×

A geometrically exact beam element based on the absolute nodal coordinate formulation. (English) Zbl 1347.74049

Summary: In this study, Reissner’s classical nonlinear rod formulation, as implemented by J. C. Simo and L. Vu-Quoc [J. Appl. Mech. 53, 849–854 (1986; Zbl 0607.73057)] by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S05 Finite element methods applied to problems in solid mechanics
70E55 Dynamics of multibody systems

Citations:

Zbl 0607.73057

Software:

BEAM189; Maple
Full Text: DOI

References:

[1] Antman, S.S.: The Theory of Rod. Handbuch der Physik. Springer, New York (1972)
[2] Bathe, K.: Finite Element Procedures in Engineering Analysis. Prentice Hall, New York (1982) · Zbl 0528.65053
[3] Dmitrochenko, O.N.: Pogorelov, D. Yu.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10, 17–43 (2003) · Zbl 1137.74435 · doi:10.1023/A:1024553708730
[4] Dufva, K., Sopanen, J., Mikkola, A.: A two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation. J. Sound Vib. 280, 719–738 (2005) · doi:10.1016/j.jsv.2003.12.044
[5] Escalona, J.L., Hussien, H.A., Shabana, A.A.: Application of the absolute nodal coordinate formulation for flexible multibody system dynamics. J. Sound Vib. 214(5), 833–851 (1998) · doi:10.1006/jsvi.1998.1563
[6] Géradin, M., Cardona, A.: Flexible Multibody Dynamics: A Finite Element Approach. Wiley, New York (2001)
[7] 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. (2008, in press). doi: 10.1016/j.jsv.2008.04.019
[8] Gerstmayr, J., Schöberl, J.: A 3D finite element method for flexible multibody systems. J. Multibody Syst. Dyn. 15, 309–324 (2006) · Zbl 1146.70318
[9] 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) · Zbl 1138.74391 · doi:10.1007/s11071-006-1856-1
[10] Ibrahimbegovíc, A.: On finite element implementation of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput. Methods Appl. Mech. Eng. 122, 11–26 (1995) · Zbl 0852.73061 · doi:10.1016/0045-7825(95)00724-F
[11] Inman, D.J.: Engineering Vibration, 2nd edn. Prentice Hall, Upper Saddle River (2001)
[12] Jeleníc, G., Crisfield, M.A.: Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput. Methods Appl. Mech. Eng. 171, 141–171 (1999) · Zbl 0962.74060 · doi:10.1016/S0045-7825(98)00249-7
[13] MAPLE 9.5, Maplesoft, Waterloo Maple Inc., 615 Kumpf Drive, Waterloo, ON, Canada (Apr. 7 2004). www.maplesoft.com
[14] 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) · doi:10.1006/jsvi.2000.3416
[15] Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. J. Appl. Math. Phys. 23, 795–804 (1972) · Zbl 0248.73022 · doi:10.1007/BF01602645
[16] Rhim, J., Lee, S.W.: A vectorial approach to computational modelling of beams undergoing finite rotations. Int. J. Numer. Meth. Eng. 41(3), 527–540 (1998) · Zbl 0902.73075 · doi:10.1002/(SICI)1097-0207(19980215)41:3<527::AID-NME297>3.0.CO;2-7
[17] Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: finite element method and absolute nodal coordinate formulation. In: Proceedings of the IDETC/CIE 2005 Long Beach, ASME, New York, Paper No. DETC2005-85104 (2005)
[18] Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1, 339–348 (1997) · Zbl 0890.73071 · doi:10.1023/A:1009740800463
[19] Simo, J.C.: A finite strain beam formulation, The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985) · Zbl 0583.73037 · doi:10.1016/0045-7825(85)90050-7
[20] 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. Mech. 53, 849–863 (1986) · Zbl 0607.73057 · doi:10.1115/1.3171870
[21] Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34(1), 53–74 (2004) · Zbl 1041.74547 · doi:10.1023/B:NODY.0000014552.68786.bc
[22] Sugiyama, H., Gerstmayr, J., Shabana, A.A.: Deformation modes of the finite element cross section. J. Sound Vib. 298, 1129–1149 (2006) · Zbl 1243.74188 · doi:10.1016/j.jsv.2006.06.037
[23] Timoshenko, S., Young, D.H., Weaver, W.: Vibration Problems in Engineering, 4th edn. Wiley, New York (1974)
[24] Yakoub, R.Y., Shabana, A.A.: Three dimensional absolute nodal coordinate formulation for beam elements. ASME J. Mech. Des. 123, 606–621 (2001) · doi:10.1115/1.1410099
[25] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, vol. 1–The Basis. Heinemann, London (2000) · Zbl 0991.74002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.