## 3 Results and Discussion
### Responses of the Compression Behavior of the SFRC: Non-Dimensional Analysis  
Presented below are the results uncovered when applying the RSM in a non-dimensional manner. Each one of the responses and factors was divided by a parameter that had the same physical dimension: the responses were divided by the value of the same variable corresponding to the concrete matrix without fibers from which it proceeded (for example, the \(f_{c}^{\circ}\) response resulted from the division of \(f_{cf}\) by \(f_{c}\)). Some factors with dimensions of length were divided by an auxiliary parameter, \(\ell_{0}\), which was the average value of the fiber lengths included in the database, which was roughly equal to 30 mm. The analysis of the \(f_{c}\) factor was eliminated from the analysis since it was actually present in the non-dimensional stresses. Besides, we sought to only study the influence of those factors that were \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \(\mathbf{\beta_{0}}\) & \(f_{c}\) **MPa** & \(d_{m}\) **mm** & \(\ell_{f}\) **mm** & \(d_{f}\) **mm** & \(\lambda\) & \(\phi_{f}\) & \(\frac{\ell_{c}}{\omega}\) & \(R_{a}^{2}\) **(\%)** \\ \hline \(f_{cf}\) (MPa) & \(-18.4\) & \(\mathbf{1.0573}\) & \(\mathbf{0.999}\) & \(-0.107\) & \(-11.6\) & \(0.020\) & \(\mathbf{194.5}\) & \(\mathbf{3.99}\) & \(90.4\) \\ \(f_{c}\) (Ca) & \(16.5\) & \(\mathbf{2.2500}\) & \(\mathbf{-1.574}\) & \(-0.380\) & \(\mathbf{67.2}\) & \(\mathbf{0.408}\) & \(\mathbf{-474}\) & \(\mathbf{-6.09}\) & \(45.9\) \\ \(\epsilon_{cf}\) & \(3894\times 10^{-6}\) & \(-4.1\times 10^{-6}\) & \(-15\times 10^{-6}\) & \(\mathbf{41.7\times 10^{-6}}\) & \(-322\times 10^{-5}\) & \(-14\times 10^{-6}\) & \(-1689\times 10^{-5}\) & \(-129\times 10^{-6}\) & \(22.4\) \\ \(W_{1f}\) (GJ/m\({}^{3}\)) & \(0.533\) & \(\mathbf{0.00181}\) & \(\mathbf{-0.00823}\) & \(\mathbf{0.01323}\) & \(-0.829\) & \(\mathbf{-587\times 10^{-5}}\) & \(\mathbf{2.468}\) & \(\mathbf{-0.02547}\) & \(72.1\) \\ \(W_{2f}\) (GJ/m\({}^{3}\)) & \(3.085\) & \(-636\times 10^{-6}\) & \(\mathbf{-0.0935}\) & \(\mathbf{0.065}\) & \(-2.59\) & \(-0.0158\) & \(\mathbf{17.63}\) & \(-0.563\) & \(70.2\) \\ \hline \end{tabular}
\end{table} 
Table 2: \(\beta_{i}\) adjustment coefficients of the \(x_{i}\) factors in the linear models with dimensional responses (bold = statistically significant). Figure 5: Residues of the compressive strength (\(f_{cf}\), **left**) and the volumetric deformation work in the post-peak branch (\(W_{2f}\), **right**). related to the materials (coarse aggregate and fiber) and not with the strength of the matrix (in the previous section, the statistical significance of \(f_{c}\) was verified in each one of the responses except in \(W_{2f}\)). The objective that was pursued with this analysis was two-fold: on the one hand, to observe how each one of the factors affected the increase or the decrease of the responses and, on the other hand, to identify those factors that had a statistical significance that may be used in the proposed model, which describes the compression strength of the SFRC [42]. The responses were adjusted by means of the linear model. Table 3 presents the linear coefficients of the \(\beta_{i}\) adjustment of the \(x_{i}\) factors for each one of the responses (all are expressed in a non-dimensional form). From this analysis, it follows that \(E_{c}^{\circ}\) did not have a significant relationship with any factor. This result was in line with that demonstrated by the scientific literature regarding the disparity of conclusions concerning the effect that the addition of fiber has on the variability of the elastic modulus. Once again, as a consequence of the number of factors that were introduced in the analysis, the VIFs obtained were higher in some terms (only \(\phi_{f}\) maintained a VIF \(\simeq 5\), as occurred in variables with dimensions). This fact may conceal factors that are significant and that do not appear as such and vice versa.