This paper investigates the behaviour of concrete failing under high stress levels and subjected to different types of loading. The aim of this investigation is to clarify the development of linear and nonlinear creep strains and how they relate to material damage and eventual failure. This research is supported on the results of a new experimental programme performed on concrete cylinders tested in uniaxial compression under varying strain and stress rates. The results of this programme allow investigating the influence of the loading history on the material response in terms both of its strength and deformation capacity. On this basis, a failure criterion related to the inelastic strain capacity of concrete is defined. Such failure criterion, showing consistent agreement for all types of loading histories, allows calculating in a simple manner the reduction of the strength for a long-term loading situation and also its associated deformation capacity. On that basis, a comprehensive method for predicting failure of concrete under different long-term loading patterns is proposed and validated.