Numerical modeling of softening hinges in thin Euler–Bernoulli beams

This paper presents new finite elements for thin Euler–Bernoulli beams that incorporate the softening hinges observed at failure. The proposed methods rely crucially on the identification of the classical notion of inelastic hinge with strong discontinuities of the generalized displacements describing the beam’s deformation. The development of a multi-scale framework that effectively incorporates the localized dissipative mechanisms associated with these discontinuous solutions into the large-scale problem of the beam, or general frame system, defines a crucial step undertaken here. This framework defines the setting of its numerical implementation by finite elements enhanced with the singular strains corresponding to the discontinuities. A general procedure is presented that leads, in particular, to finite elements free of stress-locking and that resolve exactly the kinematics of the hinge. Several numerical simulations are presented to illustrate the performance of the proposed methods.