Ductile-to-brittle transitions in RC and PC beams: Scale effects on minimum and maximum reinforcements

In the framework of Non-linear Fracture Mechanics, the Cohesive/Overlapping Crack Model is able to take into account in a comprehensive way the damage processes developing in both the tension and the compression zones of reinforced concrete (RC) or pre-stressed concrete (PC) beam sections. Constitutive laws, which are able to simulate energy dissipation over a cross-sectional domain, allow this model to capture the size effects on the ductility characterising the global sectional behaviour. During the loading process, the Cohesive/Overlapping Crack Model can capture local mechanical instabilities such as snap-back or snap-through occurring after concrete cracking, as well as, for over-reinforced or pre-stressed concrete elements, the catastrophic drops in the bearing capacity due to the unstable growth of damage in the compression zone of the beam cross-section. In the present work, by means of the Cohesive/Overlapping Crack Model, an effective estimation of the minimum reinforcement percentage for RC beams is performed, emphasizing its h-0.15 power-law scale dependence, where h is the beam depth. In addition, numerical studies are carried out in order to confirm how the extension of the plastic plateau developing after steel yielding is inversely proportional to beam depth, h, and reinforcement ratio. This evidence provides the scale-dependent maximum reinforcement percentage with a h-0.25 power law, beyond which the structure is unable to develop any ductility due to concrete crushing occurring prior to steel yielding. For PC beams, the application of the Cohesive/Overlapping Crack Model demonstrates how these structures should likely face concrete crushing rather than steel yielding, suggesting that a proper scale-dependent maximum reinforcement percentage should be considered as a strict requirement.