Golden ratio in the crack pattern of reinforced concrete structures

Among all the proportions, the golden ratio has been taken into consideration for its geometrical and morphological properties, which can be found in huge number of natural patterns and, therefore, has been always considered as a model of beauty. Nevertheless, as discussed for the first time in the present paper, also the cracking phenomenon of quasi-brittle materials calls the golden ratio into play. In particular, such an irrational number appears when the average crack spacing of reinforced concrete ties and beams is evaluated at different scales. This conjecture is corroborated by results of a tension-stiffening model capable of predicting both the crack width and the crack spacing measured by tests. In other words, it can be argued that the centrality of golden ratio in the crack pattern of concrete members has profound physical meanings, and reveals the existence of a size-effect law of crack spacing. The practical interest of this law lies on the possibility of predicting the crack pattern of large structures without knowing material performances, but testing prototypes of lower dimensions.