Skip to main content
SpringerLink
Log in

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

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, New York (2006)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  11. De Miranda, S., Ubertini, F.: A simple hybrid stress element for shear deformable plates. Int. J. Numer. Methods Eng. 65, 808–833 (2006)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  16. Liu, G.R., Nguyen-Thoi, T.: Smoothed Finite Element Methods. CRC Press, New York (2010)

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  22. Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  32. Bordas, S., Natarajan, S.: On the approximation in the smoothed finite element method (SFEM). Int. J. Numer. Methods Eng. 81, 660–670 (2010)

    MathSciNet  MATH  Google Scholar 

  33. Dasgupta, G.: Integration within polygonal finite elements. J. Aerosp. Eng. 16, 9–18 (2003)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  38. Morley, L.S.D.: Skew Plates and Structures. Pergamon Press, New York (1963)

  39. Abassian, F., Hawswell, D.J., Knowles, N.C.: Free Vibration Benchmarks Softback. Atkins Engineering Sciences, Glasgow (1987)

    Google Scholar 

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

    Google Scholar 

  41. Leissa, A.W.: Vibration of Plates. ASA Press, New York (1993)

  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)

    Article  Google Scholar 

  43. Irie, T., Yamada, G., Aomura, S.: Natural frequencies of Mindlin circular plates. J. Appl. Mech. 47, 652–655 (1980)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grants 11272118, 11372106) and Fundamental Research Fund of the Central Universities (Grant 227201401203)

Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, China

    Gang Yang, De’an Hu & Shuyao Long

Authors
  1. Gang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. De’an Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuyao Long
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Gang Yang or De’an Hu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, G., Hu, D. & Long, S. A reconstructed edge-based smoothed DSG element based on global coordinates for analysis of Reissner–Mindlin plates. Acta Mech. Sin. 33, 83–105 (2017). https://doi.org/10.1007/s10409-016-0607-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-016-0607-x

Keywords

Search

Navigation