An iso-parametric $$\pmb {\mathrm {G}^1}$$ G 1 -conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case

A geometrically exact nonlinear iso-parametric $$\mathrm {G}^1$$ -conforming finite element formulation for the analysis of Kirchhoff rods, based on the cubic Bézier curve interpolation, is presented. In this work, the formulation is restricted to the planar 2D case. Introducing the $$\mathrm {G}^1$$ -map, the interpolation preserves the continuity requirement during the deformation process of the rod. In this way, the $$\mathrm {G}^1$$ -conformity is implicitly accounted at the element formulation level.