Path-following strategy in bifurcation problems of thin-walled members by positional Finite Element Method

Resumo

In the present study, a computational framework is developed to the determination of thin-walled profi-
les’ equilibrium paths. These developments apply a geometric nonlinear formulation of the Finite Element Method

(FEM) based on the Total Lagrangian description of equilibrium. Besides, the mesh is composed of positional shell

finite element, in which the nodal degrees of freedom are positions, generalized vectors and linear rate of thick-
ness variation. The material’s behavior is represented by the Saint-Venant-Kirchhoff constitutive law. Moreover,

the precisely obtaining of the fundamental equilibrium path is executed by the Arc-Length strategy coupled with
the nonlinear solution technique based on the Newton-Raphson method. In this sense, the main purpose of the
present study is to obtain the multiple equilibrium paths starting from the same bifurcation point. Therefore, a

path-following approach is developed to induce the search for the requested path. The strategy consists of impo-
sing a perturbation on the current configuration, which is related to the eigenvector of the required mode. This

eigenvector is computed by means of a buckling analysis performed on the current configuration. The perturbation

is imposed as an external force, being applied close to the bifurcation point and removed as soon as it enters the de-
sired path. In this way, the equilibrium paths that present a bifurcation point can be completely determined without

the consideration of imperfections on the initial configuration, as it is traditionally done. Finally, one example are
compared against data available in literature, in which the accuracy, robustness and applicability of the proposed
formulation are demonstrated.