# Cracking Behavior of Fiber-Reinforced Concrete Beams Made of Waste Sand
## 5 Test Results and Their Analysis  
The \(t\)-Student distribution [75] and [PERSON]'s test were used to analyze the test results. [PERSON]'s test was employed to detect the presence of any gross errors in the data set. For each sample, it was checked whether the extreme values belonged to the population of results under consideration. Assuming a normal distribution for each tested property, the mean value, standard deviation, and coefficient of variation were determined using the \(t\)-Student test [75]. The \(t\)-test is the most commonly used method for assessing differences between the means of two groups. When samples from a given series were collected at different times due to technological constraints, the \(t\)-test was applied to determine whether the results could be combined into a single population. The \(t\)-test can also be used in the case of very small samples to determine statistical parameters, provided that the distribution of variables is assumed to be normal. A significance level of \(\alpha\) = 0.05 and a tolerance of \(\upsilon\) = 10% were assumed in the analyses. It was found that all the analyzed samples had a sufficient number of results to be considered representative. The results of the strength feature tests are presented in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Concrete**} & \multirow{2}{*}{**Series**} & \multicolumn{4}{c}{**Mechanical Properties**} \\ \cline{3-6}  & & \(\mathbf{f_{ctsp}}\) & \(\mathbf{f_{ccube}}\) & \(\mathbf{f_{c}}\) & \(\mathbf{E_{c}}\) \\ \cline{3-6}  & & **(MPa)** & **(MPa)** & **(MPa)** & **(MPa)** & **(GPa)** \\ \hline \multirow{6}{*}{with loose steel fibers 50/0.8} & A & 3.29 & 36.5 & 45.5 & 33.6 \\ \cline{2-6}  & B & 3.57 & 48.3 & 50.2 & 34.2 \\ \cline{2-6}  & C & 3.44 & 49.3 & 47.2 & 41.1 \\ \cline{2-6}  & D & 3.51 & 44.8 & 45.6 & 33.2 \\ \cline{2-6}  & G & 3.62 & 50.2 & 52.7 & 53.7 \\ \hline Plain & F & 3.82 & 59.7 & 51.9 & 35.9 \\ \hline \multirow{6}{*}{with glued steel fibers 30/0.55} & H & 2.74 & 36.2 & 34.8 & 32.1 \\ \cline{2-6}  & I & 2.69 & 35.9 & 38.1 & 28.3 \\ \cline{1-1} \cline{2-6}  & J & 2.89 & 37.6 & 38.4 & 25.9 \\ \cline{1-1} \cline{2-6}  & K & 2.94 & 38.4 & 37.5 & 30.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results from properties tests. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Series**} & \multirow{2}{*}{**Longitudinal Reinforcement Ratio (\(\rho_{\text{L}}\))**} & \multicolumn{6}{c}{**Load Phases/Reactions (kN)**} \\ \cline{3-10}  & & **II** & **III** & **IV** & **V** & **VI** & **VII** & **VIII** & **IX** & **X** \\ \hline A & 0.6 & 2.0 & 3.0 & 4.0 & 5.0 & 6.0 & 7.0 & 8.0 & 9.0 & 10.0 \\ \hline B & 0.9 & 3.0 & 4.5 & 6.0 & 7.5 & 9.0 & 10.5 & 12.0 & 13.5 & 15.0 \\ \hline C & 1.3 & 3.0 & 4.5 & 7.0 & 9.5 & 12.0 & 14.5 & 17.0 & 19.5 & 22. The mechanical properties of the ribbed steel used for the longitudinal reinforcement of the beams were also tested. The yield strength, tensile strength, and modulus of elasticity of the reinforcing bars were determined. The results are presented in Table 5. The tests indicated that the ribbed steel used in this study exhibited a distinct yield point. The primary parameter in reinforced concrete theory that describes the transition of a bent element from the first to the second phase of loading is the cracking moment. The results of tests and analyses concerning the cracking moment are presented in [50; 51]. Figure 8 provides a graphical comparison of the experimentally determined cracking moments (M\({}_{\rm{exp}}\)) with those calculated (M\({}_{\rm{cr}}\)) in according with [52; 54; 76; 77]. The studies published in [50; 51] demonstrated that, for fiber-sand concrete beams with varying longitudinal reinforcement ratios (\(\rho_{\rm{L}}\)) of 0.6%, 0.9%, and 1.3%, only slight differences in cracking moments were observed. This suggests that the magnitude of the cracking load is primarily determined by the tensile strength of the fiber-reinforced fine aggregate concrete matrix.