## 2 Materials and Methods
### Stress-Strain Model for the Uniaxial Compression Behavior of SFRC for Non-Linear Analysis  
[PERSON] et al. [42] developed a model, in a technological format, for the non-linear calculation of the compression behavior of the structural elements of SFRC. This model consists of two stress-strain curves: The first departs from the origin of the coordinates and reaches the maximum stress value and is analogous to the curve defined for plain concrete in Eurocode 2 [66] (Equation (6)). The second curve starts with maximum stress and manages to intercept the \(x\)-axis. This segment is defined by means of an inverted parabola of the vertical axis (Equation (7)), which has been calculated in such a way that the subsequent post-peak energy consumption is equal to the value of that same energy calculated in conformity with the data collected in the database. Figure 2 shows the stress-strain relationship, in reference to the SFRC and presented in a non-dimensional manner. The expression for the first curve (Stretch 1 in Figure 2) is:  
\[\sigma^{*}\,=\,\frac{\alpha\,e^{*}\,-\,e^{*2}}{1\,+\,(\alpha\,-\,2)\,e^{*}} \tag{6}\]  
where:  
* \(\sigma^{*}\): non-dimensional stress (\(\sigma^{*}=\frac{\sigma}{f_{cf}}\));
* \(f_{cf}\): compressive strength of the SFRC (cylinders \(150\times 300\) mm\({}^{2}\), diameter \(\times\) height);
* \(\alpha\): non-dimensional coefficient (\(\alpha=1.05\,e_{cf}\,\frac{E_{f}}{f_{cf}}\));
* \(\epsilon_{cf}\): critical strain corresponding to \(f_{cf}\);
* \(E_{f}\): elastic modulus of SFRC;
* \(\epsilon^{*}\): strain relative to the critical strain corresponding to \(f_{cf}\) (\(\epsilon^{*}=\frac{\epsilon}{\epsilon_{cf}}\)). Note that the light blue area below curve 1 in Figure 2 is equal to \(W_{1f}/f_{cf}\epsilon_{cf}\) (\(=W_{1}^{*}\)), which is the non-dimensional volumetric deformation work in the pre-peak branch relative to \(f_{cf}\epsilon_{cf}\). The second curve (Stretch 2 in Figure 2) can be expressed as:  
\[\sigma^{*}\,=\,1\,-\,\frac{1}{4}\,(1\,-\,\sigma^{*}_{R})\,(\epsilon^{*}\,-\,1)^{2} \tag{7}\]  
This softening branch of the model (Equation (7)) is determined by a parameter, \(\sigma^{*}_{R^{\prime}}\), which is the residual compressive strength associated with \(e^{*}=3\) (Figure 2). It is defined so that the softening parabola consumes the same energy as the actual \(\sigma\)-\(\epsilon\) curve between the peak and \(3\,\epsilon_{cf}\). Therefore, the expression to obtain \(\sigma^{*}_{R}\) from an experimental \(\sigma\)-\(\epsilon\) record is:  
\[\sigma^{*}_{R}\,=\,\frac{3\,W_{2f}}{2f_{cf}\epsilon_{cf}}-2 \tag{8}\]  
where \(W_{2f}\) is the area below the \(\sigma\)-\(\epsilon\) curve between \(\epsilon_{cf}\) and \(3\,\epsilon_{cf}\), that is the deformation work per unit volume consumed by the specimen from the peak of the curve to a strain three times larger than that of the peak. \(W_{2f}/f_{cf}\epsilon_{cf}\) (\(=W_{2}^{*}\)) is represented as the dark blue area in Figure 2. The reason for choosing \(\sigma_{R}\) as a reference is due to the fact that in all the tests included in the database that present diagrams of the \(\sigma\)-\(\epsilon\) curves, at least this value is achieved; in other words, these specimens deformed and consumed energy up to \(3\,\epsilon_{cf}\) at least. The softening curve in Equation (7) is on the safe side since the majority of the curves collected are capable of consuming more energy than that proposed in the model. In addition, \(\sigma^{*}_{R}\) is always less than the unit in such a way that the model does not take into consideration the possibility of having hardening due to compression deformation. In the case of seeking to consider it, the parabola could be replaced by a straight horizontal line (\(\sigma^{*}_{R}=1\)), limiting the ultimate strain value, \(\epsilon_{u}\).