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Free vibration analysis of plates taking into account rotary inertia and shear deformation via three alternative theories: a Lévy-type solution

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Abstract

This paper deals with the exact calculation of natural frequencies of a plate with two opposite edges simply supported considering three versions of the Uflyand–Mindlin plate theory: the original Uflyand–Mindlin plate theory, the truncated version of this model as suggested by Elishakoff, and the recently proposed Uflyand–Mindlin plate theory based on slope inertia. The comparison between the frequencies using the different models and with those found in the literature using numerical methods shows the efficiency of the models and the methods presented hereinafter.

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References

  1. Kirchhoff, G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Angew. Math. 40, 51–88 (1850) (in German)

  2. Lee, K.H., Lim, G.T., Wang, C.M.: Thick Lévy plates re-visited. Int. J. Solids Struct. 39, 127–144 (2002)

    Article  MATH  Google Scholar 

  3. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J., Appl. Mech. 12, A69–A77 (1946)

    MathSciNet  MATH  Google Scholar 

  4. Uflyand, Y.S.: The propagation of waves in the transverse vibrations of bars and plates. Akad. Nauk SSSR Prikl. Math. Mech. 12, 287–461 (1948)

    MathSciNet  Google Scholar 

  5. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Trans. ASME J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

  6. Elishakoff, I.: Generalization of the Bolotin’s dynamic edge effect method for vibration analysis of Mindlin plates. In: Cuschieri, J.M., Glegg, S.A.L., Yeager, D.M., (eds). NOISE-CON 94. Proceedings of the 1994 National Conference on Noise Control Engineering; 1994 May 01–04; Fort Lauderdale

  7. Elishakoff, I., Hache, F., Challamel, N.: Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models. Int. J. Eng. Sci. 116, 58–73 (2017)

    Article  MathSciNet  Google Scholar 

  8. Irschik, H.: Membrane-type eigenmotions of mindlin plates. Acta Mech. 55, 1–20 (1985)

    Article  MATH  Google Scholar 

  9. Irschik, H., Heuer, R., Ziegler, F.: Statics and dynamics of simply supported polygonal Reissner–Mindlin plates by analogy. Arch. Appl. Mech. 70, 231–244 (2000)

    Article  MATH  Google Scholar 

  10. Brunelle, E.J., Roberts, S.R.: Initially stressed Mindlin plates. AIAA J. 12, 1036–1045 (1974)

    Article  MATH  Google Scholar 

  11. Brunelle, E.J.: Buckling of transversely isotropic Mindlin plates. AIAA J. 9, 1018–1022 (1971)

    Article  MATH  Google Scholar 

  12. Sharma, A., Sharda, H.B., Nath, Y.: Stability and Vibration of Mindlin sector plates: an analytical approach. AIAA J. 4, 1109–1116 (2005)

    Article  Google Scholar 

  13. Liew, K.M., Wang, C.M., Xiang, Y., Kitipornchai, S.: Vibration of Mindlin plates: programming the p-Version Ritz method. Elsevier, Oxford, UK, Oxford (1998)

    MATH  Google Scholar 

  14. Zenkour, A.M.: Buckling and free vibration of elastic plates using simple and mixed shear deformation theories. Acta Mech. 146, 183–197 (2001)

    Article  MATH  Google Scholar 

  15. Naumenko, K., Altenbach, J., Altenbach, H., Naumenko, V.K.: Closed and approximate analytical solutions for rectangular Mindlin plates. Acta Mech. 147, 153–172 (2001)

    Article  MATH  Google Scholar 

  16. Eftekhari, S.A., Jafari, A.A.: A simple and accurate Ritz formulation for free vibration of thick rectangular and skew plates with general boundary conditions. Acta Mech. 224, 193–209 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rayleigh, J.W.: Theory of Sound, vol. 1. Macmillan, London (1877)

    MATH  Google Scholar 

  18. Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. Angew. Math. 135, 1–61 (1909) (in German)

  19. Leissa, A.W.: Vibration of Plates. U.S Government Printing Office, NASA SP-160, reprinted by the Acoustical Society of America (1969)

  20. Bhat, R.B.: Natural frequencies of rectangular plates using characteristic orthogonal polynomials in the Rayleigh-Ritz method. J. Sound Vib. 102, 493–499 (1985)

