Acta Mechanica Sinica

, Volume 33, Issue 1, pp 83–105 | Cite as

A reconstructed edge-based smoothed DSG element based on global coordinates for analysis of Reissner–Mindlin plates

Research Paper

Abstract

A reconstructed edge-based smoothed triangular element, which is incorporated with the discrete shear gap (DSG) method, is formulated based on the global coordinate for analysis of Reissner–Mindlin plates. A symbolic integration combined with the smoothing technique is implemented to calculate the smoothed finite element matrices, which is integrated along the boundaries of each smoothing cell. Numerical results show that the proposed element is free from shear locking, and its results are in good agreement with the exact solutions, even for very thin plates with extremely distorted elements. The proposed element gives more accurate results than the original DSG element without smoothing, and it can be taken as an alternative element for analysis of Reissner–Mindlin plates. The prominent feature of the present element is that the integration scheme is unified in the smoothed form for all of the finite element matrices.

Keywords

Reissner–Mindlin plate DSG method Symbolic integration Smoothing technique Global coordinate Distorted element 

References

  1. 1.
    Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, New York (2006)Google Scholar
  2. 2.
    Wu, F., Liu, G.R., Li, G.Y., et al.: A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin Plates. Comput. Mech. 54, 865–890 (2014)CrossRefMATHGoogle Scholar
  3. 3.
    Le, C.V.: A stabilized discrete shear gap finite element for adaptive limit analysis of Mindlin–Reissner plates. Int. J. Numer. Methods Eng. 96, 231–246 (2013)MathSciNetMATHGoogle Scholar
  4. 4.
    Mackerle, J.: Finite element linear and nonlinear, static and dynamic analysis of structural elements, an addendum: A bibliography (1999–2002). Eng. Comput. 19, 520–594 (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Zienkiewicz, O.C., Taylor, R.L., Too, J.M.: Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Methods Eng. 3, 275–290 (1971)CrossRefMATHGoogle Scholar
  6. 6.
    Hughes, T.J.R., Cohen, M., Haroun, M.: Reduced and selective integration techniques in the finite element analysis of plates. Nucl. Eng Des. 46, 203–222 (1978)CrossRefGoogle Scholar
  7. 7.
    Hughes, T.J.R., Tezduyar, T.E.: Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J. Appl. Mech. 48, 587–596 (1981)CrossRefMATHGoogle Scholar
  8. 8.
    Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29, 1595–1638 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bathe, K.J., Dvorkin, E.N.: A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int. J. Numer. Methods Eng. 21, 367–383 (1985)CrossRefMATHGoogle Scholar
  10. 10.
    Tessler, A., Hughes, T.J.R.: A three-node Mindlin plate element with improved transverse shear. Comput. Methods Appl. Mech. Eng. 50, 71–101 (1985)CrossRefMATHGoogle Scholar
  11. 11.
    De Miranda, S., Ubertini, F.: A simple hybrid stress element for shear deformable plates. Int. J. Numer. Methods Eng. 65, 808–833 (2006)CrossRefMATHGoogle Scholar
  12. 12.
    Cen, S., Long, Y.Q., Yao, Z.H., et al.: Application of the quadrilateral area co-ordinate method: a new element for Mindlin–Reissner plate. Int. J. Numer. Methods Eng. 66, 1–45 (2006)CrossRefMATHGoogle Scholar
  13. 13.
    Nguyen-Thoi, T., Phung-Van, P., Nguyen-Xuan, H., et al.: A cell-based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner–Mindlin plates. Int. J. Numer. Methods Eng. 91, 705–741 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Nguyen-Thoi, T., Bui-Xuan, T., Phung-Van, P., et al.: An edge-based smoothed three-node Mindlin plate element (ES-MIN3) for static and free vibration analyses of plates. KSCE J. Civil Eng. 18, 1072–1082 (2014)CrossRefGoogle Scholar
  15. 15.
    Bletzinger, K.U., Bischoff, M., Ramm, E.: A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput. Struct. 75, 321–334 (2000)CrossRefGoogle Scholar
  16. 16.
    Liu, G.R., Nguyen-Thoi, T.: Smoothed Finite Element Methods. CRC Press, New York (2010)CrossRefGoogle Scholar
  17. 17.
    Cen, S., Shang, Y., Li, C.F., et al.: Hybrid displacement function element method: a simple hybrid-Trefftz stress element method for analysis of Mindlin-Reissner plate. Int. J. Numer. Methods Eng. 98, 203–234 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shang, Y., Cen, S., Li, C.F., et al.: An effective hybrid displacement function element method for solving the edge effect of Mindlin-Reissner plate. Int. J. Numer. Methods Eng. 102, 1449–1487 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, T., Qi, Z.H., Ma, X., et al.: High-order assumed stress quadrilateral element for the Mindlin-Reissner plate bending problem. Struct. Eng. Mech. 54, 393–417 (2015)CrossRefGoogle Scholar
