Refined Gradient Inelastic Flexibility-Based Formulation for Members Subjected to Arbitrary Loading

Abstract

This paper advances the gradient inelastic (GI) flexibility-based (FB) frame element formulation, which focused on monotonic loading conditions, to capture member responses to arbitrary (nonmonotonic) loading conditions. The GI formulation is a generalization of the strain gradient elasticity theory to inelastic continua. Contrary to nonlocal/gradient damage or plasticity models, in the GI theory, nonlocality is strictly decoupled from the constitutive relations; as a result, the GI theory can incorporate any material constitutive law (plastic, hardening, or softening), whereas for linear elastic materials, the GI theory reduces to the strain gradient elasticity theory. In the GI theory, alleviation of strain localization and response objectivity (i.e., convergence with mesh refinements) are achieved through a localization condition applied to the strain field at strain localization locations during strain softening. Strain localization locations are not a priori known and are identified through a localization criterion. This paper advances/refines the GI theory and GI FB element formulation by (1) eliminating (unintended and unphysical) discontinuities in the temporal response of section strains introduced during application of the localization condition; and (2) alleviating false identifications of localization locations in the case of arbitrary and combined (axial/flexural/shear) loading through a new robust localization criterion. Furthermore, this paper assesses (1) various mathematically admissible end boundary conditions to the section strain fields—proper selection of which has been a challenge in higher order gradient theories—in terms of their physical rationale and mesh convergence properties; and (2) higher-order nonlocality relations in terms of the spatial characteristics of the resulting section strain fields and mesh convergence properties. The performance of the proposed formulation is evaluated through examples on frame members subjected to monotonic, cyclic, and seismic loading, and comparisons with experimental data from quasi-static cyclic testing of a reinforced concrete column.