Fracture mechanics approach to minimum reinforcement design of fibre-reinforced and hybrid-reinforced concrete beams

The problem of the minimum reinforcement condition in fibre-reinforced and hybrid-reinforced concrete flexural elements is addressed in the framework of fracture mechanics by means of the Updated Bridged Crack Model (UBCM). The model describes the crack propagation process occurring in the critical cross-section of the reinforced member, by assuming the composite as a multiphase material, whereby the toughening contribution of the cementitious matrix and of the reinforcements are independently evaluated. The key-point of the discussion is that, when the influence of the matrix nonlinearities on the response is neglected, the minimum reinforcement condition is defined by a linear relationship between the critical values of two dimensionless numbers: (i) the bar- reinforcement brittleness number, NP , proportional to the steel-bar area percentage, ρ; (ii) the fibre- reinforcement brittleness number, NP,f, proportional to the fibre volume fraction, Vf. The model is applied to several experimental campaigns of the literature, in order to assess its suitability in the minimum reinforcement design of reinforced members in a unified fracture mechanics-based framework.