Geometrically nonlinear analysis of planar Kirchhoff rods through a position-based finite element formulation

This contribution addresses the nonlinear static analysis of planar Kirchhoff rods, accounting for large displacements and finite strains. To this aim, the position-based finite element formulation developed by Valvo for isoparametric elements is extended to include rotational degrees of freedom. Accordingly, the nodal positions and rotations in the current configuration are chosen as the main unknowns. A curved planar Kirchhoff rod element is formulated adopting an interpolation scheme based on Hermite polynomials, as first proposed by Armero. Our formulation stands out for the following advantages: it is a positional model that includes rotational degrees of freedom; it can be used for both finite and infinitesimal displacements, in the latter case after a consistent linearisation; it furnishes simple analytical expressions of the secant and tangent stiffness matrices; the secant stiffness matrix turns out to be symmetric; the model can be applied to both straight and curved rods. It is also noteworthy that any hyper-elastic constitutive law can be easily implemented. In this work, for the sake of illustration, the constitutive law derived from classical laminated beam theory is adopted, potentially allowing for elastic couplings to be considered. The governing equations are solved using an incremental-iterative procedure based on Newton-Raphson’s method. The formulation is effective in both linear and nonlinear cases, as shown by the excellent agreement with both analytical solutions and Abaqus results.