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Acta Mechanica

, Volume 230, Issue 3, pp 929–952 | Cite as

Spatial ANCF/CRBF beam elements

  • Shubhankar Kulkarni
  • Ahmed A. ShabanaEmail author
Original Paper
  • 87 Downloads

Abstract

In this paper, the consistent rotation-based formulation (CRBF) is used to develop new three-dimensional beam elements starting with the absolute nodal coordinate formulation (ANCF) kinematic description. While the proposed elements employ orientation parameters as nodal coordinates, independent rotation interpolation is avoided, leading to unique displacement and rotation fields. Furthermore, the proposed spatial ANCF/CRBF-based beam elements adhere to the noncommutative nature of the rotation parameters, allow for arbitrarily large three-dimensional rotation, and eliminate the need for using co-rotational or incremental solution procedures. Because the proposed elements have a general geometric description consistent with computational geometry methods, accurate definitions of the shear and bending deformations can be developed and evaluated, and curved structures and complex geometries can be systematically modeled. Three new spatial ANCF/CRBF beam elements, which use absolute positions and rotation parameters as nodal coordinates, are proposed. The time derivatives of the ANCF transverse position vector gradients at the nodes are expressed in terms of the time derivatives of rotation parameters using a nonlinear velocity transformation matrix. The velocity transformation leads to lower-dimensional elements that ensure the continuity of stresses and rotations at the element nodal points. The numerical results obtained from the proposed ANCF/CRBF elements are compared with the more general ANCF beam elements and with elements implemented in a commercial FE software.

