Vol. 6, No. 2, pp. 767-787 (2010)
A MIXED CO-ROTATIONAL 3D BEAM ELEMENT FORMULATION
FOR ARBITRARILY LARGE ROTATIONS
Z.X. Li 1,* and L. Vu-Quoc 2
1 Associate professor, Department of Civil Engineering, Zhejiang University, Hangzhou 310058, China
2 Professor, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville FL 32611, USA
*(Corresponding author: E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.)
Received: 16 May 2009; Revised: 8 August 2009; Accepted: 11 August 2009
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ABSTRACT
A new 3-node co-rotational element formulation for 3D beam is presented. The present formulation differs from existing co-rotational formulations as follows: 1) vectorial rotational variables are used to replace traditional angular rotational variables, thus all nodal variables are additive in incremental solution procedure; 2) the Hellinger-Reissner functional is introduced to eliminate membrane and shear locking phenomena, with assumed membrane strains and shear strains employed to replace part of conforming strains; 3) all nodal variables are commutative in differentiating Hellinger-Reissner functional with respect to these variables, resulting in a symmetric element tangent stiffness matrix; 4) the total values of nodal variables are used to update the element tangent stiffness matrix, making it advantageous in solving dynamic problems. Several examples of elastic beams with large displacements and large rotations are analysed to verify the computational efficiency and reliability of the present beam element formulation.
KEYWORDS
Co-rotational method; vectorial rotational variable; 3D beam element; locking-free; Hellinger-Reissner functional; assumed strain.
REFERENCES
[1] Rankin, C.C. and Brogan, F.A., “An Element Independent Corotational Procedure for the Treatment of Large Rotation”, Journal of Pressure Vessel Technology-Transactions of The ASME, 1986, Vol. 108, No. 2, pp. 165-174.
[2] Crisfield, M.A., “Nonlinear Finite Element Analysis of Solid and Structures”, John Wiley & Sons, Chichester, 1996, Vol. 2.
[3] Yang, H.T.Y., Saigal, S., Masud, A. and Kapania, R.K., “Survey of Recent Shell Finite Elements”, International Journal for Numerical Methods in Engineering, 2000, Vol. 47, No.1, pp. 101-127.
[4] Wempner, G., “Finite Elements, Finite Rotations and Small Strains of Flexible Shells”, International Journal of Solids and Structures, 1969, Vol. 5, No. 2, pp. 117-153.
[5] Belytschko, T. and Hseih, B.J., “Non-linear Transient Finite Element Analysis with Convected Co-ordinates”, International Journal for Numerical Methods in Engineering, 1973, Vol. 7, No. 3, pp. 255-271.
[6] Belytschko, T. and Glaum, L.W., “Application of Higher Order Corotational Stretch Theories to Nonlinear Finite Element Analysis”, Computers & Structures, 1979, Vol. 10, No. 1-2, pp. 175-182.
[7] Argyris, J.H., Bahner, H., Doltsnis, J., et al., “Finite Element Method - the Natural Approach”, Computer Methods in Applied Mechanics and Engineering, 1978, Vol.17/18, Part 1, pp. l-106.
[8] Oran, C., “Tangent Stiffness in Plane Frames”, Journal of the Structural Division, ASCE, 1973, Vol. 99, ST6, pp. 973-985.
[9] Oran, C., “Tangent Stiffness in Space Frames”, Journal of the Structural Division, ASCE, 1973, Vol. 99, ST6, pp. 987-1001.
[10] Stolarski, H., Belytschko, T. and Lee, S.H., “Review of Shell Finite Elements and Corotational Theories”, Computational Mechanics Advances, 1995, Vol. 2, No. 2, pp. 125-212.
[11] Crisfield, M.A. and Moita, G.F., “A Unified Co-rotational Framework for Solids, Shells and Beams”, International Journal of Solids and Structures, 1996, Vol. 33, No. 20-22, pp. 2969-2992.
