Free Vibration Analysis of Isotropic Plates by Alternative Hierarchical Finite Element Method Based on Reddy’s C1 HSDT

Authors

  • Nadjib Sidi Mohammed Serdoun University of Tlemcen
  • S. M. Hamza-Cherif

Keywords:

Free vibration, Thick isotropic plates, hierarchical finite element method, third order C1 HSDT.

Abstract

This paper presents the free vibration analysis of isotropic thick rectangular plates, based on higher order shear deformation theory (HSDT). The plate theory ensures a zero shear-stress condition at the top and bottom surfaces of the plate, and do not requires a shear correction factor. The model requires inter-element C1 continuity for the transverse displacement. To overcome this hindrance, a new hierarchical p-element with six degrees of freedom per node is developed and used to find natural frequencies of thick plates. Convergence studies and comparison have been carried out for with different boundaries conditions. It is shown that the present element enables rapid convergence.

Author Biography

Nadjib Sidi Mohammed Serdoun, University of Tlemcen

Tlemcen

References

[1] A. E. H. Love. (1888) On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 17 491–549. Available: http://www.jstor.org/stable/90527
[2] Mindlin, R.D. Schacknow, A. Deresiewicz (1955, Jun), Flexural vibrations of rectangular plates, ASME J. Appl. Mec, 23 (1956) 430–436.
[3] R.B. Nelson, DR. Lorch. (1974, Mar), A refined theory for laminated orthotropic plates. ASME, J. Appl. Mech, 41 177-183.Available:http://appliedmechanics.asmedigitalcollection.asme.org.sci-hub.org/article.aspx?articleid=1401537
[4] K.H Lo, R.M. Christen, E.m Wu. (1977, Dec), A higher order rheory of plate deformation – Part 1: Homogeneous plates, ASME, J. Appl. Mech, 44 663-668. Available:http://appliedmechanics.asmedigitalcollection.asme.org/article.aspx?articleid=1403567
[5] M. Levinson. (1980), An accurate simple theory of the statics and dynamics of elastic plates, Mech. Res. Commun. 7 343-350.
[6] M.V.V. Murty. (1981), An improved transverse shear deformation theory for laminated anistropic plates. NASA Technical Paper 1903.
[7] J. N. Reddy.(1984, Dec), A simple higher-order theory for laminated composite plates, ASME J. Appl. Mech. 51 745-752.Available:http://appliedmechanics.asmedigitalcollection.asme.org /article.aspx?articleid=1407769
[8] T. Kant, J.H. Varaiya, C.P. Arora, Finite element transient analysis of composite and sandwich plates based a refined theory and implicit time integration shemes, Comput. Struct, 36 (1990) 401-420.Available:http://www.sciencedirect.com/science/article/pii/004579499090279B
[9] A.K. Nayak, S.S.J. Moy, R.A. Shenoi , Free vibration analysis of composite sandwich plates based on reddy's higherorder theory, Composites Part B: Engineering. 33 (7) (2002) 505-519. Available: http://www.sciencedirect.com/science/article/pii/S1359836802000355
[10] A.K. Nayak, R.A. Shenoi ,S.S.J Moy , Transient response of composite sandwich plates, Comput. Struct, 64 (2004) 249-267. Available: http://www.sciencedirect.com/science/article/pii/S0263822303001351
[11] A.H. Sheikh, A. Chakrabarti, A new plate bending element based on higher order shear deformation theory for the analysis of composite plates, Finite. Elem. Anal. Des, 39 (2003) 137-155. vailable:http://www.sciencedirect.com/science/article/pii/S0168874X02001373
[12] R.C. Batra, S. Aimmanee ,Vibrations of thick isotropic plates with higher order shear and normal deformable plate theories, Comput. Struct, 83 (2005) 934-955. Available: http://www.sciencedirect.com/science/article/pii/S0045794905000507?np=y
[13] S. Kapuria, SD Kulkarni, An improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory for static analysis of composite and sandwich plates, J. Numer. method. Eng, 69 (2007)1948-1981. Available: http://onlinelibrary.wiley.com/doi/10.1002/nme.1836/abstract
[14] B.A. Szabo, and G.J. Sahrmann, Hierarchical plate and shells models based on p extension, Int. J. Numer.method. Eng, 26 (1988) 1855-1881. Available:http://onlinelibrary.wiley.com/doi/10.1002/nme.1620260812/abstract;jsessionid=3DCB6A45626B1106E11AB1A7F34B4282.f04t04
[15] B.A. Szabo , I. Babuska., Finite Element Analysis, (1990) Wiley-lnterscience, New York, 1991. [On-line]. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471502731.html
[16] S. M. Hamza-Cherif, Free vibration analysis of rotating cantilever plates using the p-version of the finite element method, Structural Engineering and Mechanics, 22 (2006) 151-167.
[17] A.W. Leissa , The free vibration of rectangular plates, J. Sound Vib. 31 (1973) 257–293. Available: http://www.sciencedirect.com/science/article/pii/S0022460X73803712
[18] C. W. Lim a, K.M. Liew b, S. Kitipornchai, a Numerical aspects for free vibration of thick Part I: Formulation and verification plates, Comput. Methods Appl. Mech. Eng, 156 (1998) 15-29. Available: http://www.sciencedirect.com/science/article/pii/S0045782597001977
[19] S. Srinivas, C.V. JogaRao, A.K. Rao, An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates, J. Sound Vib. 12(2) (1970) 187-199. Available: http://www.sciencedirect.com/science/article/pii/0022460X70900891

