Abstract
Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper.
In many applications, a nonlinear large deformation beam element with bending, axial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner’s virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of conventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully parameterized ANCF element.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Gams, M., Planinc, I., Saje, M.: The strain-based beam finite elements in multibody dynamics. J. Sound Vib. 305, 194–210 (2007)
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, 461–487 (2008)
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)
Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45(1–2), 109–130 (2006)
Irschik, H., Gerstmayr, J.: A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli–Euler beams. Acta Mech. 206, 1–21 (2009)
Irschik, H., Gerstmayr, J.: A hyperelastic Reissner-type model for non-linear shear deformable beams. In: Troch, I., Breitenecker, F. (eds.) Proceedings of the Mathmod 09 Vienna (2009)
Kerkkänen, K.S., Sopanen, J.T., Mikkola, A.M.: A linear beam finite element based on the absolute nodal coordinate formulation. ASME J. Mech. Des. 127, 621–630 (2005)
Matikainen, M.K., von Hertzen, R., Mikkola, A.M., Gerstmayr, J.: Elimination of high frequencies in the absolute nodal coordinate formulation. In: Proceedings of the Institution of Mechanical Engineers, Part K, Journal of Multibody Dynamics (2009)
Mikkola, A.M., Garcia-Vallejo, D., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50, 249–264 (2007)
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)
Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. J. Appl. Math. Phys. 23, 795–804 (1972)
Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1(3), 339–348 (1997)
Shabana, A.A.: Dynamics of multibody systems (3rd edn). Cambridge University Press, New York (2005)
Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions—the plane case: Part I and II. J. Appl. Math. 53, 849–863 (1986)
Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34, 53–74 (2003)
Yakoub, R.Y., Shabana, A.A.: Three-dimensional absolute nodal coordinate formulation for beam elements. ASME J. Mech. Des. 123, 606–621 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Nachbagauer, K., Pechstein, A.S., Irschik, H. et al. 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, 245–263 (2011). https://doi.org/10.1007/s11044-011-9249-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11044-011-9249-8