Keywords: non-linear dynamic analysis, large amplitude transverse vibrations, finite element method, geometric effects, sandwich beams.

The influence of large displacements on the vibration behaviour of beams, subjected to bending from transverse disturbances, is analysed taking into account that the structure stiffness matrix depends on the current structural configuration at each time step. This large displacement stiffness matrix is evaluated with a second order strain field where a simple linearization technique is used.

The fundamental algorithm for direct time integration is the iterative formula of the Newmark constant acceleration algorithm [1,2], which is used herein. It is quite appropriate for non-linear dynamic problems as well as for pseudo-dynamic applications [3,4]. A variational formulation is developed and applied to obtain a beam finite element for moderate deformations, based upon the Timoshenko beam element [5] with transverse and axial deformations.

The stationarity of the total potential leads to a non-linear system of equations that contains quadratic and cubic terms in the displacements, causing the system to be non-symmetric, which leads to specific problems with equation solvers for the finite element analysis. The Picard method [1] is a direct iteration process applied herein to the linearized static solution. As the shape and orientation of the beam element is changing during its deformation, a Lagrangian reference analysis is used at any time step during the load and deformation incremental process.

Structural applications demonstrated the good performance of the algorithm for both static and dynamic analysis cases. The static application addressed the large displacement deformation of a hypothetical cantilever beam under a tip lateral load, which took into account the change of the angle of load orientation during the deformation process. The theoretical dynamic application also on a cantilever beam, addressed the influence of the magnitude of tip lateral force and of the accompanying tip axial force, showing the ability to also model non-linear geometric behaviour under dynamic loads. The theoretical and experimental dynamic calibration and validation used results for a sandwich cantilever beam constructed from aluminium with an interior cork core, subjected to a prescribed displacement at the free end.

1

J.N. Reddy, "The Finite Element Method", McGraw-Hill, 2nd Edition, New York, 1993.

2

K.J. Bathe, "Finite Element Procedures", Prentice-Hall, 2nd edition, Englewood Cliffs, New Jersey, 1996.

3

F.Q. Melo, J.O. Carneiro, H.R. Lopes, J.F.D. Rodrigues, J.F.S. Gomes, "Use of Pseudo-Dynamic Techniques in the Structural Analysis of Piping Systems under Seismic Loads", Journal of Strain Analysis, Vol 36, No 5, pp 441-451, 2001.

4

R.F. Almeida, H.P. Caraslindas, R.C. Barros, "Considerações sobre a Implementação de um Sistema Integrado para Ensaios Pseudo-Dinâmicos de Estruturas", Journal 'Mecanica Experimental', APAET, LNEC, Lisboa, Portugal, No 9, pp 75-83, 2003, (in Portuguese).

5

S. Timoshenko, G. MacCullogh, "Elements of Strength of Materials", Van Nostrand, New York, 1949.

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