Download Minimum Reinforcement in Concrete Members by Alberto Carpinteri easily in PDF format for free.
The central topic discussed in the committee is that of the minimum reinforcement in concrete members. The minimum amount of reinforcement is defined as that for which “peak load at first concrete cracking” and “ultimate load after steel yielding” are equal. In this way, any brittle behaviour is avoided as well as any localized failure, if the member is not over-reinforced. In other words, there is a reinforcement percentage range, depending on the size-scale, within which the plastic limit analysis may be applied with its static and kinematic theorems.
If we assume, as is usual in the codes of practice, a reinforcement amount proportional to the beam height h, the plastic bending moment, Mp, is proportional to the square of the beam height: Mp « }i’. On the other hand, if the reinforcement is assumed to react rigidly up to the crack propagation and to flow plastically only afterwards, the bending moment of crack propagation, MQ , is proportional to the beam height rised to 3/2: MQ « /Z . As a matter of fact, if steel-cover thickness and initial crack depth are assumed to be proportional to the beam height, e.g. both equal to 0.1 h, the stress-intensity factor K^ is proportional to the nominal stress and to the square root of beam height : K^ « aV/? ~ (MJh^ ^h. Therefore, at the critical condition, when Xj = K^Q we have MQ « K^Q h .
Since Mp « /z^ is a quantity of higher rank with respect to MQ « h^^^, for smaller sizes we have Mp < MQ and, for larger sizes, Mp > MQ . This means that, with the usual criteria, small beams tend to be under-reinforced, as well as large beams tend to be over-reinforced. Carpinteri, Ferro, Bosco and El-Katieb propose a LEFM model, according to which reinforcement reactions are applied directly on the crack surfaces and a compatibility condition is locally imposed on the crack opening displacement in correspondence with the reinforcement.
The theoretical model is found to provide a satisfactory estimate of the minimum percentage of reinforcement that depends on the scale and enables the element in flexure to prevent brittle failure. While the minimum steel percentage provided by Eurocode 2 and ACI are independent of the beam depth, the relationship established by the brittleness number N^ calls for decreasing values with increasing beam depths. Lange-Kornbak and Karihaloo compare experimental observations with approximate nonlinear fracture mechanics predictions of the ultimate capacity of three-point bend, singly-reinforced concrete beams without shear reinforcement.
called effective slip-length model, where the concrete fracture is described as a cohesive crack and the effect of reinforcement bond-slip is incorporated. Although the beams considered are of reduced size, the properties of the microconcrete were selected so that the behaviour observed is representative of beams of ordinary size made of ordinary concrete.