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Free vibration analysis of orthotropic plates with variable thickness resting on non-uniform elastic foundation by element free Galerkin method

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Abstract

This study intends to investigate the vibration behavior of a thin square orthotropic plate resting on non-uniform elastic foundation and its thickness varying in one or two directions. By using the classical plate theory and employing element free Galerkin method, it is shown that the fundamental frequency coefficients obtained are in good agreement with available results in the literature. The effects of thickness variation, foundation parameter and boundary conditions on frequency are investigated. The results show that the method converges very fast regardless of parameters involved.

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References

  1. Y. P. Xu and D. Zhou, Three-dimensional elasticity solution for simply supported rectangular plates with variable thickness, Journal of Strain Analysis for Engineering Design, 43 (2008) 165–176.

    Article  Google Scholar 

  2. Y. P. Xu and D. Zhou, Three-dimensional elasticity solution of functionally graded rectangular plates with variable thickness, Composite Structures, 91 (2009) 56–65.

    Article  Google Scholar 

  3. Y. Xu, D. Zhou and K. Liu, Three-dimensional thermoelastic analysis of rectangular plates with variable thickness subjected to thermo-mechanical loads, Journal of Thermal Stresses, 33 (2010) 1136–1155.

    Article  Google Scholar 

  4. Y. K. Cheung and D. Zhou, Free vibrations of tapered rectangular plates using a new set of beam functions in Rayleigh-Ritz method, Journal of Sound and Vibration 223 (1999) 703–722.

    Article  Google Scholar 

  5. Y. K. Cheung and D. Zhou, Eigen frequencies of tapered rectangular plates with line supports, International Journal of Solids and Structures, 36 (1999) 143–166.

    Article  MATH  Google Scholar 

  6. D. Zhou, Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions, International Journal of Mechanical Sciences, 44 (2002) 149–164.

    Article  MATH  Google Scholar 

  7. Y. K. Cheung and D. Zhou, Vibrations of tapered Mindlin plates in terms of static Timoshenko beam functions, Journal of Sound and Vibration, 260 (2003) 693–709.

    Article  Google Scholar 

  8. S. K. Malhotra, N. Ganesan and M. A. Veluswami, Vibrations of orthotropic square plates having variable thickness (parabolic variation), Journal of Sound and Vibration, 119 (1987) 184–188.

    Article  Google Scholar 

  9. C. W. Bert and M. Malik, Free vibration analysis of tapered rectangular plates by differential quadrature method: a semi-analytical approach, Journal of Sound and Vibration, 190 (1996) 41–63.

    Article  Google Scholar 

  10. D. V. Bambill, C. A. Rossit, P. A. A. Laura and R. E. Rossi, Transverse vibrations of an orthotropic rectangular plate of linearly varying thickness and with a free edge, Journal of Sound and Vibration, 235 (2000) 530–538.

    Article  Google Scholar 

  11. A. S. Ashour, A semi-analytical solution of the flexural vibration of orthotropic plates of variable thickness, Journal of Sound and Vibration, 240 (2001) 431–445.

    Article  MATH  Google Scholar 

  12. M. Huang, X. Q. Ma, T. Sakiyama, H. Matuda and C. Morita, Free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions, Journal of Sound and Vibration, 288 (2005) 931–955.

    Article  Google Scholar 

  13. N. Gajendra, Large amplitude vibration of plates on elastic foundations, International Journal of Non-linear Mechanics, 2 (1967) 163–172.

    Article  Google Scholar 

  14. S. Datta, Large amplitude free vibration of irregular plates placed on elastic foundation, International Journal of Nonlinear Mechanics, 11 (1976) 337–345.

    Article  MATH  Google Scholar 

  15. D. Zhou, S. H. Lo, F. T. K. Au and Y. K. Cheung, Threedimensional free vibration of thick circular plates on Pasternak foundation, Journal of Sound and Vibration, 292 (2006) 726–741.

    Article  Google Scholar 

  16. Bhaskar and P. C. Dumir, Non-linear vibration of orthotropic thin rectangular plates on elastic foundations. Journal of Sound and Vibration, 125 (1988) 1–11.

    Article  Google Scholar 

  17. M. H. Omurtag and F. Kadioglu, Free vibration analysis of orthotropic plates resting on Pasternak foundation by mixed finite element formulation, Computers and Structures, 67 (1998) 253–265.

    Article  MATH  Google Scholar 

  18. P. Gupta and N. Bhardwaj, Vibration of rectangular orthotropic elliptic plates of quadratically varying thickness resting on elastic foundation, Journal of Vibration and Acoustics, 126 (2004) 132–140.

