# Cracking Behavior of Fiber-Reinforced Concrete Beams Made of Waste Sand
## 5 Test Results and Their Analysis
The graph in Figure 15 demonstrates that the method for calculating crack spacing included in EC2 [52] yields values that are, on average, 39% higher than the measured values. The method proposed by [PERSON] [41] results in a slight overestimation, averaging 20% higher than the measured values. The proposal outlined in [53] is closer to the measured crack spacings, differing by an average of 19%. In contrast, the method included in EC2 [54] provides values that are, on average, 100% higher than the measured ones. However, it is important to note that the crack spacing determined according to [54] represents the maximum spacing, rather than the average spacing, as is the case with the other methods [41; 52; 53]. The process of crack formation in elements with dispersed reinforcement involves changes in microstructure. Initially, so-called microcracks are formed, which later connect, resulting in the development of larger cracks. The function of the fibers is to reduce stress concentration in the concrete and to transfer the load across the cracks. According to [80], the phenomenon of microcracking should not be considered as cracking in any case. Therefore, in this article, the minimum crack width is defined as 0.01 mm. The observed phenomenon of cracking was noted along the entire length of the beam (Figures 10-14). However, the analysis focused on the section between the applied forces (of pure bending), where only cracks perpendicular to the longitudinal axis of the element formed. Figure 16 illustrates the ratios of the maximum crack width at a given phase to the maximum width (\(\mathrm{w}_{\mathrm{k}}{}^{\mathrm{EXP}}\)) observed in the last tenth load phase (X) within the section of constant bending moment. The increase in the maximum crack width due to changes in loading is different for beams with extreme tensile reinforcement ratios. (\(\mathrm{\rho}_{\mathrm{L}}\)). The higher the reinforcement ratio, the more nonlinear the relationship presented in Figure 16 becomes. This is because, as the tensile reinforcement ratio (\(\mathrm{\rho}_{\mathrm{L}}\)) increases, the number of cracks also increases, and consequently, their maximum crack width decreases. This phenomenon can be disrupted by the simultaneously increasing diameter of the tensile reinforcement, which leads to a reduction in the number of cracks [83]. Another significant parameter in the analysis of cracking is the ratio of the maximum crack width to the average value. This relationship was defined by [PERSON] [79] for ordinary concrete beams with a probability of 95% and a coefficient of variation of 0.4. Figure 16: Change in maximum crack width as a function of moment. In the studied beams, the variability index of crack widths measured for all load phases ranged from 0.3 to 0.5 (except for beam F-1, for which a value of 0.6 was obtained). Figure 17 shows the ratios of the maximum crack widths to the average widths for the different load phases of fiber-reinforced sand concrete beams, and the value of the coefficient \(\beta\) adopted in [79] for ordinary concrete is indicated. The relationship between the average (\(\mathrm{w_{k}}\)) and maximum crack widths (\(\mathrm{w_{k}}^{\mathrm{max}}\)) along the section of constant bending moment was examined. A correlation coefficient of 0.95 was obtained, indicating a strong linear correlation. Subsequently, using the least squares method, a relationship between the average and maximum crack opening widths was determined, resulting in a linear equation with a slope coefficient of 1.55 and a y-intercept of 0.005. This form suggests that, at zero average crack spacing, there is a maximum crack equal to 0.005 mm. Therefore, it was checked whether this value is statistically significantly different from zero. For this purpose, an appropriate significance test based on Student's \(t\)-statistic [75] was used. It showed that the slope coefficient of the line is not significantly different from zero at a 10% significance level.