Comparison between different sets of interpolation functions for Timoshenko frame elements

Resumo

The objective of this paper is to discuss, in the context of a second-order geometric non-linear analysis, the differences in results considering different sets of interpolation functions for frame elements. The usual way to obtain interpolation functions for Timoshenko frame elements is using cubic Hermitian polynomials since they are the solution for the fourth-order differential equation which represents the bending behavior of the infinitesimal element. When dealing with more complex problems (geometrical and/or physical nonlinearities), the usual strategy is to subdivide the elements, which circumvents the limitation of the interpolation functions. Since discretization can sometimes be unwanted, especially for undergraduate students who still do not grasp this concept, a solution which overcomes this is interesting from a didactic point of view. This work proposes the use of different sets of shape functions to interpolate displacements, rotations and bending moments along the elements length, to account for the nonlinearities that arise from the change in geometry during loading. Shape functions obtained directly from the solution of the differential equation of an axially loaded deformed infinitesimal element and traditional Hermitian polynomials are used. Comparisons were made against analytical and numerical solutions using the two-cycle approach. Initial results indicate the ability of the formulation to capture the nonlinear behavior without the need to over discretize the domain.