Application of finite elements in time to structural dynamics problems with geometric nonlinearities

The application of the finite element method to the temporal, as well as the spatial, part of a general structural dynamics problem is investigated. A pair of variational formulations (Hamilton's Law and Hamilton's Principle) are presented and their use with specific problems is demonstrated. Geometric nonlinearities are included in the deformation model so that large deformations, but not large strains, may be accommodated in the dynamic response predictions.

Arguments are presented to justify the application of Hamilton's Principle to initial value problems in elastodynamics. This includes specific limitations on the time interval over which Hamilton's Principle is applied.

A scheme for generating discrete approximations with either of the two principles is developed. This includes the formulation of a beam element which satisfies correct kinematic constraints in spite of large deformations. An algorithm is written and programmed to provide discrete approximations to general problems in structural dynamics. The numerical properties of the algorithm (stability, dissipation, and period error) are analyzed and discussed. The algorithm is used to obtain predictions for six example problems, which assist in the evaluation of the formulations.

The Hamilton's Law formulation is shown to be highly dissipative. This property is found to be aggravated when multiple time elements are used. The failure of this formulation's predictions to converge as time elements are added is explained by this behavior.

The Hamilton's Principle formulation is shown to provide predictions which conserve energy through time and which cause minimal degradation of the natural period. It is also used to solve an example problem that involves geometric nonlinearities.