Skip to main content
SpringerLink
Log in

Three- and four-noded planar elements using absolute nodal coordinate formulation

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

This paper investigates two new types of planar finite elements containing three and four nodes. These elements are the reduced forms of the spatial plate elements employing the absolute nodal coordinate approach. Elements of the first type use translations of nodes and global slopes as nodal coordinates and have 18 and 24 degrees of freedom. The slopes facilitate the prevention of the shear locking effect in bending problems. Furthermore, the slopes accurately describe the deformed shape of the elements. Triangular and quadrilateral elements of the second type use translational degrees of freedom only and, therefore, can be utilized successfully in problems without bending. These simple elements with 6 and 8 degrees of freedom are identical to the elements used in conventional formulation of the finite element method from the kinematical point of view. Similarly to the famous problem called “flying spaghetti” which is used often as a benchmark for beam elements, a kind of “flying lasagna” is simulated for the planar elements. Numerical results of simulations are presented.

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

Similar content being viewed by others

References

  1. Song, J.O., Haug, E.J.: Dynamic analysis of planar flexible mechanisms. Comput. Methods Appl. Mech. Eng. 24, 359–381 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Shabana, A.A., Wehage, R.A.: Coordinate reduction technique for transient analysis of special substructures with large angular rotations. J. Struct. Mech. 11(3), 401–431 (1983)

    Article  Google Scholar 

  3. Belytschko, T., Hsieh, B.J.: Nonlinear transient finite element analysis with convected coordinates. Int. J. Numer. Methods Eng. 7, 255–271 (1973)

    Article  MATH  Google Scholar 

  4. Rankin, C.C., Brogan, F.A.: An element independent corotational procedure for the treatment of large rotations. J. Press. Vessel Technol. 108, 165–174 (1986)

    Article  Google Scholar 

  5. Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem, Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985)

    Article  MATH  Google Scholar 

  6. Simo, J.C., Vu-Quoc, L.: A three-dimensional finite strain rod model, Part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)

    Article  MATH  Google Scholar 

  7. Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1, 189–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1(3), 339–348 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hughes, T.J.R.: The Finite Element Method. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  10. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123(4), 606–613 (2001)

    Article  Google Scholar 

  11. Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dyn. 45(1–2), 109–130 (2006)

    Article  MATH  Google Scholar 

  12. Von Dombrowski, S.: Analysis of large flexible body deformation in multibody systems using absolute coordinates. Multibody Syst. Dyn. 8(4), 409–432 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. J. Multibody Syst. Dyn. 20, 359–384 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: classical finite element formulation and absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 5(1), 011010 (2010) (10 pages)

    Article  Google Scholar 

  15. Nachbagauer, K., Gruber, P.G., Vetyukov, Yu., Gerstmayr, J.: A spatial thin beam finite element based on the absolute nodal coordinate formulation without singularities. In: Proc. of the ASME 2011 Int. Design Eng. Techn. Conf. & Computers and Information in Eng. Conf. IDETC/CIE 201. Washington, DC, USA, August, pp. 28–31 (2011)

    Google Scholar 

  16. Mikkola, A.M., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn. 9(3), 283–309 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dmitrochenko, O.N., Pogorelov, D.Yu.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10, 17–43 (2003)

    Article  MATH  Google Scholar 

  18. Omar, M.A., Shabana, A.A.: A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243(3), 565–576 (2001)

    Article  Google Scholar 

  19. Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34(1), 53–74 (2003)

    Article  MATH  Google Scholar 

  20. Garcia-Vallejo, D., Mikkola, A.M., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50(1–2), 249–264 (2007)

    Article  MATH  Google Scholar 

  21. Nachbagauer, K., Pechstein, A., Irschik, H., Gerstmayr, J.: New locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. J. Multibody Syst. Dyn. 26(3), 245–263 (2011)

    Article  MATH  Google Scholar 

  22. Kubler, L., Eberhard, P., Geisler, J.: Flexible multibody systems with large deformations and nonlinear structural damping using absolute nodal coordinates. Nonlinear Dyn. 34(1–2), 31–52 (2003)

    Article  MathSciNet  Google Scholar 

  23. Gerstmayr, J., Schoberl, J.: A 3D finite element method for flexible multibody systems. Multibody Syst. Dyn. 15(4), 309–324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Olshevskiy, A., Dmitrochenko, O.: Three-dimensional solid elements employing slopes in the absolute nodal coordinate formulation. In: Proc. of the 24th Nordic Seminar on Computational Mechanics, Helsinki, 2011.11.3–4, pp. 162–165 (2011)

