Study of geometric non-linear instability of 2D frame structures
Keywords:
Geometric instability, post-buckling, dynamic, geometrically exact beamAbstract
In this work, we deal with the geometric instability problem of the two-dimensional (2D) elastic frame structures undergoing large overall motion. The geometrically exact beam model with total Lagrangian formulation is used to obtain the solution to non-linear instability problems with large prebuckling displacements. We propose, in particular, a study of dynamic analysis that can deal with instability problems of this kind with no need for any load decrease. The dynamics approach provides a more realistic post-buckling behaviour for the case of snap-through or snap-back. The material damping is necessary when a classical time integration scheme like Newmark is used. The principal novelty in this work is to consider non-linear damping to avoid the vibration around the equilibrium point when a classical scheme as Newark is used. The efficiency of the damping model and methodology analysis are illustrated by a number of numerical simulations.
Downloads
References
Armentani E, Cali C, Cricrí G, Caputoc F, & Esposito, R. (2006). Numerical solution techniques
for structural instability problems. Journal of Achievements in Materials and Manufacturing
Engineering, 19, 53–64.
Bathe, K. J., & Noh, G. (2012). Insight into an implicit time integration scheme for structural
dynamics. Computers & Structures, 98–99, 1–6.
Bathe, K. J., Ramm, E., & Wilson, E. L. (1976). Finite element formulations for large deformation
dynamic analysis. International Journal for Numerical Methods in Engineering, 9, 353–386.
Belytschko, T., & Hsieh, B. J. (1973). Non-linear transient finite element analysis with convected
co-ordinates. International Journal for Numerical Methods in Engineering, 7, 255–271.Cescotto S. (1977). Study of large displacement and high strain by finite element method (Phd
thesis). Belgium: LIEGE.
Chung, J., & Hulbert, G. (1993). A time integration algorithm for structural dynamics with
improved numerical dissipation: The generalized-α method. Journal of Applied Mechanics,
, 371–375.
Cichoń, C. (1984). Large displacements in-plane analysis of elastic-plastic frames. Computers
& Structures, 5, 737–745.
Crisfield M. A. (1991).Nonlinear finite element analysis of solids and structures. Vol. 1. Chichester:
Jonh Wiley Chichester.
Haisler, W. E., Stricklin, J. H., & Key, J. E. (1977). Displacement incrementation in non-linear
structural analysis by the self-correcting method. International Journal for Numerical
Methods in Engineering, 11, 3–10.
Hilber, H. M., Hughes, T. J. R., & Taylor, R. L. (1977). Improved numerical dissipation for
time integration algorithms in structural dynamics. Earthquake Engineering & Structural
Dynamics, 5, 282–292.
Ibrahimbegović, A. (1995). On finite element implementation of geometrically nonlinear
Reissner’s beam theory: Three-dimensional curved beam elements. Computer Methods in
Applied Mechanics and Engineering, 122, 11–26.
Ibrahimbegović, A., Al Mikdad, M. (2000). Quadratically convergent direct calculation of
critical points for 3d structures undergoing finite rotations. Computer Methods in Applied
Mechanics and Engineering, 189, 107–120.
Ibrahimbegovic A. (2009). Nonlinear solid mechanics: Theoretical formulations and finite element
solution methods. New York, NY: Springer.
Ibrahimbegović, A., & Frey, F. (1993). Finite element analysis of linear and non-linear planar
deformations of elastic initially curved beams. International Journal for Numerical Methods
in Engineering, 36, 3239–3258.
Ibrahimbegovic, A., Knopf-Lenoir, C., Kucerova, A., & Villon P. (2004). Optimal design
and optimal control of structures undergoing finite rotations and elastic deformations.
International Journal for Numerical Methods in Engineering, 61, 2428–2460.
Ibrahimbegović, A., & Mamouri, S. (1999). Nonlinear dynamics of flexible beams in planar
motion: Formulation and time-stepping scheme for stiff problems. Computers & Structures,
, 1–22.
Ibrahimbegovic, A., & Mamouri, S. (2002). Energy conserving/decaying implicit time-stepping
scheme for nonlinear dynamics of three-dimensional beams undergoing finite rotations.
Computer Methods in Applied Mechanics and Engineering, 191, 4241–4258.
Ibrahimbegović, A., Shakourzadeh, H., Batoz, J. L., AI Mikdad M, & Guo Y. Q. (1996). On the
role of geometrically exact and second-order theories in buckling and post-buckling analysis
of three-dimensional beam structures. Computers & Structures, 61, 1101–1114.
Kuo Mo Hsiao, K. M. (1987). Nonlinear finite element analysis of elastic frames. Computers
& Structures, 26, 693–701.
Mamouri S, & Ibrahimbegovic A. (2001). Modified HHT dissipation scheme for beams
undergoing finite rotations. European Journal of Computational Mechanics, 11, 121–143.
Marsden, J. E., & Hughes, T. J. R. (1983). Mathematical foundation of elasticity. New York, NY:
Dover Publications.
Meek, J. L., & Xue, Q.. (1996). A study on the instability problem for 2D-frames. Computer
Methods in Applied Mechanics and Engineering, 136, 347–361.
Meek, J. L., & Qiang Xue (1998). A study on the instability problem for 3D-frames. Computer
Methods in Applied Mechanics and Engineering, 158, 235–254.
Newmark, N. M. (1959). A method of computational for structural dynamics. Journal of
Engineering Mechanics Division ASCE, 85, 67–94.Papadrakakis, M. (1981). Post-buckling analysis of spatial structures by vector iteration
methods. Computers & Structures, 14, 393–402.
Schwarz B., & Richardson M. (2013). Proportional Damping from Experimental Data.
Proceedings of the 31st IMAC, a conference on structural dynamics, Conference Proceedings
of the Society for Experimental Mechanics Series 45, Topics in Modal Analysis, Volume 7,
Chapter 17, R. Allemang et al. (eds.).
Tan H. S. (1985). Finite element analysis of the elastic, non-linear response of frames, plates
and arbitrary shells to static loads (Phd thesis). Brisbane: University of Queensland.
Timoshenko, S., & Gere, J. M. (1961). Theory of elastic stability (2nd ed.). New York, NY:
McGraw Hill.
Williams, F. W. (1964). An approach to the non-linear behaviour of the members of a rigid
jointed plane framework with finite deflections. Quarterly Journal of Mechanics and Applied
Mathematics, 17, 456–469.
Wood R. D., & Zeinkiewisz O. C. (1977). Geometrically nonlinear finite element analysis of
beams, frames, arches and axisymmetric shells. Computers & Structures, 7,725–735.
Zienkiewicz O. C., & Taylor R. L. (1991). Solid and fluid mechanics, dynamics and nonlinearity.
London: Mc Graw-Hill.
