A Gradient Inelastic Flexibility-Based Frame Element Formulation

Abstract

This paper introduces a novel gradient inelastic beam theory and a corresponding flexibility-based (FB) frame element formulation to treat major weaknesses of existing FB formulations in the presence of softening section constitutive relations (or softening material response). These weaknesses include (1) strain localization and loss of objectivity and (2) instabilities and convergence failures of the numerical solution algorithms. These weaknesses are shown to emanate from the pathogenies of Navier’s beam theory in the presence of softening section constitutive relations. These pathogenies include (1) loss of solution uniqueness; and (2) loss of spatial continuity of the strain field, under a continuous force field. The proposed gradient inelastic beam theory is derived by enriching Navier’s beam theory with nonlocal section deformation variables through gradient-based equations. For the proposed theory, a FB formulation is developed. This FB formulation (1) results in a single set of algebraic equations, which, in its discretized form, can be solved by a Newton-Raphson iterative algorithm, as opposed to other formulations that use complicated solution algorithms employing nested iterative loops; (2) does not impose any restrictions on the type/form of the selected stress-strain constitutive laws, which are solely expressed in terms of local variables; and (3) is general, such that Navier’s (local) beam theory is a special case of it. The proposed FB formulation is shown to alleviate strain localization and loss of objectivity, and reduce instabilities and convergence failures of the numerical solution algorithm. However, solution uniqueness is not achieved.