A fracture mechanics approach to the design of hybrid-reinforced concrete beams

The Updated Bridged Crack Model (UBCM) is proposed as a fracture mechanics tool to thoroughly predict the structural behaviour of hybrid-reinforced concrete (HRC) beams. The cementitious matrix is assumed to be linear-elastic perfectly-brittle, whereas nonlinear constitutive laws are used to describe the toughening action of the reinforcing phases, which is due to the yielding of steel-bars and to the pull-out of the short fibres. Under these assumptions, the model predicts different HRC post-cracking structural behaviours that can be synthetically described by three scaling dimensionless numbers: the bar-reinforcement brittleness number, NP, which is directly related to the steel-bar reinforcement percentage, ρ; the fibre-reinforcement brittleness number, NP,f, which depends on the fibre volume fraction, Vf; and the pull-out brittleness number, Nw, which is a function of the critical embedded length of the fibre-reinforcement, wc. An effective relationship between the two reinforcement brittleness numbers is proved to define the minimum reinforcement conditions for HRC beams. All the other parameters being the same, it can be translated into a relationship between the steel-bar and the short-fibre reinforcement ratios, thus providing an effective and straightforward tool for the minimum reinforcement design of HRC structures.