## 3 Results and Discussion
### Stress-Strain Model of the Compression Response for SFRC
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathbf{\beta_{0}}\) & \(\mathbf{\ell_{f}^{*}}\) & \(\lambda\) & \(\mathbf{\phi_{f}}\) & \(\mathbf{R_{a}^{2}}\) **(\%)** & **VIF** \\ \hline \(f_{c}^{\circ}\) & \(0.980\) & \(0.0151\) & \(189\times 10^{-6}\) & \(\mathbf{2.94}\) & \(5.3\) & \([1.3\)–\(5.3]\) \\ \(E_{c}^{\circ}\) & — & — & — & — & — & — \\ \(\epsilon_{c}^{\circ}\) & \(0.996\) & \(-\mathbf{0.0959}\) & \(\mathbf{185\times 10^{-5}}\) & \(\mathbf{28.23}\) & \(36.1\) & \([1.5\)–\(2.1]\) \\ \(W_{1}^{\circ}\) & \(0.929\) & \(0.143\) & \(-251\times 10^{-5}\) & \(\mathbf{40.97}\) & \(31.6\) & \([1.3\)–\(5.1]\) \\ \(W_{2}^{\circ}\) & \(0.770\) & \(-0.048\) & \(\mathbf{1444\times 10^{-5}}\) & \(\mathbf{96.90}\) & \(50.8\) & \([1.6\)–\(4.6]\) \\ \(\sigma_{R}^{2}\) & \(-0.131\) & \(-0.0717\) & \(\mathbf{609\times 10^{-5}}\) & \(\mathbf{21.72}\) & \(29.7\) & \([1.0\)–\(1.7]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: \(\beta_{i}\) adjustment coefficients of the \(x_{i}\) factors and the non-dimensional responses, calculated for the \(\sigma\)-\(\epsilon\) model for SFRC proposed by [PERSON] et al. [42] (bold = statistically significant). Figure 6: Pareto diagram and response surface of \(f_{c}^{\circ}\). Figure 7: Pareto diagram and response surface of \(\epsilon_{c}^{\circ}\). Once again, the absence of the statistically significant relationship of \(E_{\epsilon}^{\circ}\) with any factor was verified, as occurred in the prior analysis, which included a majority of factors in the model, thereby reinforcing the idea of the absence of the relationship between the variability of the elasticity modulus and the inclusion of the fiber in the concrete. The physical reason behind this fact maybe that, as the fiber content increases, the SFRC porosity also increases, which in turn leads to a decrease in the modulus of the base concrete that compensates the stiffening effect due to the presence of fibers. Insofar as \(f_{\epsilon}^{\circ}\), the SFRC compression strength, it was only significantly affected by \(\phi_{f}\), thereby demonstrating a slight increase as fiber was added. Despite the low value of \(R_{a}^{2}\) (5.3%), the statistical significance, demonstrated by \(\phi_{f}\), and the positive value of its adjustment coefficient in the model indicated the tendency of the increase of \(f_{\epsilon}^{\circ}\), in mean values. This is how this effect must be interpreted, as observed in Figure 10. The value of \(f_{\epsilon}^{\circ}\) may be easily measured through the uniaxial strength test [67] or calculated by means of Equation (10), which was obtained from a new linear correlation between \(f_{\epsilon}^{\circ}\) and \(\phi_{f}\) alone:  
\[f_{\epsilon}^{\circ}\,=\,1\,+\,3.80\,\phi_{f}\quad(R_{a}^{2}\,=\,4.5\%) \tag{10}\]  
The RSM can also provide the characteristic value, understood as the value corresponding to a cumulative probability of 5% in the distribution curves, as shown in Figure 10. The result is:  
\[f_{\mathrm{ck}}^{\circ}\,=\,0.825\,+\,3.75\,\phi_{f} \tag{11}\]  
Furthermore, \(\epsilon_{\epsilon}^{\circ}\) may be experimentally measured with the uniaxial compression test, although it also may be estimated with the coefficients of Table 5.  
Figure 8: Pareto diagram and response surface of \(W_{1}^{\circ}\). Figure 9: Pareto diagram and response surface of \(W_{2}^{\circ}\). The residual compressive strength, \(\sigma_{R}^{*}\), defined in the model of [PERSON] et al. [42] (equal to the non-dimensional compressive stress associated with a deformation \(\epsilon^{*}\) = 3) is calculated in Equation (12):  
Figure 6:0  
In the analysis performed, \(\phi_{f}\) and \(\lambda\) were statistically significant, for which reason a new analysis was executed again with the result that both of these factors were once again significant. \(\phi_{f}\) and \(\lambda\) were then related to \(\sigma_{R}^{*}\) by means of Equation (13):  
Figure 6:1  
Figure 11 represents the Pareto diagram and the response surface of the statistically significant factors of \(\sigma_{R}^{*}\).