# Revisiting Cracking in Reinforced Concrete Beams: An Updated Analysis
## 2 Initial Considerations  
### Crack Initiation  
The main codes of practice are unanimous in establishing that the strength of concrete in tension is important in cracking considerations, i.e., when ultimate stress of concrete in tension is reached, cracking starts. Furthermore, they all evaluate crack width as a function of steel strain. For example, about the bending of RC beams, ACI [2] advocates that flexural and flexure-shear cracking is initiated by flexural cracking. When flexural cracking occurs, the shear stresses in the concrete above the crack are increased. The flexure-shear crack develops when the combined shear and tensile stress exceeds the tensile strength of the concrete. Web-shear cracking begins from an interior point in a member when the principal tensile stresses exceed the tensile strength of the concrete. Thus, it is common to make the beginning of cracking correspond to a certain load (cracking load \(\mathrm{P_{cr}}\)) which determines the boundary between the known State I and State II of the behavior of RC structures. State I corresponds to the non-cracked phase, in which the behavior of the beam is essentially ensured by the concrete. Associated with \(\mathrm{P_{cr}}\) is the respective cracking moment \(M_{cr}\). Considering a rectangular cross section of a RC beam, of width \(b\), height \(h\), whose tensile strength is \(f_{ct}\), the cracking moment \(M_{cr}\) is determined by [24; 25; 26]:  
\[M_{cr}=f_{ct}\frac{b.h^{2}}{6} \tag{1}\]  
For the value of \(f_{ct}\): (i) although not explicitly indicating it, the value \(1.25f_{ctm}\) is assumed by ACI [2]; (ii) the EC2 Handbook [24] indicates the mean value of axial tensile strength of concrete \(f_{ctm}\); (iii) the CEB manual [25] indicates the characteristic 5% value of axial tensile strength of concrete \(f_{c\mathrm{f}\pm 0.05}=0.7f_{ctm}\); and (iv) the Structural Concrete [26] indicates the mean flexural tensile strength of concrete \(f_{ctm,fl}\geq f_{ctm}\). These are substantially different values. For the purposes of this work, the mean flexural tensile strength of reinforced concrete members \(f_{ctm,fl}\) was adopted as the value for \(f_{ct}\), defined by EC2 [1]:  
\[f_{ctm,fl}=max\{(1.6-h/1000).f_{ctm};f_{ctm}\} \tag{2}\]  
where \(h\) is the total member depth (in mm). Alternatively, consider the stress-strain diagram of concrete tensile behavior shown in Figure 1. This diagram is qualitatively equivalent to that shown in Volume 1 of MC2010 [27] in a schematic representation of the stress-strain and stress-crack width relation for uniaxial tension (according to fib Bulletin 42). Here, it is important to define the beginning of cracking. Does cracking start at point A? Or at point B? Or at some intermediate point of the line AB? Considering what is defined in Equation (1), cracking starts at point B. However, if many other contexts are considered (micro-cracks for example), cracking should start at point A. The MC2010 [27] indicates that, at tensile stresses of about 90% of the tensile strength \(f_{ct}\) (point A), micro-cracking starts to reduce the stiffness in a small failure zone. For what follows, this is not important. It is necessary to focus the attention on the strains. Regarding the strains, MC2010 [27] indicates that the stiffness of the line OA is defined by modulus of elasticity \(\mathrm{E_{ci}}\) (at concrete age of 28 days). Therefore, by establishing the k1 coefficient and the \(\mathrm{f_{ct}}\) value, the \(\mathrm{\varepsilon_{ct1}}\) strain is defined. Then, it is important to define the strain value at the point of maximum tensile strength \(\mathrm{\varepsilon_{ctu}}\). MC2010 [27] defines \(\mathrm{\varepsilon_{ctu}}=0.15\%\). [PERSON] [8] proposes a value in the range {0.1-0.12%}. In this context, note that [PERSON] [6], in 1973, indicated values 2 to 3 times higher.