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A new locking-free finite element method based on more consistent version of Mindlin plate equation

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Abstract

A finite element (FE) approach is presented for the dynamic analysis of the Mindlin plates considering both shear deformation and rotary inertia effects. The model is based on the consistent version of the Mindlin equations, which neglects the higher-order time derivative contribution. The approach provides a new class of interdependent Hermite shape polynomials by the definition of a fictitious deflection that takes into account the effective interdependence between the generalized displacements in both the continuous and FE discretized schemes. This implies that the proposed approach is free-shear-locking and is characterized by a good accuracy even for low-order FEs. Several examples are considered whose results are compared with analogous ones proposed in the literature with other approaches.

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References

  1. Reddy J.N.: A penalty plate bending element for the analysis of laminated anisotropic composite plates. Int. J. Numer. Methods Eng. 15, 1187–1206 (1980)

    Article  MATH  Google Scholar 

  2. Averill R.C., Reddy J.N.: On the behavior of plate elements based on the first-order shear deformation theory. Eng. Comput. 7, 57–74 (1990)

    Article  Google Scholar 

  3. Zienkiewicz O.C., Taylor R.L., To J.M.: Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Methods Eng. 3, 275–290 (1971)

    Article  MATH  Google Scholar 

  4. Hughes T.J.R., Taylor R.L., Kanoknukulchai W.: Simple and efficient element for plate bending. Int. J. Numer. Methods Eng. 11, 1529–1543 (1977)

    Article  MATH  Google Scholar 

  5. Reddy J.N.: An Introduction to the Finite Element Method, 2nd edn. McGraw-Hill, New York (1993)

    Google Scholar 

  6. Lee S.W., Wong C.: Mixed formulation finite elements for Mindlin theory plate bending. Int. J. Numer. Methods Eng. 18, 1297–1311 (1982)

    Article  MATH  Google Scholar 

  7. Auricchio F., Taylor R.L.: A triangular thick plate finite element with an exact thin limit. Finite Elem. Anal. Des. 19, 57–68 (1995)

    Article  MATH  Google Scholar 

  8. Lovadina C.: Analysis of a mixed finite element method for the Reissner–Mindlin plate problems. Comput. Methods Appl. Mech. Eng. 163, 71–85 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hughes T.J.R., Tezduyar T.: Finite elements based upon Mindlin plate theory with particular reference to the four-node isoparametric element. J. Appl. Mech. 48, 587–596 (1981)

    Article  MATH  Google Scholar 

  10. Bathe K.J.: Finite Element Procedures. Prentice-Hall/MIT, Englewood Cliffs (1996)

    Google Scholar 

  11. Zienkiewicz O.C., Taylor R.L.: The Finite Element Method, 5th edn. Butterworth-Heinemann, Oxford (2000)

    Google Scholar 

  12. Bletzinger K., Bischoff M., Ramm E.: A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput. Struct. 75, 321–334 (2000)

    Article  Google Scholar 

  13. Nguyen-Xuan H., Liu G.R., Thai-Hoang C., Nguyen-Thoi T.: An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates. Comput. Methods Appl. Mech. Eng. 199, 471–489 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu G.R., Nguyen-Thoi T., Lam Y.K.: An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. J. Sound Vib. 320, 1100–1130 (2009)

    Article  Google Scholar 

  15. Falsone G., Settineri D.: A Kirchhoff-like solution for the Mindlin plate model: a new finite element approach. Mech. Res. Commun. 40, 1–10 (2012)

    Article  Google Scholar 

  16. Falsone G., Settineri D.: An Euler–Bernoulli-like finite element method for Timoshenko beams. Mech. Res. Commun. 38, 12–16 (2011)

    Article  MATH  Google Scholar 

  17. Stephen N.G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 80, 578–582 (1982)

    Article  Google Scholar 

  18. Stephen N.G.: The second spectrum of Timoshenko beam theory—further assessment. J. Sound Vib. 292, 372–389 (2006)

    Article  MATH  Google Scholar 

  19. Stephen N.G.: Mindlin plate theory: best shear coefficient and higher spectra validity. J. Sound Vib. 202, 539–553 (1997)

    Article  Google Scholar 

  20. Levinson M.: Free vibrations of a simply supported, rectangular plate: an exact elasticity solution. J. Sound Vib. 98, 289–298 (1998)

    Article  Google Scholar 

  21. Elishakoff, I.: An equation both more consistent and simpler than the Bresse–Timoshenko equation. In: Gilat, R., Banks-Sills, L. (eds.) Advances in Mathematical Modeling and Experimental Methods for Materials and Structures, Solid Mechanics and Its Applications. Springer, Berlin, pp. 249–254 (2010)

  22. Nesterenko V.V.: A theory for transverse vibrations of a Timoshenko beam. PMM-J. Appl. Math. Mech. 57, 669–677 (1993)

    Article  MathSciNet  Google Scholar 

  23. Timoshenko S.P.: On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philos. Mag. Ser. 6(41), 744–746 (1921)

    Article  Google Scholar 

  24. 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.) National Conference on Noise Control Engineering. New York, pp. 911–916 (1994)

  25. Kaplunov J.D., Kossovich L.Y., Nolde E.V.: Dynamics of Thin Walled Elastic Bodies. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  26. Reddy J.N.: On locking free shear deformable beam elements. Comput. Methods Appl. Mech. Eng. 149, 113–132 (1997)

    Article  MATH  Google Scholar 

  27. Averill R.C., Reddy J.N.: An assessment of four-noded plate finite element based on a generalized third-order theory. Int. J. Numer. Methods Eng. 33, 1553–1572 (1992)

    Article  MATH  Google Scholar 

  28. Hashemi S.H., Arsanjani M.: Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. Int. J. Solids Struct. 42, 819–853 (2005)

    Article  MATH  Google Scholar 

  29. Roberts D.B.: Formulas for Natural Frequency and Mode Shape. Van Nostrand Reinhold, New York (1979)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile, Informatica, Edile, Ambientale e Matematica Applicata, Università di Messina, C.da Di Dio, 98166, Messina, Italy

    G. Falsone & D. Settineri

  2. Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, 33431-0991, USA

    I. Elishakoff

Authors
  1. G. Falsone
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  2. D. Settineri
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  3. I. Elishakoff
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Correspondence to D. Settineri.

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Falsone, G., Settineri, D. & Elishakoff, I. A new locking-free finite element method based on more consistent version of Mindlin plate equation. Arch Appl Mech 84, 967–983 (2014). https://doi.org/10.1007/s00419-014-0842-1

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  • DOI: https://doi.org/10.1007/s00419-014-0842-1

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