## 5 Comparisons between Numerical Analyses and Theoretical Formulations
### Strengthened Walls
The comparisons between the theoretical and numerical predictions are shown again in Figure 6: the theoretical values are calculated by using the two procedures, A and B, for the assessment of the effective stresses in the FRCM reinforcement, while the numerical ones are the same considered in Figure 5, i.e., calculated according to the three different approaches represented by Equations (7), (10) and (13). It is worthy of note that, in all cases, the theoretical predictions of shear strength clearly reduce respect to the values plotted in Figure 4. In particular, for approach 3, a densification of the points along the bisector of the quadrant can be observed for all the types of reinforcement, emphasizing a not excessive gap between the theoretical predictions and the FEM simulations. Moreover, the cases of the walls strengthened with steel cords, which resulted strongly overestimated in Figure 4, are more reliably predicted by the theoretical formulations, thanks to the significant reduction of strains (see Figure 5c). Conversely, for approaches 1 and 2, the use of effective strains leads all theoretical predictions to become lower than the numerical ones, evidencing an excessive underestimation, especially for approach 2. It is worthy of note that, based on Figure 5, higher values of the theoretical shear strength are obtained when procedure B was used. This result means that procedure A, i.e., the one suggested by the Italian guidelines for calculating the contribute of the FRCM reinforcement, neglecting the contribute of mortar matrix, is reasonably on the safe side. The statistical parameters MAPE, MSE, R\({}^{2}\), and \(\beta\) were calculated again for the strengthened walls, for all the approaches and for both \"A\" and \"B\" procedures, and are listed in Tables 4 and 5. Figure 6: Comparison between theoretical shear strength and FEM predictions for strengthened walls: (a) Approach 1; (b) Approach 2; (c) Approach 3. As observed for the unreinforced walls, similar considerations can also be made for the strengthened ones: when approach 1 is used (Figure 5(a)), the theoretical provisions for the procedure \"A\" are sensibly lower than FEM, i.e., \(\beta_{\rm A}=0.50\), while procedure \"B\" appears to be slightly better, since \(\beta\) is 0.62. For approach 2 (Figure 5(b)), similar considerations to approach 1 can be made: all points are in safe area, too far below the bisector for the procedure \"A\" (\(\beta_{\rm A}=0.40\)), while, for procedure \"B\", the average difference is slightly reduced (\(\beta_{\rm B}=0.50\)). Finally, for approach 3 (Figure 5(c)) and procedure \"A\", the theoretical predictions are lower than the numerical ones, in most cases, with low differences, i.e., \(\beta_{\rm A}=0.75\), while, for procedure \"B\", fewer points are in safe area and, consequently, the differences are further reduced, i.e., \(\beta_{\rm B}=0.93\). On the statistical error indicators, as observed for the unreinforced walls, approach 3 is characterized by lower values of MAPE and MSE than those of approaches 1 and 2, which are also safe, but too much, as previously discussed. Furthermore, looking at the slope of the best fitting lines, approach 3 seems to be again more reliable than the others, since, despite the fact that the two fitting lines are in the safe area like as for approaches 1 and 2, they are closest to the bisector line, i.e., \(\beta_{\rm A}=0.75\) and \(\beta_{\rm B}=0.93\); thus, they have to be preferred over the other approaches. Finally, all the previous considerations lead to considering approach 3 as the most suitable one both for the un-strengthened and strengthened walls.