Minimum reinforcement condition in fibre-reinforced and hybrid-reinforced concrete beams

The problem of the minimum reinforcement condition in fibre-reinforced and hybrid-reinforced concrete elements subjected to bending is addressed in the framework of fracture mechanics by means of the Updated Bridged Crack Model (UBCM). Within the model assumptions, the scale-dependent minimum reinforcement is described by two dimensionless numbers: the bar-reinforcement brittleness number, Np, which is proportional to the steel-bar percentage, p, and the fibre-reinforcement brittleness number, Np,f, which is proportional to the fibre volume fraction, Vf. The critical values of Np and Np,f are found to be connected by a linear relationship in a design nomograph, which is proposed as a robust tool to quantify the amount of fibres required to a partial (or total) replacement of ordinay steel-bars. The effectiveness of the approach is discussed on the basis of several experimental results that can be found in the scientific literature.