[12] Felippa, C.A. and Haugen, B., “A Unified Formulation of Small-strain Corotational Finite Elements, I. Theory”, Computer Methods in Applied Mechanics and Engineering, 2005, Vol. 194, No. 21-24, pp. 2285-2335.
[13] Urthaler, Y. and Reddy, J.N., “A Corotational Finite Element Formulation for the Analysis of Planar Beams”, Communications in Numerical Methods in Engineering, 2005, Vol. 21, No. 10, pp. 553–570.
[14] Galvanetto, U. and Crisfield, M.A., “An Energy-conserving Co-rotational Procedure for the Dynamics of Planar Beam Structures”, International Journal for Numerical Methods in Engineering, 1996, Vol. 39, No. 13, pp. 2265-2282.
[15] Iura, M., Suetake, Y. and Atluri, S.N., “Accuracy of Co-rotational Formulation for 3-D Timoshenko's Beam”, CMES-Computer Modeling In Engineering & Sciences, 2003, Vol. 4, No. 2, pp. 249-258.
[16] Pajot, J.M. and Maute, K., “Analytical Sensitivity Analysis of Geometrically Nonlinear Structures Based on the Co-rotational Finite Element Method”, Finite Elements in Analysis and Design, 2006, Vol. 42, No. 10, pp. 900-913.
[17] 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, 1986, Vol. 58, No. 1, pp. 79-116.
[18] Jelenic, G. and Crisfield, M.A., “Problems Associated with the Use of Cayley Transform and Tangent Scaling for Conserving Energy and Momenta in the Reissner–Simo Beam Theory”, Communications in Numerical Methods in Engineering, 2002, Vol. 18, No. 10, pp.711-720.
[19] McRobie, F.A. and Lasenby, J., “Simo-Vu Quoc rods using Clifford algebra”, International Journal for Numerical Methods in Engineering, 1999, Vol. 45, No. 4, pp. 377-398.
[20] Crisfield, M.A., “Consistent Co-rotational Formulation for Non-linear, Three-dimensional, Beam-elements”, Computer Methods in Applied Mechanics and Engineering, 1990, Vol. 81, No. 2, pp. 131-150.
[21] Simo, J.C., “(Symmetric) Hessian for Geometrically Nonlinear Models in Solid Mechanics, Intrinsic Definition and Geometric Interpretation”, Computer Methods in Applied Mechanics and Engineering, 1992, Vol. 96, No. 2, pp. 189-200.
[22] Li, Z.X., “A Mixed Co-rotational Formulation of 2D Beam Element Using Vectorial Rotational Variables”, Communications in Numerical Methods in Engineering, 2007, Vol. 23, No. 1, pp. 45-69.
[23] Li, Z.X., “A Co-rotational Formulation for 3D Beam Element Using Vectorial Rotational Variables”, Computational Mechanics, 2007, Vol. 39, No. 3, pp. 293-308.
[24] Li, Z.X. and Vu-Quoc, L., “An Efficient Co-rotational Formulation for Curved Triangular Shell Element”, International Journal for Numerical Methods in Engineering, 2007,Vol.72, No. 9, pp. 1029-1062.
[25] Li, Z.X., Izzuddin, B.A. and Vu-Quoc, L., “A 9-node Co-rotational Quadrilateral Shell Element”, Computational Mechanics, 2008, Vol. 42, No. 6, pp. 873-884.
[26] Lee, K., “Analysis of Large Displacements and Large Rotations of Three-dimensional Beams by Using Small Strains and Unit Vectors”, Communications in Numerical Methods in Engineering, 1997, Vol. 13, No. 12, pp. 987-997.
[27] Bathe, K.J. and Bolourchi, S., “Large Displacement Analysis of Three-dimensional Beam Structures”, International Journal for Numerical Methods in Engineering, 1979, Vol. 14, No.7, pp. 961-986.
[28] Wen, R.K. and Rahimzadeh, J., “Nonlinear Elastic Frame Analysis by Finite Element”, Journal of the Structural Division, ASCE, 1983, Vol. 109, No. 8, pp. 1951-1971.