Non-linear beam vibration problems and simplifications in finite element models

When finite element formulations are used to study the non-linear vibration problems, some simplifications, which are not consistent with the governing variational principles, are commonly employed. Three such simplifications are critically reviewed here, through beam finite element models. The first one, equivalent/ quasi-linearisation technique is shown to have a reduced non-linear stiffness. The second, where in neglect of in plane displacements takes place, is seen to register an excessive non-linear stiffness. Thirdly, when both these simplifications are introduced together, they produce results closer to those of variationally correct ones,rather fortuitously. The objective of this paper is to highlight the necessity of formulating this class of problems in a variationally correct and consistent manner. Numerical computations are performed systematically, using two different beam finite element models for various commonly studied boundary conditions and suitable conclusions are drawn.