Reference Coordinate System of Nonlinear Beam Element Based on the Geometrically Exact Formulation under Large Spatial Rotations and Deformations — Oak Academic Publishing
Research ArticleOpen AccessGoogle Scholar indexed
Reference Coordinate System of Nonlinear Beam Element Based on the Geometrically Exact Formulation under Large Spatial Rotations and Deformations
Analysis of slender beam structures in a three-dimensional space is widely applicable in mechanical and civil engineering. This paper presents a new procedure to determine the reference coordinate system of a beam element under large rotation and elastic deformation based on a newly introduced physical concept: the zero twist sectional condition, which means that a non-twisted section between two nodes always exists and this section can reasonably be regarded as a reference coordinate system to calculate the internal element forces. This method can avoid the disagreement of the reference coordinates which might occur under large spatial rotations and deformations. Numerical examples given in the paper prove that this procedure guarantees the numerical exactness of the inherent formulation and improves the numerical efficiency, especially under large spatial rotations.
E. Reissner, “On One-Dimensional Finite-Strain Beam Theory: The Plane Problem,” Journal of Applied Mathematics and Physics (ZAMP), Vol. 23, No. 5, 1972, pp. 795-804. doi:10.1007/BF01602645
J. C. Simo and L. Vu-Quoc, “A Three-Dimensional Finite-Strain Rod Model. Part II: Computa-tional Aspects,” Computer Methods in Applied Mechanics and Engineering, Vol. 58, No. 1, 1986, pp. 79-116. doi:10.1016/0045-7825(86)90079-4
A. A. Shabana and R. Y. Yakoub, “Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory,” Journal of Mechan-ical Design, Vol. 123, No. 4, 2001, pp. 606-613. doi:10.1115/1.1410100
R. Y. Yakoub and A. A. Shabana, “Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications,” Journal of Mechanical Design, Vol. 123, No. 4, 2001, pp. 614-621. doi:10.1115/1.1410099
I. Romero, “A Comparison of Finite Elements for Nonlinear Beams: The Absolute Nodal Coordinate and Geometrically Exact Formulations,” Multibody System Dynamics, Vol. 20, No. 1, 2008, pp. 51-68. doi:10.1007/s11044-008-9105-7
M. A. Crisfield, “A Consistent Co-Rotational Formulation for Non-Linear, Three-Dimensional, Beam-Elements,” Computer Methods in Applied Mechanics and Engineering, Vol. 81, No. 2, 1990, pp. 131-150. doi:10.1016/0045-7825(90)90106-V
M. A. Crisfield, “Non-linear Finite Element Analysis of Solids and Structures,” John Wiley & Sons, Hoboken, 1997.
S. R. Kuo, Y. B. Yang and J. H. Chou, “Nonlinear Analysis of Space Frames with Finite Rotations,” Journal of Structural Engineering, Vol. 119, No. 1, 1993, pp. 1-15. doi:10.1061/(ASCE)0733-9445(1993)119:1(1)
Y. B. Yang and S. R. Kuo, “Thoery & Analysis of Nonlinear Framed Structures,” Prentice Hall, Upper Saddle River, 1994.
J. Argyris, “An Excursion into Large Rotations,” Computer Me-thods in Applied Mechanics and Engineering, Vol. 32, No. 1-3, 1982, pp. 85-155. doi:10.1016/0045-7825(82)90069-X
M. A. Crisfield, “Non-linear Finite Element Analysis of Solids and Structures,” John Wiley & Sons, Hoboken, 1991.
A. R. Klumpp, “Singularity-Free Extraction of a Quaternion from a Direction-Cosine Matrix,” Journal of Spacecraft and Rockets, Vol. 13, No. 2, 1976, pp. 754-755. doi:10.2514/3.27947
R. A. Spurrier, “Comment on ‘Singularity-free Extraction of a Quaternion from a Direction-Cosine Matrix’,” Journal of Spacecraft and Rockets, Vol. 15, No. 4, 1978, pp. 255-255. doi:10.2514/3.57311
J. H. Argyris, et al., “Finite Element Method: The Natural Approach,” Computer Methods in Applied Mechanics and Engineering, Vol. 17-18, 1979, pp. 1-106. doi:10.1016/0045-7825(79)90083-5
M. Y. Kim, S. P. Chang and S. B. Kim, “Spatial Postbuckling Analysis of Nonsymmetric Thin-Walled Frames II: Geometrically Nonlinear FE Procedures,” Journal of Engineering Mechanics, Vol. 127, No. 8, 2001, pp. 779- 790. doi:10.1061/(ASCE)0733-9399(2001)127:8(779)
M. Y. Kim, S. P. Chang and H. G. Park, “Spatial Postbuckling Analy-sis of Nonsymmetric Thin-Walled Frames. I: Theoretical Con-siderations Based on Semitangential Property,” Journal of En-gineering Mechanics, Vol. 128, No. 8, 2001, pp. 769-778.doi:10.1061/(ASCE)0733- 9399(2001)127:8(769)
S. B. Kim and M. Y. Kim, “Improved Formulation for Spatial Stabil-ity and Free Vibration of Thin-Walled Tapered Beams and Space Frames,” Engineering Structures, Vol. 22, No. 5, 2000, pp. 446-458. doi:10.1016/S0141- 0296(98)00140-0
H. H. Chen, W. Y. Lin and K. M. Hsiao, “Co-Rotational Finite Ele-ment Formulation for Thin-Walled Beams with Generic Open Section,” Computer Methods in Applied Mechanics and Engi-neering, Vol. 195, No. 19-22, 2006, pp. 2334-2370. doi:10.1016/j.cma.2005.05.011
K. M. Hsiao and W. Y. Lin, “A Co-Rotational Finite Element Formulation for Buckling and Postbuckling Analyses of Spatial Beams,” Computer Methods in Applied Mechanics and Engineering, Vol. 188, No. 1-3, 2000, pp. 567-594. doi:10.1016/S0045-7825(99)00284-4
W. Y. Lin and K. M. Hsiao, “Co-Rotational Formulation for Geometric Nonlinear Analysis of Doubly Symmetric Thin-Walled Beams,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 45, 2001, pp. 6023-6052. doi:10.1016/S0045-7825(01)00212-2
K. M. Hsiao and W. Y. Lin, “A Co-Rotational Formulation for Thin-Walled Beams with Monosymmetric Open Section,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 8-10, 2000, pp. 1163-1185. doi:10.1016/S0045-7825(99)00471-5
M. Y. Kim, N. I. Kim and H. T. Yun, “Exact Dynamic and Static Stiffness Matrices of Shear Deformable Thin-Walled Beam-Columns,” Journal of Sound and Vibration, Vol. 267, No. 1, 2003, pp. 29-55. doi:10.1016/S0022-460X(02)01410-4
K.-J. Bathe and S. Bolourchi, “Large Displacement Analysis of Three-Dimensional Beam Structures,” International Journal for Numerical Methods in Engineering, Vol. 14, No. 7, 1979, pp. 961-986. doi:10.1002/nme.1620140703
A. Cardona and M. Gera-din, “A Beam Finite Element Non-Linear Theory with Finite Rotations,” International Journal for Numerical Methods in Engineering, Vol. 26. No. 11, 1988, pp. 2403-2438. doi:10.1002/nme.1620261105