Stability and convergence of mixed methods for elastic rods of arbitrary geometry

A Timoshenko’s small-strain model for elastic rods with arbitrary geometry is analyzed using mixed finite element methods based on the Hellinger–Reissner principle. After presenting the mathematical model and commenting on some drawbacks of standard finite element approximations, a stabilized mixed formulation is derived by adding to the Galerkin formulation least squares residual of the equilibrium equations. Stability, uniform convergence and error estimates are proved and results of numerical experiments are presented illustrating the behavior of the finite element approximations, confirming the predicted rates of convergence and attesting the robustness of the stabilized mixed formulation.