## 5 Comparisons between Numerical Analyses and Theoretical Formulations
### Strengthened Walls  
For the strengthened walls, Figure 4 shows the comparisons between the theoretical shear strength, \(V_{t,th}\), given by Equations (1) and (3), and the numerical values, \(V_{t,FEM}\), calculated according to Equations (7), (10) and (13) based on the maximum compression force, \(P_{FEM}\), given by the FE analyses, i.e., the data listed in Table 2. Again, provisions given by Equation (16) are not plotted in the graphs since they are too low in comparison with the other ones. In general, Figure 4 shows that for the basalt and glass reinforcements the comparisons between numerical and theoretical results are always better than those related to the steel cords for whatever approach; the theoretical predictions for the steel FRCM reinforcement are, indeed, always significantly higher than the numerical ones, whatever the equation used for calculating the shear strength. Such a strong difference in terms of reliability for the three types of FRCM reinforcement makes it difficult to  
Figure 4: Theoretical shear strength vs. FEM predictions for strengthened walls: (**a**) Approach 1; (**b**) Approach 2; (**c**) Approach 3. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{**Approaches**} & \multirow{2}{*}{**Shear Force**} & \multicolumn{5}{c}{**Parameters**} \\ \cline{3-6}  & & **MAPE** & **MSE** & **R\({}^{2}\)** & \(\beta\)** \\ \hline Approach 1 & \(V_{t}=0.71\)\(P\) & 0.43 & 3876 & 0.60 & 0.57 \\ Approach 2 & \(V_{t}=0.88\)\(P\) & 0.54 & 8395 & 0.60 & 0.46 \\ Approach 3 & \(V_{t}=0.50\)\(P\) & 0.34 & 953 & 0.60 & 0.86 \\ Approach 4 & \(V_{t}=0.33\)\(P\) & 0.37 & 990 & 0.60 & 1.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical error indicators for the different approaches applied to the un-strengthened walls. evaluate which is the best fitting approach and can be related to the actual working strains of fibers, which are probably lower than the ultimate ones. Note, indeed, that the values of \(V_{t,th}\) listed in Table 2 and plotted in Figure 4 are calculated by using the experimental ultimate strain of the fibers obtained in the tensile tests, but these values could not have been attained in the diagonal compression test on the strengthened walls. Therefore, the calculation of the shear capacity of the strengthened walls was again evaluated by using strain values as close to the real numerical ones as possible. In order to correctly assess the actual strain values in the fibers when the walls fail, from the FE analysis carried out with DIANA ([PERSON] et al., 2019 [15]), the average fiber strain at the peak force for each simulated wall along its diagonal was identified. Figure 5 shows and justifies what emerged previously: the values of the strains attained in the fibers at the peak force in the strengthened walls are indicated on the experimental constitutive laws of each FRCM reinforcement. It is evident that, for the case of basalt fibers (Figure 5a), with the exception of a couple of points, the strains are higher than 50% of the ultimate values. Conversely, for the glass fibers (Figure 5b), the strains are sufficiently close to the ultimate values. For the case of steel cords (Figure 5c), on the other hand, it is worthy of note that the strains at the peak load are very low, on average, and, therefore, very far from the ultimate values. Analyzing the individual cases in detail, it clearly emerges that the range of variation of strains with reference to their ultimate value are:  
1. for the case of basalt fibers, excluding two isolated cases: 53-68%;
2. for glass fibers: 61-93%;
3. for steel cords: 12-5%. Figure 5: Evaluation of fibers strains in the strengthened walls at failure and comparisons with the constitutive laws: (**a**) basalt; (**b**) glass; (**c**) steel. These results justify that, if the ultimate strains of the constitutive laws are used, as previously done, despite the reduction related to the use of \(\alpha_{\rm t}=0.8\), the contributions to the shear strengths of the strengthened walls are clearly overestimated.