Failure-mode scale transitions in RC and PC beams

Current design Standards for reinforced concrete beams prescribe to respect a minimum, ρ_min, and a maximum, ρ_max, reinforcement ratio in the design of structures. Below ρ_min a brittle failure due to unstable crack propagation is expected. On the other hand, for ρ > ρ_max a brittle failure due to concrete crushing is obtained. In this framework, a reinforced concrete element with ρ_min < ρ < ρ_max presents yielded steel at Ultimate Limite State (ULS) with a stable behaviour and no catastrophic loss of bearing capacity. Design Standards define ρ_min and ρ_max limits on the basis of the Bernoulli’s hypothesis of plane sections, and completely disregard size-scale effects. Within the present paper, Dimensional Analysis is used to determine the Brittleness Numbers that govern the behaviour of reinforced concrete (RC) as well as of prestressed reinforced concrete (PC) beams. Therefore, parametric analyses carried out by means of the Cohesive/Overlapping Crack Model (COCM) are used to study the ductile-to-brittle transitions in RC and PC beams, and to highlight the size-scale dependency of the two above-mentioned reinforcement limits.