Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Analysis of non-homogeneous orthotropic plates using EDQM

  • Rajasekaran, S. (Department of Civil Engineering, PSG College of Technology)
  • Received : 2016.04.27
  • Accepted : 2016.12.19
  • Published : 2017.01.25
⟨ Previous Next ⟩

Abstract

Element based differential quadrature method (EDQM) has been applied to analyze static, stability and free vibration of non-homogeneous orthotropic rectangular plates of variable or stepped thickness. The Young's modulus and the density are assumed to vary in exponential form in X-direction whereas the thickness is assumed to vary linear, parabolic or exponential variation in one or two directions. In-plane loading is assumed to vary linearly. Various combinations of clamped, simply supported and free edge conditions (regular and irregular boundary) have been considered. Continuous plates could also be handled with ease. In this paper, formulation for equilibrium, buckling and free vibration problems is discussed and several numerical examples are solved using EDQM and compared with the published results.

Keywords

References

  1. Akiyama, K. and Kuroda, M. (1997), "Fundamental frequencies of rectangular plates with linearly varying thickness", J. Sound Vib., 205(3), 380-384. https://doi.org/10.1006/jsvi.1997.1058
  2. Bellman, R. and Casti, J. (1971), "Differential quadrature and long term integration", J. Math. Anal. Appl., 34, 235-238. https://doi.org/10.1016/0022-247X(71)90110-7
  3. Bert, C.W. and Malik, M. (1996), "Differential quadrature methods in computational mechanics: a review", Appl. Mech. Rev., 49(1), 1-28. https://doi.org/10.1115/1.3101882
  4. Bert, C.W. and Malik, M. (1996), "Free vibration analysis of tapered rectangular plate by differentia quadrature method-a semi-analytical approach", J. Sound Vib., 190(1), 41-63. https://doi.org/10.1006/jsvi.1996.0046
  5. Cal, R. and Saini, F. (2013), "Buckling and vibration of nonhomogeneous rectangular plates subjected to linearly varying in-plane force", Shock Vib. Dig., 20, 879-894. https://doi.org/10.1155/2013/579813
  6. Cheung, Y.K. and Zhou, D. (1989), "The free vibrations of tapered rectangular plates using a new set of beam functions with the Rayleigh-Ritz method", J. Sound Vib., 223, 703-722.
  7. Civalek, O. (2004), "Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns", Eng. Struct., 26(2), 171-186. https://doi.org/10.1016/j.engstruct.2003.09.005
  8. Civalek, O. (2006), "Harmonic differential quadratic finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation , J. Sound Vib., 294. 966-980. https://doi.org/10.1016/j.jsv.2005.12.041
  9. Civalek, O. (2009), "Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method", Appl. Math. Model., 33, 3825-3835. https://doi.org/10.1016/j.apm.2008.12.019
  10. Civalek, O. and Ulker, M. (2004), "Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates", Struct. Eng. Mech., 17(1), 1-14. https://doi.org/10.12989/sem.2004.17.1.001
  11. Civalek, O. and Ulker, M. (2005), "HDQ-FD integrated methodology for nonlilnear static and dynamic response of doubly curved shallow shells", Struct. Eng. Mech., 19(5), 535-550. https://doi.org/10.12989/sem.2005.19.5.535
  12. Civalek, O., Korkmaz, A. and Demir, C. (2010), "Discrete singular convolution approach for buckling analysis of rectangular Kirhhoff plates subjected to compressive loads on two- opposite edges", Adv. Eng. Softw., 41(4), 557-560. https://doi.org/10.1016/j.advengsoft.2009.11.002
  13. Dokainish, M.A. and Kumar, K. (1973), "Vibration of orthotropic parallelogram plate with variable thickness", AIAA J., 11(12), 1618-1621. https://doi.org/10.2514/3.50658
  14. Eisenberger, M. and Alexandrov, A. (2003), "Buckling loads of variable thickness thin isotropic plates", Thin Wall. Struct., 41(9), 871-889 https://doi.org/10.1016/S0263-8231(03)00027-2
