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PREFACE:

Experimental observations and numerical simulations are compared with theoretical results based on 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 o f the minimum percentage o f 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 o f the beam depth, the relationship established by the brittleness number calls fo r decreasing values with increasing beam depths.

Experimental observations and numerical simulations are compared with theoretical results based on 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 o f the minimum percentage o f 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 o f the beam depth, the relationship established by the brittleness number calls fo r decreasing values with increasing beam depths.

subjected to remarkable size effects. An extensive experimental research was proposed by ESIS Technical Committee 9 on Concrete in order to obtain a rational and unified explanation fo r the transitions usually observed between the above mentioned collapse mechanisms. The influence o f size on the inelastic rotational capacity has not been completely clarified (and demonstrated) yet. In fact the experimental data available up to a fe w years ago, mostly obtained by load-controlled tests on reinforced concrete beams with high ductility bars, show a considerable scatter. 

On the other hand, some numerical evaluations, assuming strain localization in the compression zone, indicate tha t plastic rotation depends on the scale (i.e. beam depth) and the experimental tests recently carried out seem to validate this dependence [6 , 7] . The attention is focused onto the transition between failure mechanisms (from reinforcement failure to concrete crushing) and onto the minimum percentage o f reinforcement that depends on the scale [8 ] and enables the element in flexure to prevent brittle failure.

 First o f all, a presentation o f the theoretical model based on LEFM is reported, in which the reinforcement reactions are applied directly to the crack surfaces and the compatibility condition is locally imposed to the crack opening displacement in correspondence with the reinforcement. 

Such a theoretical approach appears to be very useful fo r estimating the minimum amount o f reinforcement for members in flexure, assuming simultaneous concrete cracking and steel yielding (transitional condition). 

In the second part o f the chapter, the experimental results o f three point bending tests performed on 45 reinforced concrete beams at the Department o f Structural Engineering o f the Politecnico o f Torino, are presented. From these results, a confirmation o f the empirical formula fo r the critical value o f the brittleness number Np can be evidenced. In addition, a numerical simulation o f the tests was performed.