    Article  Google Scholar 

  19. M. H. Hsu, Vibration analysis of orthotropic rectangular plates on elastic foundations, Composite Structures, 92 (2010) 844–852.

    Article  Google Scholar 

  20. M. F. Liu, T. P. Chang and Y. H. Wang, Free vibration analysis of orthotropic rectangular plates with tapered varying thickness and winkler spring foundation, Mechanics Based Design of Structures and Machines, 39 (2011) 320–333.

    Article  Google Scholar 

  21. W. Yan, W. Zhong-min and R. Miao, Element-free Galerkin method for free vibration of rectangular plates with interior elastic point, supports and elastically restrained edges, Journal of Shanghai University (English Edition), 14(3) (2010) 187–195.

    Article  MathSciNet  Google Scholar 

  22. X. L. Chen, G. R. Liu and S. P. Lim., An element free Galerkin method for the free vibration analysis, of composite laminates of complicated shape, Composite Structures, 59 (2003) 279–289.

    Article  Google Scholar 

  23. K. Y. Dai, G. R. Liu, K. M. Lim and X. L. Chen, A meshfree method for static and free vibration analysis of shear deformable laminated composite plates, Journal of Sound and Vibration, 269 (2004) 633–652.

    Article  Google Scholar 

  24. T. Belytschko, Y. Krongauz, M. Fleming, D. Organ and K. S. Wing Liu, Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational and Applied Mathematics, 74(1–2) (1996) 111–126.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Dolbow and T. Belytschko, An introduction to programming the meshless element free galerkin method, Archives of Computational Methods in Engineering, 5(3) (1998) 207–242.

    Article  MathSciNet  Google Scholar 

  26. G. R. Liu, Mesh free methods: Moving beyond the finite element method, CRC Press, New York (2002).

    Book  Google Scholar 

  27. T. Zhu and S. N. Atluri, A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free galerkin method, Computational Mechanics, 21(3) (1998) 211–222.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Krysl and T. Belytschko, Analysis of thin plates by the element-free Galerkin method, Computational Mechanics, 17 (1995) 26–35.

    Article  MathSciNet  MATH  Google Scholar 

  29. D. Zhou, Y. K. Cheung, S. H. Lo and F. T. K. Au, Threedimensional vibration analysis of rectangular thick plates on Pasternak foundation, International Journal of Numerical Methods in Engineering, 59 (2004) 1313–1334.

    Article  MATH  Google Scholar 

  30. J. M. Ferreira, C. M. C. Roque, A. M. A. Neves, R. M. N. Jorge and C. M. M. Soares, Analysis of plates on Pasternak foundations by radial basis functions, Computational Mechanics, 46 (2010) 791–803.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. H. Omurtag, A. Ozutok and A. Y. Akoz, Free vibration analysis of Kirchhoff plates resting on elastic foundation by mixed finite element formulation based on gateaux differential, International Journal of Numerical Methods in Engineering, 40 (1997) 295–317.

    Article  MathSciNet  Google Scholar 

  32. Y. Xiang, C. M. Wang and S. Kitipornchai, Exact vibration solution for initially stressed Mindlin plates on Pasternak foundation, International Journal of Mechanic Science, 36 (1994) 311–316.

    Article  MATH  Google Scholar 

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Correspondence to Ehsan Bahmyari.

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Recommended by Editor Yeon June Kang

Rahbar Ranji Ahmad received his B.S. in Civil Engineering from Tehran University, Iran, in 1989. He finished his M.Sc. in Naval Architecture from Technical University of Gdansk, Poland. in 1992 and Ph.D. from Yokohama National University, Japan, in 2001. Dr. Rahbar is currently an Assistant Professor at Department of ocean engineering, AmirKabir university of Technology in Tehran, Iran.

Bahmyari Ehsan received his Diploma in Maths & Physics from National Organization for Development of Exceptional Talents, Iran, in 2005. He finished his B.S in Naval Architecture from Persian Gulf University of Boushehr, Iran, in 2009 and M.Sc. in Ocean Engineering from Amirkabir University of Technology, Tehran, Iran, in 2011.

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Bahmyari, E., Rahbar-Ranji, A. Free vibration analysis of orthotropic plates with variable thickness resting on non-uniform elastic foundation by element free Galerkin method. J Mech Sci Technol 26, 2685–2694 (2012). https://doi.org/10.1007/s12206-012-0713-z

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  • DOI: https://doi.org/10.1007/s12206-012-0713-z

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