Vibration of Rectangular Plates on Elastic Foundations by Finite Grid Solution.

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Vibration of Rectangular Plates on Elastic Foundations by Finite Grid Solution

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ABSTRACT

Assessment of stress conditions created by vertical or horizontal forces to the supporting medium is a frequent problem of design. In engineering practice, beside static case often dynamic effects must be taken into consideration for plate design problems. Plate vibration solutions have been available for regular geometries for a long time, but it will be necessary to describe the governing equation of motion in a general mathematical form. This is not easy. The intention of this study is to extend analytical solutions of the discrete one-dimensional beam elements resting on elastic foundation for solution of plate vibration problems. The solution can be stated as an extension of the so-called discrete parameter approach where the physical domain is broken down into discrete sub-domains, each endowed with a response suitable for the purpose of mimicking problem at hand.

KEYWORDS

Grillage of beams, vibration, elastic foundation, finite grid solution

REFERENCES

[1] A.P.S. Selvadurai, Elastic analysis of soil – foundation interaction, Elsevier Scientific Publishing Co. Amsterdam, Netherlands, 1979. [1] A.P.S. Selvadurai, Elastic analysis of soil – foundation interaction, Elsevier Scientific Publishing Co. Amsterdam, Netherlands, 1979.

[2] M. Hete´nyi, Beams on elastic foundation, Ann Arbor, MI: University of Michigan Press, 1964.

[3] A.D. Kerr, “A study of a new foundation model,” Acta Mech, vol.1, no.2, pp.135– 47,1965.

[4] F.R. Scott, Foundation analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1981.

[5] M.H. Omurtag and F. Kadıoğlu, “Free vibration analysis of orthotropic plates resting on Pasternak foundation by mixed finite element formulation,” Computers and Structures, vol.67, no.4, pp.253-265, 1988.

[6] W.Q. Chen, C.F. Lü and Z.G. Bian, “A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation,” Applied Mathematical Modeling, vol.28, no.10, pp. 977-890, 2004.

[7] P.Ye. Tovstik, “The vibrations and stability of a prestressed plate on an elastic foundation,” Journal of Applied Mathematics and Mechanics, vol.73, no.1, pp. 77-87, 2009.

[8] M.K.Singha, T. Prakash, and M. Ganapathi, “Finite element analysis of functionally graded plates under transverse load,” Finite Elements in Analysis and Design, vol. 47, no. 4, pp. 453– 460, 2011.

[9] K. Frydrýšek, R. Jančo, and H. Gondek, “Solutions of Beams, Frames and 3D Structures on Elastic Foundation Using FEM,” International Journal of Mechanics, , vol. 7, no. 4, pp., 362-369, 2013.

[10] P. Gagnon, C. Gosselin and L. Cloutier, “A finite strip element for the analysis of variable thickness rectangular thick plates,” Computers and Structures, vol.63 no.2, pp.349-362, 1997.

[11] M.A El -Sayad, A.M.Farag, “Semi-Analytical solution based on strip method for buckling and vibration of isotropic plate,” Journal of Applied Mathematics, vol.2013, p.p. 1-10, 2013.

[12] M.H. Huang and D.P. Thambiratnam, “Analysis of plate resting on elastic supports and elastic foundation by finite strip method,” Computers and Structures, vol.79, no. 29-30, 2547-2557, 2001.

[13] R. Szilard, Theory and analysis of plates: classical and numerical methods, Prentice Hall, Englewood Cliffs, New Jersey, 1974.

[14] S.S. Tameroğlu, “Vibrations of clamped rectangular plates on elastic foundations subjected to uniform compressive forces,” Journal of Engineering Mechanics, vol.122, no.8, pp.714-718, 1996.

[15] S. Hurlebaus, L. Gaul, J.T. Wang, ”An exact series solution for calculating the natural frequencies of orthotropic plates with completely free boundary,” Journal of Sound and Vibration, Vol.244, no.5, pp.747–759, 2001.

[16] I.E. Avramidis and K. Morfidis, “Bending of beams on three-parameter elastic foundation,” International Journal of Solids and Structures, vol.43 no. 2, 357–375, 2006.

[17] S. H. Hashemi, K. Khorshidi, and H. Rokni Damavandi Taher, “Exact acoustical analysis of vibrating rectangular plates with two opposite edges simply supported via Mindlin plate theory,” Journal of Sound and Vibration, vol. 322, no. 4-5, pp. 883–900, 2009.

[18] X. Kai, W. Jiufa, L. Qiuhong, W. Weiyuan, and W. Ping, ”An exact series solution for the vibration of Mindlin rectangular plates with elastically restrained edges,” Shock and Vibration, Vol. 2014, pp.1-11,2014.

[19] S. Motaghian, M. Mofid, and J.E. Akin, “On the free vibration response of rectangular plates, partially supported on elastic foundation,” Applied Mathematical Modelling, vol.36 no.9, pp.4473– 4482, 2012.

[20] A. Karasin, “Extension the matrices of one dimensional beam elements for solution of rectangular plates resting on elastic foundation problems,” International Journal of Mechanics, vol.9, pp.189-197, 2015.

[21] M. Eisenberger and J. Clastornik, “Beams on variable two parameter elastic foundations,” ASCE Journal of Engineering Mechanics, vol.113, no.10, pp. 1454-1466,1987.

[22] J. Ying, C.F. Lü and W.Q. Chen, “Twodimensional elasticity solutions for functionally graded beams resting on elastic foundations,” Composite Structures, vol.84, no.3, pp. 209– 219, 2008.

[23] M. Simsek, “Vibration analysis of a functionally graded beam under a moving mass by using different beam theories,” Composite Structure, vol.92 no.4, pp. 904–917, 2010.

[24] B. Öztürk and S.B. Coşkun, “Analytical solution for free vibration analysis of beam on elastic foundation with different support conditions,” Journal of Applied Mathematics, vol.2013, pp.1-7, 2013.

[25] A. Karasin, P. Gülkan and G. Aktas, “A finite grid solution for circular plates on elastic foundations,” KSCE Journal of Civil Engineering, vol.19, no.4), 1157-1163, 2015.

[26] A. Karasin and P. Gulkan, “Finite grid analogy for Levy plates on generalized foundations,” Turkish Journal of Engineering and Environment Science, vol. 31 no.3, pp.157- 170, 2007.

[27] A. Karasin, M.E. Oncu, M. Suer, “Extension the Consistent Mass Matrices of Beam Elements for Vibration Problems of Rectangular Plates on Winkler Foundation,” 17th International Conference on Mathematical and Computational Methods in Science and Engineering, 23-25 April 2015, Kuala Lumpur, Malaysia, p.101-106, 2015.

Cite this paper

Abdulhalim Karasin. (2016) Vibration of Rectangular Plates on Elastic Foundations by Finite Grid Solution. International Journal of Mathematical and Computational Methods, 1, 140-145