  20. 20.
    Chen, J.S., Wu, C.T., Yoon, S., et al.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001)CrossRefMATHGoogle Scholar
  21. 21.
    Liu, G.R., Nguyen-Thoi, T., Lam, K.Y.: 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)CrossRefGoogle Scholar
  22. 22.
    Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)CrossRefMATHGoogle Scholar
  23. 23.
    Nguyen-Thoi, T., Liu, G.R., Nguyen-Xuan, H., et al.: Adaptive analysis using the node-based smoothed finite element method (NS-FEM). Int. J. Numer. Methods Biomed. Eng. 27, 198–218 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Nguyen-Thoi, T., Liu, G.R., Vu-Do, H.C., et al.: A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh. Comput. Methods Appl. Mech. Eng. 198, 3479–3498 (2009)Google Scholar
  25. 25.
    Liu, G.R., Nguyen-Xuan, H., Nguyen-Thoi, T.: A variationally consistent \(\alpha \)-\(\text{ FEM }\,(\text{ VC }\alpha \text{ FEM }\)) for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements. Int. J. Numer. Methods Eng. 85, 461–497 (2011)CrossRefMATHGoogle Scholar
  26. 26.
    Nguyen-Xuan, H., Rabczuk, T., Bordas, S., et al.: A smoothed finite element method for plate analysis. Comput. Methods Appl. Mech. Eng. 197, 1184–1203 (2008)CrossRefMATHGoogle Scholar
  27. 27.
    Nguyen-Xuan, H., Nguyen-Thoi, T.: A stabilized smoothed finite element method for free vibration analysis of Mindlin–Reissner plates. Commun. Numer. Methods Eng. 25, 882–906 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nguyen-Thoi, T., Phung-Van, P., Luong-Van, H., et al.: A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates. Comput. Mech. 51, 65–81 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Nguyen-Xuan, H., Liu, G.R., Thai-Hoang, C., et al.: 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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Nguyen-Xuan, H., Rabczuk, T., Nguyen-Thanh, N., et al.: A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates. Comput. Mech. 46, 679–701 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Nguyen-Thanh, N., Rabczuk, T., Nguyen–Xuan, H., et al.: An alternative alpha finite element method with discrete shear gap technique for analysis of isotropic Mindlin–Reissner plates. Finite Elements Anal. Design 47, 519–535 (2011)Google Scholar
  32. 32.
    Bordas, S., Natarajan, S.: On the approximation in the smoothed finite element method (SFEM). Int. J. Numer. Methods Eng. 81, 660–670 (2010)MathSciNetMATHGoogle Scholar
  33. 33.
    Dasgupta, G.: Integration within polygonal finite elements. J. Aerosp. Eng. 16, 9–18 (2003)CrossRefGoogle Scholar
  34. 34.
    Liew, K.M., Wang, J., Ng, T.Y., et al.: Free vibration and buckling analyses of shear-deformable plates based on FSDT meshfree method. J. Sound Vib. 276, 997–1017 (2004)CrossRefGoogle Scholar
  35. 35.
    Lyly, M., Stenberg, R., Vihinen, T.: A stable bilinear element for the Reissner-Mindlin plate model. Comput. Methods Appl. Mech. Eng. 110, 343–357 (1993)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Bischoff, M., Bletzinger, K.U.: Stabilized DSG plate and shell elements. In: Conference of Trends in Computational Structural Mechanics, Barcelona, May 20–23 (2001)Google Scholar
  37. 37.
    Taylor, R.L., Auricchio, F.: Linked interpolation for Reissner–Mindlin plate elements: Part II–A simple triangle. Int. J. Numer. Methods Eng. 36, 3057–3066 (1993)CrossRefMATHGoogle Scholar
  38. 38.
    Morley, L.S.D.: Skew Plates and Structures. Pergamon Press, New York (1963)Google Scholar
  39. 39.
    Abassian, F., Hawswell, D.J., Knowles, N.C.: Free Vibration Benchmarks Softback. Atkins Engineering Sciences, Glasgow (1987)Google Scholar
  40. 40.
    Robert, D.B.: Formulas for Natural Frequency and Mode Shape. Van Nostrand Reinhold, New York (1979)Google Scholar
  41. 41.
    Leissa, A.W.: Vibration of Plates. ASA Press, New York (1993)Google Scholar
  42. 42.
    Al-Bermani, F.G.A., Liew, K.M.: Natural frequencies of thick arbitrary quadrilateral plates using the pb-2 Ritz method. J. Sound Vib. 196, 371–385 (1996)CrossRefGoogle Scholar
  43. 43.
    Irie, T., Yamada, G., Aomura, S.: Natural frequencies of Mindlin circular plates. J. Appl. Mech. 47, 652–655 (1980)CrossRefMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle EngineeringHunan UniversityChangshaChina

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