Under significant seismic loading conditions, the response of concentrically braced frames largely depends on the behaviour of the diagonal braces, which represent the key energy dissipating zones. Although the hysteretic response of steel braces under cyclic axial loading has been examined in previous studies, there is a need for further assessments that focus on quantifying failure. This paper describes the development of detailed finite-element models of hollow sections subjected to cyclic axial loading. The effects of initial imperfections and cyclic hardening are taken into consideration and the models are validated against data from 19 tests. A method to predict the fracture life of bracing members under cyclic loading is also described. Using the numerical models, parametric studies are undertaken to assess the influence of global and local slendernesses on the performance of the braces – both are found to affect the occurrence and severity of local buckling under cyclic loading, which causes high localised strains at the corner areas of sections, leading to fatigue fracture. A predictive equation addressing the coexisting influence of global slenderness and local slenderness on displacement ductility is presented. The observations in the current study are compared with conclusions from other experimental programmes, and the discrepancy between the findings is discussed.