Upper and lower bounds of crack width of R.C. Tie - Simplified method

A simplified model that takes account of the effects of the so-called Goto cracks, or internal secondary cracks, is here set up to study the mechanical behaviour of a reinforced concrete tie subjected to monotonic loading, at the crack formation stage and the stabilized cracking stage. This approach is based on the assumption that bond stresses are linearly distributed along the transmission length. Bond stresses and slips are correlated through the fib Model Code 2010 bond law only at the section where bond stresses reach the maximum value. In the other sections, bond stresses and slips are uncorrelated from fib Model Code 2010 bond law, but in compliance with the second-order differential equation of cracking behaviour. In particular, at the stabilized cracking stage, under the assumption of maximum crack spacing, it is shown that the relative mean strain formula proposed in fib Model Code 2010 and Eurocode 2 can be improved by introducing an η-coefficient, depending only on the acting axial forceto- cracking force ratio, which allows the influence of the internal secondary cracks on the tensionstiffening effect to be pointed out. Moreover, at the stabilized cracking stage, the simplified model also allows the minimum crack width to be determined. Owing to the practical difficulty of predicting the expected value of the crack spacing or the crack width, a comparison has here been made between the upper and lower bounds of the crack width and the experimental data available in the literature.