Stress-based formulation for non-linear analysis of curved beams

The linear and non-linear analysis of curved beams is a very classical but still very discussed topic in literature. The most commonly used approach is the displacement approach, which is affected, as it is well known by locking phenomena. A viable alternative to overcome the problem of locking encountered is represented by equilibrium formulation in static linear analysis and mixed formulations in dynamic and non-linear analysis. In particular, in non linear analysis of beams formulations based on a Hellinger-Reissner variational principle or modified Hu Washizu principle have been proposed in literature. In this paper a valid alternative is proposed for the non-linear analysis of an arbitrarily curved, extensible, shear flexible, elastic planar beam. The proposed formulation is based on a new variational principle expressed in terms of stress components. In particular, the unknowns are represented by the bending moment and by a variable from which the shear and the axial force can be expressed and by the generalized forces at the extremities of the bar. The Euler-Lagrange equations of this principle are the elasto-kinematic relations related to the curvature and the moment equilibrium equation. The effectiveness of the approach is illustrated through numerical examples. A comparison in terms of displacements and stress with the other formulations available in literature ends the paper.