## 3 Results and Discussion  
### Responses of the Compression Behavior of the SFRC: Analysis with Physical Magnitudes  
From the information of the database, the responses that describe the compressive strength of the SFRC, which are \(f_{cf}\), \(E_{f}\), \(\epsilon_{cf}\), \(W_{1f}\), and \(W_{2f}\), have been studied. Each one of them has been analyzed by means of the following factors: the compressive strength of the base concrete or matrix without fibers, \(f_{c}\), the maximum size of the aggregate, \(d_{m}\), the parameters that define the geometry of the fiber \(\ell_{f}\), \(d_{f}\), and \(\lambda\), the fiber volume ratio, \(\phi_{f}\), and a parameter that takes into account the relationship between the sizes of fiber and the aggregate, \(\ell_{f}/d_{m}\). Figure 3 shows a scheme of the process followed to apply the RSM to the results from the database. All the responses were initially adjusted to two models of complete first and second degree polynomic regression. However, it was verified that the second degree models did not correctly predict the responses in the definition domain of the levels of the factors, for which reason they were disregarded for their use in the analysis. Furthermore, these models had excessively high VIF values in the majority of their terms, which indicated the presence of a strong multicollinearity. As stated above, this phenomenon caused the loss of precision in the estimation of the coefficients of the factors, as well as alterations in the detection of their statistical significance. It bears emphasizing that the fiber factor, \(\phi_{f}\lambda\), was not found to be significant in the two degree polynomial models, neither in the quadratic nor in the combined adjustments, as shown in the Pareto diagrams in Figure 4. The corresponding VIFs were 3.85 and 2.85, respectively. Thus, it can be assumed that the level of multicollinearity for \(\phi_{f}\lambda\) in both cases was low, and consequently, the low significances given by ANOVA were reliable. Therefore, the analysis of the responses has been performed exclusively with first degree models (Equation (9)). After verifying that the linear models correctly predicted the responses in the domain of the definition of factors, the study of the VIF revealed that they were greater than 10 in the majority of factors, except for \(f_{c}\) and \(\phi_{f}\), for which VIF \(\simeq 1\). This fact demonstrated that multicollinearity still existed, for which reason it would be appropriate to make a selection and reduce the number of factors with which the responses are analyzed. The factors that were statistically significant were determined by making use of the Pareto diagrams. Table 2 contains coefficients of the \(\beta_{i}\) adjustment of the \(x_{i}\) factors in the linear models for each one of the responses, as well as the terms that were statistically significant. Each response can be calculated as:  
\[y=\beta_{0}+\sum_{i=1}^{n}\beta_{i}\,x_{i}+\xi \tag{9}\]  
When making an adjustment among the variables that have different units, as is the case, some of the coefficients of the \(\beta_{i}\) factors also require units to conserve the dimensional stability. Thus, the \(\beta_{0}\) coefficients will have the corresponding response units (for example, for the response \(f_{cf}\), \(\beta_{0}\) is expressed in MPa, and for the \(\epsilon_{cf}\) response, \(\beta_{0}\) does not have any units). For the remaining \(\beta_{i}\) coefficients of the factors, the definition of their units is as follows: the numerator corresponds to the response unit, and the denominator corresponds to the unit of the factor to which it is associated (for example, for the \(E_{f}\) response, the \(\beta_{2}\) coefficient, associated with the \(d_{m}\) factor, has GPa/mm as  
Figure 4: Pareto diagrams for the quadratic (**left**) and combined (**right**) adjustments of the residual compressive strength. Figure 3: Diagram of the steps followed in the RSM calculation process. a unit).