    Google Scholar 

  25. Dmitrochenko, O., Mikkola, A.: A formal procedure and invariants of a transition from conventional finite elements to the absolute nodal coordinate formulation. Multibody Syst. Dyn. 22(4), 323–339 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dufva, K., et al.: Nonlinear dynamics of three-dimensional belt drives using the finite-element method. Nonlinear Dyn. 48(4), 449–466 (2007)

    Article  MATH  Google Scholar 

  27. Weed, D., Maqueda, L., Brown, M., Hussein, B., Shabana, A.: A new nonlinear multibody/finite element formulation for knee joint ligaments. Nonlinear Dyn. 60(4), 357–367 (2010)

    Article  MATH  Google Scholar 

  28. Gantoi, F., Brown, M., Shabana, A.: ANCF finite element/multibody system formulation of the ligament/bone insertion site constraints. J. Comput. Nonlinear Dyn. 5(3), (2010), 9 pages

    Google Scholar 

  29. Abdullah, M.A., Michitsuji, Y., Nagai, M., Miyajima, N.: Analysis of contact force variation between contact wire and pantograph based on multibody dynamics. J. Mech. Syst. Transp. Logist. 3(3), 552–567 (2010)

    Article  Google Scholar 

  30. Wan, H., Dong, H., Ren, Y.: Study of strain energy in deformed insect wings dynamic. In: Behavior of Materials. Conference Proceedings of the Society for Experimental Mechanics Series, vol. 1, pp. 323–328. Springer, New York (2011)

    Google Scholar 

  31. Hrennikoff, A.: Solution of problems of elasticity by the frame-work method. ASME J. Appl. Mech. 8, A619–A715 (1941)

    MathSciNet  Google Scholar 

  32. Melosh, R.J.: Structural analysis of solids. ASCE Struct. J. 4, 205–223 (1963)

    Google Scholar 

  33. Dmitrochenko, O., Mikkola, A.: Digital nomenclature code (dnc) of finite element kinematics and its modification (dncm) for absolute nodal coordinates. In: Proc. of 1st Int. Joint Conf. for Multibody System Dynamics (IMSD-2010), Lappeenranta, 25–27.05.2010 (2010)

    Google Scholar 

  34. Dufva, K., Shabana, A.: Analysis of thin plate structures using the absolute nodal coordinate formulation. J. Multibody Dyn. 219, 345–355 (2005). Proceedings of the Institution of Mechanical Engineers, Part K

    Google Scholar 

  35. Specht, B.: Modified shape functions for the three node plate bending element passing the patch test. Int. J. Numer. Methods Eng. 26, 705–715 (1988)

    Article  MATH  Google Scholar 

  36. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Solid Mechanics, vol. 2. Butterworth, London (2000)

    MATH  Google Scholar 

  37. Gere, J.M., Timoshenko, S.P.: Mechanics of Materials, 4th edn. PWS, Boston (1997), 912 pages

    Google Scholar 

  38. Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions—the plane case: Parts I and II. J. Appl. Mech. 53, 849–863 (1986)

    Article  MATH  Google Scholar 

  39. Pogorelov, D.: Some developments in computational techniques in modeling advanced mechanical systems. In: van Campen, D.H. (ed.) Proc. of IUTAM Symp. on Interaction between Dynamics and Control in Advanced Mech. Systems, pp. 313–320. Kluwer Academic, Dordrecht (1997)

    Chapter  Google Scholar 

Download references

Acknowledgements

This paper was supported by Konkuk University in 2012.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Konkuk University, 1 Hwanyang-Dong, Gwangjin-Gu, 143-701, Seoul, Korea

    Alexander Olshevskiy & Changwan Kim

  2. Department of Mechanical Engineering, Lappeenranta University of Technology, Skinnarilankatu 34, 53850, Lappeenranta, Finland

    Oleg Dmitrochenko

Authors
  1. Alexander Olshevskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg Dmitrochenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Changwan Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changwan Kim.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Olshevskiy, A., Dmitrochenko, O. & Kim, C. Three- and four-noded planar elements using absolute nodal coordinate formulation. Multibody Syst Dyn 29, 255–269 (2013). https://doi.org/10.1007/s11044-012-9314-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-012-9314-y

Keywords

Search

Navigation