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Analysis of thin plates by the weak form quadrature element method

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Abstract

The recently proposed weak form quadrature element method (QEM) is applied to flexural and vibrational analysis of thin plates. The integrals involved in the variational description of a thin plate are evaluated by an efficient numerical scheme and the partial derivatives at the integration sampling points are then approximated using differential quadrature analogs. Neither the grid pattern nor the number of nodes is fixed, being adjustable according to convergence need. The C1 continuity conditions characterizing the thin plate theory are discussed and the robustness of the weak form quadrature element for thin plates against shape distortion is examined. Examples are presented and comparisons with analytical solutions and the results of the finite element method are made to demonstrate the convergence and computational efficiency of the weak form quadrature element method. It is shown that the present formulation is applicable to thin plates with varying thickness as well as uniform plates.

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References

  1. Batoz J L, Bathe K J, Ho W L. A study of three-node triangular plate bending elements. Int J Numer Methods Eng, 1980, 15: 1771–1812

    Article  MATH  Google Scholar 

  2. Zienkiewicz O C, Taylor R L. The Finite Element Method for Solid and Structural Mechanics. 6th ed. Oxford: Elsevier, 2005. Chapter 11

    MATH  Google Scholar 

  3. Stricklin J A, Haisler W, Tisdale P, et al. A rapidly converging triangle plate element. J AIAA, 1969, 7: 180–181

    Article  ADS  MATH  Google Scholar 

  4. Zienkiewicz O C, Lefebvre D. Three field mixed approximation and the plate bending problem. Commun Appl Numer Methods, 1987, 3: 301–309

    Article  MATH  Google Scholar 

  5. Cook R D. Two hybrid elements for analysis of thick, thin and sandwich plates. Int J Numer Methods Eng, 1972, 5: 277–299

    Article  MATH  Google Scholar 

  6. Zhong H, Yu T. Flexural vibration analysis of an eccentric annular mindlin plate. Arch Appl Mech, 2007, 77: 185–195

    Article  MATH  Google Scholar 

  7. Zhong H, Yu T. A weak form quadrature element method for plane elasticity problems. Appl Math Modelling, 2009, 33(10): 3801–3814

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhong H, Wang Y. Weak form quadrature element analysis of Bickford beams. Eur J Mech Solids, 2010, 29(5): 851–858

    Article  Google Scholar 

  9. Zhong H, Gao M. Quadrature element analysis of planar frameworks. Arch Appl Mech, 2010, 80: 1391–1405

    Article  Google Scholar 

  10. Striz A G, Chen W L, Bert C W. High accuracy plane stress and plate elements in the quadrature element method. In: Proceedings of the 36th AIAA/SME/SCE/AHS/ASC, 1995. 957–965

  11. Striz A G, Chen W L, Bert C W. Free vibration of plates by the high accuracy quadrature element method. J Sound Vib, 1997, 202: 689–702

    Article  ADS  MATH  Google Scholar 

  12. Malik M, Bert C W. Implementing multiple boundary conditions in the DQ solution of high-order PDEs: Application to free vibration of plates. Int J Numer Methods Eng, 1996, 39: 1237–1258

    Article  MATH  Google Scholar 

  13. Wang Y L, Wang X W, Zhou Y. Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method. Int J Numer Methods Eng, 2004, 59: 1207–1226

    Article  MATH  Google Scholar 

  14. Zhong H Z. Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates. Appl Math Modeling, 2004, 28(4): 353–366

    Article  MATH  Google Scholar 

  15. Xing Y F, Liu B. High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain. Int J Numer Methods Eng, 2009, 80(13): 1718–1742

    Article  MathSciNet  MATH  Google Scholar 

  16. ANSYS. Swanson Analysis System, US, 2003

  17. Gordon W J. Blending function methods of bivariate and multivariate interpolation and approximation. SIAM J Numer Anal, 1971, 8: 158–177

    Article  MathSciNet  MATH  Google Scholar 

  18. Davis P I, Rabinowitz P. Methods of Numerical Integration. 2nd ed. Orlando: Academic Press, 1984

    MATH  Google Scholar 

  19. Bellman R E, Casti J. Differential quadrature and long term integration. J Math Anal Appl, 1971, 34: 235–238

    Article  MathSciNet  MATH  Google Scholar 

  20. Quan J R, Chang C T. New insights in solving distributed system equations by the quadrature method-I. Anal Comput Chem Eng, 1989, 13: 779–788

    Article  Google Scholar 

  21. Bert C W, Malik M. Differential quadrature method in computational mechanics: A review. Appl Mech Rev, 1996, 49: 1–27

    Article  ADS  Google Scholar 

  22. Shu C. Differential Quadrature and Its Application in Engineering. London: Springer-Verlag, 2000

    Book  MATH  Google Scholar 

  23. Hermite C. Sur la Formula d’interpolation de Lagrange(in French). J Reine Angewandte Math, 1878, 84: 70–79

    Article  Google Scholar 

  24. Timoshenko S, Woinowsky-Krieger S. Theory of Plates and Shells. 2nd ed. New York: McGraw-Hill, 1959

    Google Scholar 

  25. Timoshenko S, Young D H, Weaver W Jr. Vibration Problems in Engineering. 4th ed. New York: John Wiley & Sons, Inc, 1974

    Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  27. Duan W H, Quek S T, Wang Q. Generalized hypergeometric function solutions for transverse vibration of a class of non-uniform annular plates. J Sound Vib, 2005, 287: 785–807

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Tsinghua University, Beijing, 100084, China

    HongZhi Zhong & ZhiGuang Yue

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  1. HongZhi Zhong
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  2. ZhiGuang Yue
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Correspondence to HongZhi Zhong.

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Zhong, H., Yue, Z. Analysis of thin plates by the weak form quadrature element method. Sci. China Phys. Mech. Astron. 55, 861–871 (2012). https://doi.org/10.1007/s11433-012-4684-y

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  • DOI: https://doi.org/10.1007/s11433-012-4684-y

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