Abstract
The objective of this paper is to investigate the accuracy of the elastic force models that can be used in the absolute nodal coordinate finite element formulation. This study focuses on the description of the elastic forces in three-dimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for a three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces in the beam element. The improvement proposal is verified using several numerical examples that show that the proposed elastic force model of the beam element agrees with the analytical results as well as with the solutions obtained using existing finite element formulation. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that a beam element based on the absolute nodal coordinate formulation relaxes the assumption of a rigid cross-section and is capable of representing a distortional deformation of the cross-section. The numerical results also imply that the beam element does not suffer from the phenomenon called shear locking.
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References
Simo, J. C., ‘A finite strain beam formulation. The three-dimensional dynamic problem. Part I’, Computer Methods in Applied Mechanics and Engineering 49, 1985, 55–70.
Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model. Part II: Computational aspects’, Computer Methods in Applied Mechanics and Engineering 58, 1986, 79–116.
Cardona, A. and Geradin, M., ‘A beam finite element non-linear theory with finite rotations’, International Journal for Numerical Methods in Engineering 26, 1988, 2403–2438.
Avello, A. and García de Jalón, J., ‘Dynamics of flexible multibody systems using Cartesian coordinates and large displacement theory’, International Journal for Numerical Methods in Engineering 32, 1991, 1543–1563.
Jelenić, G. and Saje, M., ‘A kinematically exact space finite strain beam model – Finite element formulation by generalized virtual work principle’, Computer Methods in Applied Mechanics and Engineering 120, 1995, 131–161.
Ibrahimbegović, A., ‘On the finite element implementation of geometrically nonlinear Reissner's beam theory: Three-dimensional curved beam elements’, Computer Methods in Applied Mechanics and Engineering 122, 1995, 11–26.
Jelenić, G. and Crisfield, M. A., ‘Interpolation of rotational variables in nonlinear dynamics of 3D beams', International Journal for Numerical Methods in Engineering 43, 1998, 1193–1222.
Jelenić, G. and Crisfield, M. A., ‘Geometrically exact 3D beam theory: Implementation of a strain-invariant finite elements for statics and dynamics', Computer Methods in Applied Mechanics and Engineering 171, 1999, 141–171.
Géradin, M. and Cardona, A., Flexible Multibody Dynamics: A Finite Element Approach, Wiley, West Sussex, 2001.
Shabana, A. A., Dynamics of Multibody Systems, Wiley, New York, 1998.
Shabana, A. A. and Yakoub, R. Y., ‘Three dimensional absolute nodal coordinate formulation for beam elements: Theory', ASME Journal of Mechanical Design 123, 2001, 606–613.
Yakoub, R. Y. and Shabana, A. A., ‘Three dimensional absolute nodal coordinate formulation for beam elements: Implementation and application', ASME Journal of Mechanical Design 123, 2001, 614–621.
Mikkola, A. M. and Shabana, A. A., ‘A new plate element based on the absolute nodal coordinate formulation', in Proceedings of DETC'01, ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Pittsburgh, PA, September 9–12, 2001.
ANSYS User's Manual, Theory, Twelfth Edition, SAS IP Inc., 2001.
Berzeri, M. and Shabana, A. A., ‘Development of simple models for the elastic forces in the absolute nodal coordinate formulation', Journal of Sound and Vibration 235, 2000, 539–565.
Omar, M. A. and Shabana, A. A., ‘A two-dimensional shear deformation beam for large rotation and deformation', Journal of Sound and Vibration 243(3), 2001, 565–576.
Sopanen, J. T. and Mikkola, A. M., ‘Studies on the stiffness properties of the absolute nodal coordinate formulation for three-dimensional beams’, in Proceedings of DETC'03, ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, IL, 2003, DETC2003/VIB-48325.
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Sopanen, J.T., Mikkola, A.M. Description of Elastic Forces in Absolute Nodal Coordinate Formulation. Nonlinear Dynamics 34, 53–74 (2003). https://doi.org/10.1023/B:NODY.0000014552.68786.bc
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DOI: https://doi.org/10.1023/B:NODY.0000014552.68786.bc