410050031095
  46. Wang, Y.Y., Lam, K.Y, and Liu, G. R. (2001), "A Strip Element Method for the Transient Analysis of Symmetric Laminated Plates", Int. J. Solids Struct., 38(2), 241-259. https://doi.org/10.1016/S0020-7683(00)00035-4
  47. Wei, G.W. (1999), "Discrete singular convolution for the solution of the Fokker-Planck equations", J. Chem. Phys., 110, 8930-8942. https://doi.org/10.1063/1.478812
  48. Wei, G.W. (2001a), "A new algorithm for solving some mechanical problems", Comp. Meth. Appl. Mech. Eng., 190(15-17), 2017-2030. https://doi.org/10.1016/S0045-7825(00)00219-X
  49. Wei, G.W. (2001b), "Discrete singular convolution for beam analysis", Eng. Struct., 23(9), 1045-1053. https://doi.org/10.1016/S0141-0296(01)00016-5
  50. Wei, G.W., Kouri, D.J. and Hoffman, D.K. (1998), "Wavelets and distributed approximating functionals", Comp. Phys. Commun., 112(1), 1-6. https://doi.org/10.1016/S0010-4655(98)00051-4
  51. Wei, G.W., Zhao, Y.B. and Xiang, Y. (2001), "The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution", Int. J. Mech. Sci., 43(8), 1731-1746. https://doi.org/10.1016/S0020-7403(01)00021-2
  52. Wei, G.W., Zhao, Y.B. and Xiang, Y. (2002a), "Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm", Int. J. Numer. Meth. Eng., 55(8), 913-946. https://doi.org/10.1002/nme.526
  53. Wei, G.W., Zhao, Y.B. and Xiang, Y. (2002b), "A novel approach for the analysis of high-frequency vibrations", J. Sound Vib., 257(2), 207-246. https://doi.org/10.1006/jsvi.2002.5055
  54. Xiang, Y., Lai, S.K. and Zhou, L. (2010), "DSC-element method for free vibration analysis of rectangular Mindlin plates", Int. J. Mech. Sci., 52(4), 548-560. https://doi.org/10.1016/j.ijmecsci.2009.12.001
  55. Xiang, Y., Lai, S.K., Zhou, L. and Lim, C.W. (2010), "DSC-Ritz element method for vibration analysis of rectangular Mindlin plates with mixed edge supports", Eur. J. Mech. A/Solids, 29(4), 619-628. https://doi.org/10.1016/j.euromechsol.2009.12.007
  56. Xiang, Y., Zhao, Y.B. and Wei, G.W. (2002), "Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: Applications", Int. J. Numer. Meth. Eng., 55(8), 947-971. https://doi.org/10.1002/nme.527
  57. Zhao, S. and Wei, G.W. (2003), "Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher's equation", SIAM J. Sci. Comput., 25(1), 127-147. https://doi.org/10.1137/S1064827501390972
  58. Zhao, S., Wei, G.W. and Xiang, Y. (2005), "DSC analysis of free-edged beams by an iteratively matched boundarymethod", J. Sound Vib., 284(1-2), 487-493. https://doi.org/10.1016/j.jsv.2004.08.037
  59. Zhao, Y.B. and Wei, G.W. (2002), "DSC analysis of rectangular plates with non-uniform boundary conditions", J. Sound Vib., 255(2), 203-228. https://doi.org/10.1006/jsvi.2001.4150
  60. Zhao, Y.B., Wei, G.W. and Xiang, Y. (2002a), "Discrete singular convolution for the prediction of high frequency vibration of plates", Int. J. Solids Struct., 39(1), 65-88. https://doi.org/10.1016/S0020-7683(01)00183-4

Cited by

  1. Thermal post-buckling and vibration analysis of a symmetric sandwich beam with clamped and simply supported boundary conditions 2017, https://doi.org/10.1007/s00419-017-1326-x
  2. A novel approximate solution for nonlinear problems of vibratory systems vol.57, pp.6, 2016, https://doi.org/10.12989/sem.2016.57.6.1039
  3. Analytical study of nonlinear vibration of oscillators with damping vol.9, pp.1, 2015, https://doi.org/10.12989/eas.2015.9.1.221
  4. Study of complex nonlinear vibrations by means of accurate analytical approach vol.17, pp.5, 2014, https://doi.org/10.12989/scs.2014.17.5.721
  5. Nonstationary Deformation of Longitudinally and Transversely Reinforced Cylindrical Shells on an Elastic Foundation vol.52, pp.1, 2016, https://doi.org/10.1007/s10778-016-0733-y
  6. High conservative nonlinear vibration equations by means of energy balance method vol.11, pp.1, 2016, https://doi.org/10.12989/eas.2016.11.1.129
  7. Nonstationary Vibrations of Transversely Reinforced Elliptic Cylindrical Shells on an Elastic Foundation vol.52, pp.6, 2016, https://doi.org/10.1007/s10778-016-0785-z
  8. A homogenization procedure for geometrically non-linear free vibration analysis of functionally graded annular plates with porosities, resting on elastic foundations vol.7, pp.1, 2016, https://doi.org/10.1016/j.asej.2015.11.016
  9. A comparative study for bending of cross-ply laminated plates resting on elastic foundations vol.15, pp.6, 2015, https://doi.org/10.12989/sss.2015.15.6.1569