  15. Fares, M.E. and Zenkour, A.M. (1999), "Bucking and free vibration of non-homogeneous composite cross-ply laminated plates with various plate theories", Compos. Struct., 44(4), 279-287. https://doi.org/10.1016/S0263-8223(98)00135-4
  16. Ferreira, A.J.M., Roque, C.M.C., Carrera, E. and Cinefra, M. (2011), "Analysis of thick isotropic and cross-ply laminated plates by radial basis functions and a unified formulation", J. Sound Vib., 330, 771-787. https://doi.org/10.1016/j.jsv.2010.08.037
  17. Gorman, D.J. (1993), "Accurate free vibration analysis of the completely free orthotropic rectangular plates by the method of superposition", J. Sound Vib., 65, 409-420.
  18. Huang, M., Ma, X.Q., Sakiyama, T., Matuda, H. and Morita, C. (2005), "Free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions", J. Sound Vib., 287, 931-955.
  19. Kang, J.H. and Leissa, A.W. (2005), "Exact solutions for the buckling of rectangular plates having varying in-plane loading on two opposite simply supported edges", Int J. Solid. Struct., 42, 4220-38. https://doi.org/10.1016/j.ijsolstr.2004.12.011
  20. Kukreti, A.A., Farsa, J. and Bert, C.W. (1996), "Differential quadrature and Rayleigh-Ritz methods in determining the fundamental frequencies of simply supported plates with linearly varying thickness", J. Sound Vib., 189, 103-122. https://doi.org/10.1006/jsvi.1996.0008
  21. Lal, R. (2007), "Transverse vibration of non-homogeneous orthotropic rectangular plate of variable thickness: a spline technique", J. Sound Vib., 306, 203-214. https://doi.org/10.1016/j.jsv.2007.05.014
  22. Lal, R. and Gupta, U.S. (1997), "Quintic splines on the study of transverse vibrations of non-uniform orthotropic rectangular plates", J. Sound Vib., 208, 1-13. https://doi.org/10.1006/jsvi.1997.1083
  23. Lal, R., Gupta, U.S. and Goel, C. (2001), "Chebyshev polynomials in the study of Transverse vibrations of non-uniform rectangular orthotropic plates", Shock Vib. Dig., 33, 103-112. https://doi.org/10.1177/058310240103300202
  24. Laura, P.A.A. and Gutierrez, R.H. (1993), "Analysis of vibrating Timoshenko beam using the method of differential quadrature", Shock Vib. Dig., 1, 89-93. https://doi.org/10.1155/1993/124195
  25. Leissa, A.W. (1969), "Vibration of plates", NASA, SP- 160, US Govt Printing office, Washington, DC.
  26. Leissa, A.W. (1977), "Recent research in plate vibrations, 1973-1976: classical theory", Shock Vib. Dig., 9, 13-24.
  27. Leissa, A.W. (1987), "Recent studies in plate vibrations: 1981-85, Part II, complicating effect", Shock Vib. Dig., 19, 19-24.
  28. Leissa, A.W. (1987), "Recent studies in plate vibrations: Part I, classical theory", Shock Vib. Dig., 19, 11-18.
  29. Lekhnitskii, S.G. (1968), Anisotropic Plates, Gordon and Breach, New York.
  30. Liew, K.M. and Wang, C.M. (1993), "pb2-Rayleigh-Ritz method for general plate analysis", Eng. Struct., 15(1), 55-60. https://doi.org/10.1016/0141-0296(93)90017-X
  31. Lim, C.W. and Liew, K.M. (1993), "Effect of boundary constraint and thickness variation on the vibratory response of rectangular plates", Thin Wall. Struct., 17(2), 133-159. https://doi.org/10.1016/0263-8231(93)90031-5
  32. Liu, F.L. (2000), "Rectangular thick plates on Winkler foundation: differential quadrature element solution", Int. J. Solid. Struct., 37, 1743-1763, https://doi.org/10.1016/S0020-7683(98)00306-0
  33. Lopatin, A.V. and Morozov, E.V. (2009), "Buckling of the SSFF rectangular orthotropic plate under in-plane pure bending", Compos. Struct., 90, 287-2904. https://doi.org/10.1016/j.compstruct.2009.03.006
  34. Lopatin, A.V. and Morozov, E.V. (2010), "Buckling of the CCFF orthotropic rectangular plates under in-plane pure bending", Compos. Struct., 92, 1423-31. https://doi.org/10.1016/j.compstruct.2009.10.038
  35. Lopatin, A.V. and Morozov, E.V. (2011), "Buckling of the SSCF rectangular orthotropic plates subjected to linearly varying inplane loading", Compos. Struct., 93, 1900-1909. https://doi.org/10.1016/j.compstruct.2011.01.024
