## 5 Comparisons between Numerical Analyses and Theoretical Formulations
### Un-Strengthened Walls  
For the un-strengthened walls, Figure 3 shows the comparison between \(V_{t,th}\) and the shear strength given by the FE model, \(V_{t,FEM}\), evaluated according to the different approaches described above and already listed in Table 2. In particular, approach 1 corresponds to Equation (7), approach 2 to Equation (10), and approach 3 to Equation (13). The best fitting lines passing through the origin are also plotted for each approach in Figure 3. Approach 4, which corresponds to Equation (16), was not considered in Figure 3, since it provides too low values in comparison with the other approaches. In Table 3, the synthetic statistical parameters MAPE, MSE, R\({}^{2}\), and \(\beta\), previously introduced, are listed.  
Figure 3 shows that, whatever approach is used, the FE results are usually higher than the theoretical predictions given by Equation (3). In particular, for approach 2, in few cases, the theoretical predictions are higher than the FE results, and, indeed the value of \(\beta\) is the largest among the three considered approaches (i.e., \(\beta_{2}=0.86\) vs. \(\beta_{1}=0.57\) and \(\beta_{3}=0.46\)). Conversely, the values of \(R^{2}\) are the same for the three approaches since the three equations differ only for a coefficient (0.500, 0.707, 0.880); thus, the scatter of each point, \(V_{t,th,i}-V_{t,FEM,i}\), with respect to the best fitting line of each approach, is the same.  
If the results of simulations by validated FE models, i.e., numerical experiments, are considered equivalent to experimental tests (as shown in many simulations of tests on masonry with DIANA--[PERSON] et al., 2008 [28], [PERSON] et al., 2012 [29], [PERSON] et al., 2013 [30], [PERSON] et al., 2013 [31], [PERSON] et al., 2014 [32], [PERSON] et al., 2016 [33]), the theoretical values should be equal to the FE predictions. This means that the \"underestimation\" area in the graphs of Figure 3 refers to the assumption that the theoretical predictions given by the codes without safety factors are higher than the numerical results, assumed equivalent to experimental results. In this sense, they are not unsafe from a design point of view, but by looking at the predictability of the theoretical model with average values.  
However, if it is not the case, on safe side, the theoretical values (without any design safety coefficient) should be preferably lower than the FE one. This occurs in almost all cases for the three considered approaches, but approaches 1 and 2 seem to be too safe. Thus, approach 3 seems to be the most suitable approach to estimate the shear strength of the un-strengthened wall starting from the maximum compressive force attained in a diagonal compression test. Moreover, the values of MAPE and MSE for the three approaches evidence that Equation (13) provides the lowest average percentage difference between the theoretical predictions and the numerical results. Finally, looking at the slope of the best fitting lines, approach 3 seems again to be the most reliable since its fitting line is closest to the bisector because it has \(\beta=0.86\).  
Figure 3: Theoretical shear force vs. Finite Element Model (FEM) predictions for un-strengthened walls.