On the Corotational Beam Element Formulation in Large Deformation Analysis

Document Type : Research Paper


1 Amirkabir University of Technology

2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran.


This paper sheds more light on the co-rotational element formulation for beams with uniform cross-section. The co-rotational elements are commonly used in problems in which a structure undergoes a large deformation. In this study, the foregoing element obeys the Euler-Bernoulli beam assumptions. Unlike the formulations presented in the literature, in this paper, a number of local nodal coordinates are employed which makes the kinematic description of the deformed beam much easier without the need of expressing any complicated relations. In this regard, via a clamped planar beam as a case study, the methodology is implemented step-by-step, and the results are compared with the ones calculated analytically and by means of elliptic integrals. Then, the methodology is briefly formulized for 3D cases as well. At the end, as a second case study, the large deformation analysis is conducted on a simply supported planar beam as well.


Main Subjects

[1] Bisshopp, K., and Drucker, D. C., "Large Deflection of Cantilever Beams", Quarterly of Applied Mathematics, Vol. 3, No. 3, pp. 272-275, (1945).
[2] Frisch-Fay, R., "Flexible Bars", Butterworths, London, (1962).
[3] Zhang, A., and Chen, G., "A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms", Journal of Mechanisms and Robotics, Vol. 5, No. 2, pp. 021006, (2013).
[4] Wang, C.M., and Kitipornchai, S., "Shooting Optimization Technique for Large Deflection Analysis of Structural Members", Engineering Structures, Vol. 14, No. 4, pp. 231-240, (1992).
[5] Pai, P.F., and Palazotto, A.N., "Large-deformation Analysis of Flexible Beams", International Journal of Solids and Structures, Vol. 33, No. 9, pp.1335-1353, (1996).
[6] Yin, X., Lee, K.M., and Lan, C.C., "Computational Models for Predicting the Deflected Shape of a Non-uniform, Flexible Finger", In Robotics and Automation, Proceedings, ICRA'04, 2004 IEEE International Conference, New Orleans, LA, USA, Vol. 3, pp. 2963-2968, (2004).
[7] Bathe, K.J., Ramm, E., and Wilson, E.L., "Finite Element Formulations for Large Deformation Dynamic Analysis", International Journal for Numerical Methods in Engineering, Vol. 9, No. 2, pp. 353-386, (1975).
[8] Bathe, K.J., and Bolourchi, S., "Large Displacement Analysis of Three‚Äźdimensional Beam Structures", International Journal for Numerical Methods in Engineering, Vol. 14, No. 7, pp. 961-986, (1979).
[9] Pai, P.F., Anderson, T.J., and Wheater, E.A., "Large-deformation Tests and Total-Lagrangian Finite-element Analyses of Flexible Beams", International Journal of Solids and Structures, Vol. 37, No. 21, pp. 2951-2980, (2000).
 [10] Borri, M., and Merlini, T., "A Large Displacement Formulation for Anisotropic Beam Analysis", Meccanica, Vol. 21, No. 1, pp. 30-37, (1986).
[11] Urthaler, Y., and Reddy, J.N., "A Corotational Finite Element Formulation for the Analysis of Planar Beams", International Journal for Numerical Methods in Biomedical Engineering, Vol. 21, No. 10, pp. 553-570, (2005).
[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, Vol. 194, No. 21, pp. 2285-2335, (2005).
[13] Shabana, A.A., "Dynamics of Multibody Systems", Cambridge University Press, London, (2013).
[14] Shabana, A.A., "Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation", Multibody System Dynamics, Vol. 1, No. 3, pp. 339-348, (1997).
[15] Shabana, A.A., Hussien, H.A., and Escanola, J.L., "Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems", Journal of Mechanical Design, Vol. 120, No. 2, pp. 188-195, (1998).
[16] Shabana, A.A., and Yakoub, R.Y., "Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory", Journal of Mechanical Design, Vol. 123, No. 4, pp. 606-613, (2001).
[17] Gerstmayr, J., Sugiyama, H., and Mikkola, A., "Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems", Journal of Computational and Nonlinear Dynamics", Vol. 8, No. 3, pp. 031016, (2013).
[18] Fish, J., and Belytschko, T., "A First Course in Finite Elements", John Wiley and Sons, New York, (2007).
[19] Conn, Andrew R., Nicholas IM Gould, and Ph L. Toint, "Trust Region Methods", Vol. 1, Siam, (2000).