[20] D. Zhou a, Y.K. Cheung, F.T.K. Au, S.H. Lo,Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method, Int. J. Solids Struct, 63(2002) 39–6353.Available:http://www.sciencedirect.com/science/article/pii/S0020768302004602
[21] S. Wang, Vibration of thin skew fiber reinforced composite laminates, J. Sound. Vib. 201(3) (1997) 335–352.Available:http://www.sciencedirect.com/science/article/pii/S0022460X96907452
[22] S. Wang, Free vibration analysis of skew fiber-reinforced composite laminates based on first-order shear deformation plate theory, Comput. Struct, 63 (3) (1997) 525-538. Available:http://www.sciencedirect.com/science/article/pii/S0045794996003574
[23] K.M. Liew, K.C. Hung, M.K. Lim, A continuum three-dimensional vibration analysis of thick rectangular plates, Int. J. Solids Struct. 30 (24) (1993) 3357-3379. Available:http://www.sciencedirect.com/science/article/pii/002076839390089P
[24] A. Houmat, An alternative hierarchical finite element formulation applied to plate vibrations, J. Sound Vib. (1997) 206 (2) 201-215. Available: http://www.sciencedirect.com/science/article/pii/S0022460X97910762
[25] S. Hosseini-Hashemi , M. Fadaee , H. RokniDamavandiTaher, Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory, Appl. math. Model.35 (2011) 708–727. Available: http://www.sciencedirect.com/science/article/pii/S0307904X10002775
[26] S. Hosseini-Hashemi, M. Arsanjani, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates, Int. J. Solids. Struct. 47 (2005) 819–853. Available:http://www.sciencedirect.com/science/article/pii/S0020768304003671
[27] M. Malik, C.W. Bert, Three-dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method, Int. J. Solids. Struct. 35 (1998) 299–318. Available: http://www.sciencedirect.com/science/article/pii/S0020768397000735

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Published

2014-07-21

How to Cite

Serdoun, N. S. M., & Hamza-Cherif, S. M. (2014). Free Vibration Analysis of Isotropic Plates by Alternative Hierarchical Finite Element Method Based on Reddy’s C1 HSDT. American Scientific Research Journal for Engineering, Technology, and Sciences, 9(1), 1–19. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/522