# Cracking Behavior of Fiber-Reinforced Concrete Beams Made of Waste Sand
## 5 Test Results and Their Analysis
Thus, the value of the coefficient \(\beta\) can be practically accepted as equal to 1.55, bearing in mind that it pertains to fiber-reinforced sand concrete beams under loads ranging from the cracking moment to approximately 0.8 of the destructive moment. The analyzed coefficient was verified by comparing it with the research results contained in Project No. BE 97-4163 (Brite Euram BRPR-CT98-0813-2002) [84]. It examined fiber-reinforced concrete beams with \(\rho_{\mathrm{L}}\) = 0.25; 0.5; 1.0%, using RC 65/60 BN fibers at amounts of 25 and 50 kg/m\({}^{3}\) and RC 80/60 BP fibers at 60 kg/m\({}^{3}\). The considerations were limited to a maximum crack opening width of 0.3 mm, as this is the limit value provided in EC2 [52]. To determine whether the proposed value of the coefficient \(\beta\) is valid for the derived research results, a significance test based on Student's \(t\)-statistic [75] was used. It showed that at a 10% significance level, it can be accepted that the proposed value \(\beta\) = 1.55 is also appropriate for fiber-reinforced concrete beams. Figure 18 presents a comparison of the average and maximum crack opening widths obtained from the research according to Project [84] and from own studies. The graph marks the value of the coefficient \(\beta\) according to [52] and, as well as according to the author's proposal, for beams with steel fibers. Figure 17: Change in maximum crack width in the individual loading phases as a function of moment. Most of the proposals for calculating the average crack width of perpendicular cracks, at the level of the tensile reinforcement axis, are based on the condition of equality of elongations of steel and concrete over the average segment between cracks. In the phase of stabilized cracking, in sections located halfway between cracks, no mutual displacement of steel relative to concrete is observed. The difference in elongations of both materials over a segment equal to half the distance between cracks is equal in magnitude to half the average crack width. Neglecting the small elongation of concrete relative to the elongation of reinforcement and assuming the average elongation of steel over the analyzed segment (between cracks), the values of crack widths can be calculated, among other methods, as the product of the average distance between cracks and the average strain of the tensile reinforcement, or as the product of the maximum spacing and the difference in strains in steel and concrete. Another important factor affecting the crack width is therefore the value of the average stresses in the reinforcing steel. The calculated method for determining it according to EC 2 [52; 54] assumes a triangular stress diagram in the compressed zone. Figure 19 presents a comparison of the average stresses in the tensile reinforcement determined based on strain measurements with stresses calculated for successive loading levels (from the cracking moment to approximately 0.8 of the ultimate moment). Figure 19: Comparison of calculated and experimental average stress in tensile reinforcement. Figure 18: Comparison of average and maximum crack width [52; 84]. It was found that the stresses in the tensile reinforcement determined based on the tests were only slightly dependent on the reinforcement of the compressed zone of the cross-section and on the type of steel fibers used (comparison of beams A, D, and K). In the next analysis, it was checked whether the relationship between the examined stresses is linear. The correlation coefficient was determined to be 0.95, indicating a strong linear relationship. Using the least squares method, the equation of the line was determined, and it was checked whether it statistically differs from the optimal relationship (y = x) between the analyzed stresses. For this purpose, a significance test based on Student's \(t\)-statistic was applied. At the accepted significance level of 10%, it was shown that the hypothesis was correct, meaning that the difference between the analyzed stresses is statistically insignificant.