New enriched finite elements with softening plastic hinges for the modeling of localized failure in beams

We present in this work the theoretical aspect and numerical implementation of new enriched finite elements with smeared-embedded softening hinges for the analysis of Timoshenko and Euler–Bernoulli beams at failure. On one hand, similarly to the embedded hinge model, the softening hinge is described by a cohesive model between the stress resultants and the displacement jumps. On the other hand, the stress field in the element is evaluated from a constitutive relation at the cross-section level as in the classical smeared hinge model. The crucial point is the introduction of a consistent inelastic strain field dependent on both the interpolation scheme (not merely the characteristic length) of underlying finite elements and the hinge mode of interested. Furthermore, the traction continuity condition is enforced in strong form. Based on the irreversible thermodynamics, a multisurface model for the softening hinge and an inelastic constitutive relation for the cross-section of the beam are then established. Particularly, the displacement jumps are regarded as internal variables and can be determined at the material point level of the softening hinge. Therefore, it is not necessary to introduce the ad hoc penalty elastic stiffness, the extra degrees of freedom or the static condensation at the element level as in the embedded hinge model. Besides the above characteristics, the proposed method also defines a very convenient framework for its numerical implementation. Standard finite elements of beams/rods and well-developed algorithms for inelastic material models can be straightforwardly employed with minor modifications. Several representative numerical examples are presented, verifying that the proposed model is mesh-size objective, stress-locking free and frame invariant owing to its correct resolution of the kinematics and statics of the softening hinge.