Nonlocal Hardening-Damage Beam Model and Its Application to a Force-Based Element Formulation

This paper introduces a novel nonlocal hardening-damage model for beams and integrates it within a flexibility-based frame element formulation. The proposed formulation eliminates strain singularity phenomena, which are common in (local) beam theories in the presence of softening materials, and achieves objective response (i.e., convergence with mesh refinements). Strain singularity phenomena have prevented accurate simulation of framed structures experiencing structural softening and have introduced major uncertainty in predictions of structural damage and collapse. The proposed model extends classical nonlocal elastodamage models to account for hardening (as opposed to elastic) materials by combining a damage/softening factor that is a (decreasing) function of nonlocal strains together with a hardening constitutive model that is solely a function of local strains. This allows simulation of a variety of responses combining stiffness and strength deterioration (captured by the damage/softening factor) together with hysteretic/cyclic response (captured by the hardening model). Moreover, unlike common elastodamage models, this model is also capable of capturing residual deformations, which are of importance to structural engineering applications. Because the damage/softening factor is fully decoupled from the hardening model, any (local) hardening model, simple or complex, can be used, including the large variety of (local) material constitutive relations currently used in nonlinear dynamic analyses of buildings and bridges. On the basis of the proposed nonlocal hardening-damage beam model, a flexibility-based frame element formulation is developed, which allows exploration of integral and (implicit) gradient nonlocal formulations with various boundary conditions (BCs). Symmetry considerations are employed to select nonlocal models (and their BCs) that provide physically plausible response predictions. The performance of these formulations is explored for axial and bending problems in terms of response properties and convergence rate. The performance and accuracy of the proposed formulation is further assessed through comparison with experimental data from quasi-static cyclic testing of a reinforced concrete column.