Buckling of softening columns in a continuum damage mechanics perspective – Local versus non-local formulation

Abstract

This article deals with the mathematical modelling of the buckling and post-buckling response of structural systems composed of quasi-brittle materials such as concrete (as found for instance in reinforced concrete columns). More specifically, columns with softening characteristics are of concern in this study. The moment-curvature constitutive law is based on continuum damage mechanics arguments (CDM theory). Then, the instability problem can be referred to as the Elastica problem for the case with a continuum damage mechanics constitutive law, or, for short, it may be referred to as the Continuum Damage Mechanica problem. It is numerically shown for the parameters of interest, that results from geometrically exact and second-order analyses for small rotations are almost equivalent. The instability of the imperfect softening system is associated with a limit load which decreases with the imperfection considered. Such columns are shown to be typically imperfection sensitive. Furthermore, a load-imperfection relationship is derived (similar to Koiter's power law), that can be useful in structural design contexts. An asymptotic expansion is performed to obtain a closed-form analytical solution of the limit load imperfection rule. A comparison with exact numerical values for the continuous column shows good agreement with the asymptotic expression results. The need for including non-locality formulations in a damage localization zone is finally discussed for post-buckling analysis in the presence of curvature softening. The adopted local bending-curvature constitutive law leads to the unloading Wood's paradox. To cover the propagation phenomenon of localization, some non-locality is included in a generalized formulation of the principle of virtual work. However, non-locality is not necessary for limit load calculations since limit loads occur for curvatures in the local hardening regime.