Different Beam-Column Finite Element Formulations for Geometric Nonlinear Analysis with Large Displacements

The structural elements that make up the most varied structures used in structural engineering are increasingly slender. The mechanical behavior of these slender elements generally presents strongly nonlinear equilibrium paths in terms of their configuration changes, which leads to the need for specific strategies to solve the problem. In this sense, this work presents different 2D beam-column finite element formulations based on the total and updated Lagrangian and co-rotational references. As a contribution of this work, one of the co-rotational formulations considers that the length of the element is dependent on the nodal displacements, that is, a general form of the co-rotational formulations found in the literature. The objective is to verify the influence of this consideration in the analysis of plane frames with large displacements. The presented formulations are validated through the responses obtained from the analysis of different classic examples in the literature. In these analyses, it was observed that the element based on the total Lagrangian cannot represent the geometric nonlinear behavior from a displacement limit, and that the formulation considering the variation of the element length with the nodal displacements generated a small difference compared to the constant element length, justifying this simplification adopted in all formulations found in the literature. However, one of the examples analyzed shows that for high values of the inertia/area ratio, this simplification can lead to poor responses.