  36. Lopatin, A.V. and Morozov, E.V. (2014), "Approximate buckling analysis of the CCFF ;orthotropic plates subjected to in-plane bending", Int. J. Mech. Sci., 85, 38-44. https://doi.org/10.1016/j.ijmecsci.2014.05.003
  37. Malekzadeh, P. and Shahpari, S.A. (2005), "Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by differential quadrature method", Thin Wall. Struct., 43, 1037-1050. https://doi.org/10.1016/j.tws.2004.11.008
  38. Ng, S.F. and Araar, Y. (1989), "Free vibration and buckling analysis of clamped rectangular plates of variable thickness by equilibrium method", J. Sound Vib., 135(2), 263-274. https://doi.org/10.1016/0022-460X(89)90725-6
  39. Rajasekaran, S. (2013), "Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using Differential Transformation and quadrature methods", Appl. Math. Model., 37, 4440-4463. https://doi.org/10.1016/j.apm.2012.09.024
  40. Rajasekaran, S. and Wilson, A.J. (2013), "Buckling and vibration of rectangular plates of variable thickness with different conditions by finite difference technique", Struct. Eng. Mech., 46(2), 269-294. https://doi.org/10.12989/sem.2013.46.2.269
  41. Rao, G.V., Rao, B.P. and Raju, L.S. (1974), "Vibration of inhomogeneous plates using a high precision triangular element", J. Sound Vib., 34, 444-445. https://doi.org/10.1016/S0022-460X(74)80323-8
  42. Reddy, J.N. (2004), Mechanics of Laminated composite plates and Shells:Theory and Analysis, 2nd Edition, CRC Press, Boca Raton, Florida.
  43. Shu, C. (2000), Differential Quadrature and its Application in Engineering, London, Springer-Verlag.
  44. Tang, Y. and Wang, X. (2011), "Buckling of symmetrically laminated rectangular plates under parabolic edge compression", Int. J. Mech. Sci., 53, 91-97. https://doi.org/10.1016/j.ijmecsci.2010.11.005
  45. Timoshenko, S.P. and Gere, J.M. (1963), Theory of Elastic Stability, 2nd Edition, McGraw-Hill, New York.
  46. Timoshenko, S.P. and Krieger, S.W. (1959), Theory of Plates and Shells, Second Edition, McGraw-Hill Inc., USA.
  47. Tomar, J.S., Gupta, D.C. and Jain, N.C. (1984), "Free vibrations of an isotropic non-homogeneous infinite plate of parabolically varying thickness", Ind. J. Pure Appl. Math., 15, 11-220.
  48. Wang, X. (2015) Differential Quadrature and Differential Quadrature Based Element Methods: theory and Aplication, Elsevier Inc.
  49. Wilson, E.L. (2002) Three Dimensional Static and Dynamic Analysis of Structures, Computers and Structures Inc, Berkeley, California
  50. Xiang, Y. (2003), "Exact solutions for buckling of multi-span rectangular plate", ASCE J. Eng. Mech., 129, 181-187. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:2(181)
  51. Xiang, Y. and Wei, G.W. (2004), "Exact solutions for buckling and vibration of stepped rectangular Mindlin plates", Int. J. Solid. Stuct., 41, 279-294. https://doi.org/10.1016/j.ijsolstr.2003.09.007
  52. Xiang, Y.F. and Liu, B. (2009), "New exact solutions for free vibrations of thin orthotropic rectangular plates", Compos. Struct., 89, 567-574. https://doi.org/10.1016/j.compstruct.2008.11.010
  53. Zhong, H. and Gu, C. (2010), "Buckling of symmetrical cross-ply composite rectangular plats under a linearly varying in-plane loads", Int. J. Mech. Sci., 52, 819-828. https://doi.org/10.1016/j.ijmecsci.2010.01.009
  54. Zong, Z. and Zhang, Y. (2009), Advanced Differential Quadrature Methods, CRC Press.

Cited by

  1. Bending, buckling and vibration of small-scale tapered beams vol.120, 2017, https://doi.org/10.1016/j.ijengsci.2017.08.005
  2. Nonlinear bending analysis of porous FG thick annular/circular nanoplate based on modified couple stress and two-variable shear deformation theory using GDQM vol.33, pp.2, 2017, https://doi.org/10.12989/scs.2019.33.2.307
  3. Non-Linear dynamic pulse buckling of laminated composite curved panels vol.73, pp.2, 2017, https://doi.org/10.12989/sem.2020.73.2.181