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References

  1. 1.
    Bathe, K.J.: Finite Element Procedures. Prentice Hall Inc, Englewood Cliffs (1996)zbMATHGoogle Scholar
  2. 2.
    Belytschko, T., Hsieh, B.J.: Non-linear transient finite element analysis with convected co-ordinates. Int. J. Numer. Methods Eng. 7(3), 255–271 (1973)CrossRefzbMATHGoogle Scholar
  3. 3.
    Belytschko, T., Liu, W.K., Moran, B.: Nonlinear Finite Elements for Continua and Structures. Wiley, New York (2000)zbMATHGoogle Scholar
  4. 4.
    Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  5. 5.
    Campanelli, M., Berzeri, M., Shabana, A.A.: Performance of the incremental and non-incremental finite element formulations in flexible multibody problems. Trans. Am. Soc. Mech. Eng. J. Mech. Des. 122(4), 498–507 (2000)Google Scholar
  6. 6.
    Cook, R.D., Malkus, D.S., Plesha, M.E.: Concepts and Applications of Finite Element Analysis, 3rd edn. Wiley, New York (1989)zbMATHGoogle Scholar
  7. 7.
    Crisfield, M.A.: Nonlinear Finite Element Analysis of Solids and Structures, Vol. 1: Essentials. Wiley, New York (1991)zbMATHGoogle Scholar
  8. 8.
    Ding, J., Wallin, M., Wei, C., Recuero, A.M., Shabana, A.A.: Use of independent rotation field in the large displacement analysis of beams. Nonlinear Dyn. 76, 1829–1843 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dmitrochenko, O., Mikkola, A.: Digital nomenclature code for topology and kinematics of finite elements based on the absolute nodal co-ordinate formulation. IMechE J. Multibody Dyn. 225, 34–51 (2011)Google Scholar
  10. 10.
    Gerstmayr, J., Gruber, P., Humer, A.: Comparison of fully parameterized and gradient deficient elements in the absolute nodal coordinate formulation. In: ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V006T10A025–V006T10A025 (2017)Google Scholar
  11. 11.
    Gerstmayr, J., Irschik, H.: On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach. J. Sound Vib. 318(3), 461–487 (2008)CrossRefGoogle Scholar
  12. 12.
    Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 20(4), 359–384 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Greenberg, M.D.: Advanced Engineering Mathematics, 2nd edn. Prentice-Hall, Englewood Cliffs (1998)zbMATHGoogle Scholar
  14. 14.
    Gruber, P.G., Nachbagauer, K., Vetyukov, Y., Gerstmayr, J.: A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities. Mech. Sci. 4(2), 279–289 (2013)CrossRefGoogle Scholar
  15. 15.
    Hu, W., Tian, Q., Hu, H.Y.: Dynamics simulation of the liquid-filled flexible multibody system via the absolute nodal coordinate formulation and SPH method. Nonlinear Dyn. 75, 653–671 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kreyszig, E.: Differential Geometry. Dover Publications, New York (1991)zbMATHGoogle Scholar
  17. 17.
    Kulkarni, S., Pappalardo, C.M., Shabana, A.A.: Pantograph/catenary contact formulations. ASME J. Vib. Acoust. 139(1), 1–12 (2017)Google Scholar
  18. 18.
    Liu, C., Tian, Q., Hu, H.Y.: Dynamics of large scale rigid-flexible multibody system composed of composite laminated plates. Multibody Syst. Dyn. 26, 283–305 (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Matikainen, M.K., von Hertzen, R., Mikkola, A., Gerstmayr, J.: Elimination of high frequencies in the absolute nodal coordinate formulation. Proc. Inst. Mech. Eng. Part K J. Multibody Dyn. 224(1), 103–116 (2010)Google Scholar
  20. 20.
    Nachbagauer, K.: Development of shear and cross-section deformable beam finite elements applied to large deformation and dynamics problems. Ph.D. dissertation, Johannes Kepler University, Linz, Austria (2013)Google Scholar
  21. 21.
    Nachbagauer, K., Pechstein, A.S., Irschik, H., Gerstmayr, J.: A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 26(3), 245–263 (2011)CrossRefzbMATHGoogle Scholar
  22. 22.
    Nachbagauer, K., Gruber, P., Gerstmayr, J.: Structural and continuum mechanics approaches for a 3D shear deformable ANCF beam finite element: application to static and linearized dynamic examples. J. Comput. Nonlinear Dyn. 8(2), 021004-1–021004-7 (2013)Google Scholar
  23. 23.
    Nachbagauer, K.: State of the art of ANCF elements regarding geometric description, interpolation strategies, definition of elastic forces, validation and the locking phenomenon in comparison with proposed beam finite elements. Arch. Comput. Methods Eng. 21(3), 293–319 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nicolsen, B., Wang, L., Shabana, A.A.: Nonlinear finite element analysis of liquid sloshing in complex vehicle motion scenarios. J. Sound Vib. 405, 208–233 (2017)CrossRefGoogle Scholar
  25. 25.
