Among the existing types of distributed-plasticity element formulation used to simulate structural members (e.g. beams and columns), force/flexibility-based (FB) formulations are superior, because they strictly satisfy force equilibrium and section strain-displacement (compatibility) equations. In the presence of softening behavior (e.g. in RC members), however, conventional FB formulations suffer from two pathogenies caused by strain field singularity, which appears as the so-called strain localization phenomenon. These pathogenies are loss of response objectivity – i.e. divergence of responses with mesh refinements – and iterative solution algorithm convergence failures and instabilities. These pathogenies have hindered the vast use of FB elements in the collapse analysis of RC structures.

In order to tackle these issues, as part of his Ph.D. research, Dr. Salehi developed a gradient inelastic (GI) beam theory and a corresponding innovative FB GI element formulation. The GI beam theory is obtained by generalizing the so-called strain gradient elasticity models to inelastic problems and enables the generation of continuous strain fields by incorporating section strain gradients in the strain-displacement equations. The primary advantages of the GI beam theory and its corresponding FB element formulation are: (a) their ability to produce objective local (e.g. curvature distributions) and global responses (e.g. nodal force vs. nodal displacement responses); (b) allowing use of any type of material constitutive relations (e.g. elastic, plasticity and damage models) in their sections; and (c) enabling generation of section strain (e.g. curvature) distributions with arbitrary resolution along the element length.

In addition to the development of the GI element formulation, Dr. Salehi recently explored its advantages for collapse analysis of RC frames and bridges in comparison to other existing element formulations through Incremental Dynamic Analysis (IDA) – this study can be found here. Dr. Salehi has also had a chance to extend the above GI beam theory to account for finite strains and rotations (sometimes referred to as P-δ and P-Δ effects), resulting in a finite-strain GI beam theory. The FB element formulation developed on the basis of this beam theory, unlike existing finite-strain formulation, enables simulation of structural members incorporating both material softening and geometric nonlinearities. To achieve the finite-strain GI beam theory, Dr. Salehi derived a full set of geometrically consistent equilibrium and section strain-displacement equations and combined those with the existing gradient nonlocality relations. The detailed descriptions of the finite-strain GI beam theory and its corresponding FB element formulation are found here. In addition to this extension, Dr. Salehi recently developed a multi-node GI element formulation, which allows positioning intermediate nodes over a GI element length, while strain field continuity is preserved. Dr. Salehi has implemented all the above element formulations in OpenSees and the basic small-strain two-node element formulation is now available to the public.