Scale-dependent minimum and maximum reinforcement ratios in rc and pc beams

Reinforced concrete (RC) and prestressed concrete (PC) beams need to be designed in order to guarantee large plastic deformations and to avoid any loss in the load bearing capacity. In order to ensure this stable behavior, several codes of practice impose to respect minimum, ρmin, and maximum, ρmax, reinforcement percentages. Below ρmin, brittle failure due to a vertical drop in the bearing capacity can be detected, revealing a hyperstrength behaviour. On the other hand, beyond ρmax, no ductility is developed in the structural element, since the failure mechanism is mainly governed by concrete crushing rather than steel yielding. Within the design standards, ρmin and ρmax are determined on the basis of classic σ-ε constitutive laws and on the hypothesis of plane sections. This approach leads to completely disregard scale effects, which are proved to be crucial in the study of concrete structures. In this framework, Non-linear Fracture Mechanics is able to describe in a thorough way cracking and crushing instability phenomena occurring in RC and PC structural elements subjected to bending. These behaviours result to be governed by dimensionless parameters, such as the matrix brittleness number, s, and the reinforcement brittleness number, NP, which are functions of the material properties, the reinforcement characteristics, and the structural size-scale. Several numerical investigations carried out by means of the Cohesive/Overlapping/Bridged Crack Model highlight the scale effect concerning ρmin for RC with a power law proportional to h−0.15, where h is the beam depth, and the scale effect concerning ρmax for both RC and PC beams with a power law proportional to h−0.25 . In this way, a rational and comprehensive framework for the design optimization and safety assessment of reinforced or prestressed concrete structures can be provided.