    Orzechowski, G.: Analysis of beam elements of circular cross-section using the absolute nodal coordinate formulation. Arch. Mech. Eng. 59, 283–296 (2012)CrossRefGoogle Scholar
  26. 26.
    Orzechowski, G., Frączek, J.: Integration of the equations of motion of multibody systems using absolute nodal coordinate formulation. Acta Mech. Autom. 6, 75–83 (2012)Google Scholar
  27. 27.
    Orzechowski, G., Frączek, J.: Nearly incompressible nonlinear material models in the large deformation analysis of beams using ANCF. Nonlinear Dyn. 82, 1–14 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pappalardo, C.M.: A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems. J. Nonlinear Dyn. 81, 1841–1869 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pappalardo, C.M., Wallin, M., Shabana, A.A.: A new ANCF/CRBF fully parametrized plate finite element. ASME J. Comput. Nonlinear Dyn. 12(3), 1–13 (2017)Google Scholar
  30. 30.
    Patel, M.D., Orzechowski, G., Tian, Q., Shabana, A.A.: A new multibody system approach for tire modeling using ANCF finite elements. Proc. Inst. Mech. Eng. Part K J. Multibody Dyn. 230, 1–16 (2015)Google Scholar
  31. 31.
    Patel, M., Shabana, A.A.: Locking alleviation in the large displacement analysis of beam elements: the strain split method. Acta Mech. (2018) (accepted for publication)Google Scholar
  32. 32.
    Rankin, C.C., Brogan, F.A.: An Element independent corotational procedure for the treatment of large rotations. ASME J. Pressure Vessel Technol. 108, 165–174 (1986)CrossRefGoogle Scholar
  33. 33.
    Roberson, R.E., Schwertassek, R.: Dynamics of Multibody Systems. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  34. 34.
    Shabana, A.A.: Uniqueness of the geometric representation in large rotation finite element formulations. J. Comput. Nonlinear Dyn. 5(4), 044501-1–44501-5 (2010)CrossRefGoogle Scholar
  35. 35.
    Shabana, A.A.: Dynamics of Multibody Systems, 4th edn. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  36. 36.
    Shabana, A.A.: ANCF consistent rotation-based finite element formulation. ASME J. Comput. Nonlinear Dyn. 11(1), 014502-1–014502-4 (2016)MathSciNetGoogle Scholar
  37. 37.
    Shabana, A.A., Patel, M.: Coupling between shear and bending in the analysis of beam problems: planar case. J. Sound Vib. 419, 510–525 (2018)CrossRefGoogle Scholar
  38. 38.
    Shabana, A.A., Yakoub, R.Y.: Three-dimensional absolute nodal coordinate formulation for beam elements: theory. Trans. Am. Soc. Mech. Eng. J. Mechv Des. 123(4), 606–613 (2001)Google Scholar
  39. 39.
    Shampine, L., Gordon, M.: Computer Solution of ODE: The Initial Value Problem. Freeman, San Francisco (1975)zbMATHGoogle Scholar
  40. 40.
    Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions—the plane case, part I. J. Appl. Mech. 53, 849–863 (1986)CrossRefzbMATHGoogle Scholar
  41. 41.
    Takahashi, Y., Shimizu, N.: Study on elastic forces of the absolute nodal coordinate formulation for deformable beams. In: Proceedings of ASME International Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Las Vegas, NV (1999)Google Scholar
  42. 42.
    Tian, Q., Chen, L.P., Zhang, Y.Q., Yang, J.Z.: An efficient hybrid method for multibody dynamics simulation based on absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 4, 021009-1–021009-14 (2009)CrossRefGoogle Scholar
  43. 43.
    Tian, Q., Sun, Y.L., Liu, C., Hu, H.Y., Paulo, F.: Elasto-hydro-dynamic lubricated cylindrical joints for rigid-flexible multibody dynamics. Comput. Struct. 114–115, 106–120 (2013)CrossRefGoogle Scholar
  44. 44.
    Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill Book Co. Inc., New York (1970)zbMATHGoogle Scholar
  45. 45.
    Wittenburg, J.: Dynamics of Multibody Systems, 2nd edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  46. 46.
    Yakoub, R.Y., Shabana, A.A.: Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. Trans. Am. Soc. Mech. Eng. J. Mech. Des. 123(4), 614–621 (2001)Google Scholar
  47. 47.
    Zheng, Y., Shabana, A.A.: A two-dimensional shear deformable ANCF consistent rotation-based formulation beam element. Nonlinear Dyn. 87(2), 1031–1043 (2017)CrossRefGoogle Scholar
  48. 48.
    Zheng, Y., Shabana, A.A., Zhang, D.: Curvature expressions for the large displacement analysis of planar beam motions. ASME J. Comput. Nonlinear Dyn. 13, 011013-1–011013-12 (2018)Google Scholar
  49. 49.
    Zienkiewicz, O.C.: The Finite Element Method, 3rd edn. McGraw Hill, New York (1977)zbMATHGoogle Scholar
  50. 50.
    Zienkiewicz, O.C, Taylor, R.L.: The Finite Element Method, Vol. 2: Solid Mechanics, 5th edn. Butterworth Heinemann, Boston (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of Illinois at ChicagoChicagoUSA

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