## 3 Results and Discussion
### Scale Effect
#### 3.3.2 Travertine  
As with the limestone, the composition of travertine is calcium carbonate and, despite the fact that the structure is not the same (mainly due to the distribution of pores and, secondly, to the mineralogical composition being calcite and aragonite), the correlations presented in Table 1 were applied. The results obtained are plotted in Figure 9, in which it can be seen that the main trend of the curve was an ascendent model again, but according to the scale effect of the point load index (Figure 3d), the model should be descending. In accordance with the above, Figure 9 shows the area of the results of two investigations in which only travertine was used, those of [PERSON] et al. [60] and [PERSON] et al. [61]. In them, \(\sigma_{c}\) intervals of 30-60 and 20-50 MPa were recorded, respectively, for a \(d\) of 54 mm in both cases. Despite the fact that the model must be descending, according to the experimental results of \(\sigma_{c}\) from the point load test, added to the results of [PERSON] et al. [60] and [PERSON] et al. [61], these results support the theory of the descending model and not the ascending one. It has to be taken into account that limestone appeared in all the correlations but not for travertine. In this sense, the curve of the scale effect of travertine (Figure 3d) began with a \(d_{e}\) of 46.7 mm, so it could not be ruled out with certainty that for lower values the model was ascending and it turned into a mixed (ascending-and-descending) model. Based on this, two curves would be estimated, but due to a lack of data and the dispersion of the existing ones, it was decided that it was too complex (due to uncertainty) and the ascending model would be estimated directly. To try to shed more light on the trend of the curve, Figure 10 shows the results of \(\sigma_{c}\) estimated from the correlations with \(R_{L}\) and \(R_{N}\) in Table 5. The plotted points (colored squares) did not provide a clear trend (ascending or descending) if the experimental \(\sigma_{c(31)}\) (red triangle) was taken into account as the reference value (91.72 MPa from Table 4). However, if it is supposed that the model must be descending, Equations (20) and (24)--proposed by [PERSON] and [PERSON] [10] and [PERSON] and [PERSON] [25], respectively--would provide reliability to the estimated curve of the scale effect. Figure 8: Scale effect curve from the estimated \(\sigma_{c}\) of the limestone. In Figure 11, all the pairs of points from Figures 9 and 10 were plotted, discarding those that are outside of the downward trend. The discarded values corresponded to the  
Figure 10: Estimation of \(\sigma_{c}\) from the correlations with \(R_{N}\) and \(R_{L}\) in Table 2 for traverting. correlations for the highest values of \(d\), and are in the range of 54-60 mm. Therefore, the estimates of Equations (1), (2), (4), (6), (11), (19) and (21) were used, together with the \(\sigma_{c(31)}\) value (91.72 MPa). After discarding values as necessary, a potential least squares adjustment was carried out, which yielded values of \(k_{1}\) and \(\sigma_{c(50)}\) of +0.858 and 71.70 MPa, respectively. Finally, the scale effect curve, represented in Figure 12, is given by the following equation:  
\[\frac{\sigma_{c}}{\sigma_{c(50)}}=\left(\frac{50}{d}\right)^{0.8579}, \tag{39}\]  
where the value of \(k_{1}\) (0.8579) differs from the 0.18 proposed by [PERSON] and [PERSON] [33; 34], and from the value of \(k_{2}\) of +1.160 obtained in the point load test. As discussed in the case of limestone, [PERSON] [16] and [PERSON] et al. [26] do not indicate that the two coefficients should be equal, but it is assumed that the two coefficients should be similar. To support this assertion, it could be thought that the experimental \(k_{1}\) value would have more weight due to its relatively low dispersion (\(R^{2}\) = 0.6198), but in the absence of more data the experimental interval was between 0.8579 and 1.160. Figure 11: Least squares adjustment to estimate the \(\sigma_{c(50)}\) of the travertine scale effect.