On the Influence of Warping, Shear and Longitudinal Displacements on the Nonlinear Vibrations of Beams

Abstract

The geometrically nonlinear vibrations of beams which may experience longitudinal, torsional and bending deformations in any plane are investigated by the p-version finite element method. Bernoulli-Euler or Timoshenkos beam theories are considered for bending and Saint-Venantss for torsion. A warping function is included in the model. The geometrical nonlinearity is taken into account by considering the Greens strain tensor and the longitudinal displacements of quadratic order, which are most often neglected in the strain displacement relation, are considered here. Generalised Hookes law is used and the equation of motion is derived by the principle of virtual work. Comparisons of both models, Bernoulli-Euler and Timoshenko, and comparison of models including and neglecting the quadratic terms of longitudinal displacements are presented. It is shown that Timoshenkos theory gives better results than Bernoulli-Eulers when the bending and torsion motions are coupled and the nonlinear terms become important. This is explained by the fact that when bending and torsion are coupled, the rotations along the transverse axes of the beam cannot be approximated by the respective derivatives of the transverse displacement functions as is assumed in BernoulliEulers theory. The importance of warping is also analysed for different rectangular cross sections, and it is shown that its consideration can be fundamental to obtain correct results.