    Article  Google Scholar 

  21. Bhat, R.B.: Flexural vibration of polygonal plates using characteristics orthogonal polynomials in two-variables. J. Sound Vib. 114, 65–71 (1987)

    Article  Google Scholar 

  22. Liew, K.M., Lam, K.Y.: Application of two-dimensional orthogonal plate function to flexural vibration of skew plates. J. Sound Vib. 139, 241–252 (1990)

    Article  Google Scholar 

  23. Leissa, A.W.: The free vibration of rectangular plates. J. Sound Vib. 31, 257–293 (1973)

    Article  MATH  Google Scholar 

  24. Gorman, D.J.: Free Vibration Analysis of Rectangular Plates. Elsevier-North Holland Publishing Co, New York (1982)

    MATH  Google Scholar 

  25. Gorman, D.J., Ding, W.: Accurate free vibration analysis of point supported Mindlin plates by the superposition method. J. Sound Vib. 219, 265–277 (1999)

    Article  MATH  Google Scholar 

  26. Hashemi, S.H., Arsanjani, M.: Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick plates. Int. J. Solids Struct. 45, 819–853 (2005)

    Article  MATH  Google Scholar 

  27. Hashemi, S.H., Khorshidi, K., Amabili, M.: Exact solution for linear buckling of rectangular Mindlin plates. J. Sound Vib. 315, 318–342 (2008)

    Article  Google Scholar 

  28. Mindlin, R.D.: In: Yang, J. (ed.) An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. World Scientific Publishing Co, Singapore (2006) (based on the monograph written by R.D. Mindlin in 1955)

  29. Navier, C.L.M.H.: Extrait des recherches sur la flexion des plans élastiques. Bull. Sci. Soc. Philomarhiques de Paris 5, 95–102 (1823)

    Google Scholar 

  30. Chen, W.C., Liu, W.H.: Deflections and free vibration of laminated plates—Lévy-type solutions. Int. J. Mech. Sci. 32, 779–793 (1990)

    Article  Google Scholar 

  31. Szilard, R.: Theory and Analysis of Plates. Prentice-Hall, Englewood Cliffs, New Jersey (1974)

    MATH  Google Scholar 

  32. Jomehzadeh, E., Saidi, A.R.: A Lévy type solution for free vibration analysis of a nano-plate considering the small scale effect. Recent Adv. Vib. Anal. 47–58 (2011)

  33. Wang, C.M., Lim, G.T., Reddy, J.N., Lee, K.H.: Relationships between bending solutions of Reissner and Mindlin plate theories. Eng. Struct. 23, 838–849 (2001)

    Article  Google Scholar 

  34. Wang, C.M.: Natural frequencies formula for simply supported Mindlin plates. J. Vib. Acoust. 116, 536–540 (1994)

    Article  Google Scholar 

  35. Mindlin, R.D., Schacknow, A., Deresiewicz, H.: Flexural vibrations of rectangular plates. Trans. ASME J. Appl. Mech. 23, 430–436 (1956)

    MathSciNet  MATH  Google Scholar 

  36. Wang, C.Y., Wang, C.M.: Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates. CRC Press, Boca Raton (2014)

    Google Scholar 

  37. Srinivas, S.R., Joga Rao, C.V., Rao, A.K.: An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates. J. Sound Vib. 12, 187–199 (1970)

    Article  MATH  Google Scholar 

  38. Dawe, D.J.: Finite strip models for vibration of Mindlin plates. J. Sound Vib. 59, 441–452 (1978)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, 33481-0991, USA

    F. Hache & I. Elishakoff

  2. Université de Bretagne Sud, FRE CNRS 3744 IRDL, Centre de Recherche, Rue de Saint Maudé, BP 92116, 56321, Lorient, France

    F. Hache & N. Challamel

Authors
  1. F. Hache
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  2. I. Elishakoff
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  3. N. Challamel
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Correspondence to N. Challamel.

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Hache, F., Elishakoff, I. & Challamel, N. Free vibration analysis of plates taking into account rotary inertia and shear deformation via three alternative theories: a Lévy-type solution. Acta Mech 228, 3633–3655 (2017). https://doi.org/10.1007/s00707-017-1890-8

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  • DOI: https://doi.org/10.1007/s00707-017-1890-8

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