## 4 Summary of FE Analyses
### Un-Strengthened Walls  
Un-strengthened masonry walls were simulated in order to check the influence of the mechanical properties of masonry on the global behavior of the walls in terms of strength and ductility. Table 1 shows the mechanical parameters of the materials varied in the un-strengthened walls analyzed in DIANA ([PERSON]'s modulus, \(E\), compressive strength\({}_{t}\),\(f_{c}\), tensile strength,\(f_{t}\), fracture energy in tension, \(G_{t}\), of the masonry substrate).  
Three types of masonry, which are identified by the compressive strength, were considered, and they correspond to typical values of a natural stone, tuff (indicated with \"T\" in the specimen's label), very common in the South Italy, and of two types of clay bricks (B1 and B2). The values of \(f_{c}\) and \(E\) adopted for the different types of tuff and brick masonry reported in Table 1 are related to the values suggested in the commentary document joined to the Italian Code (IBC, 2018 [35], IBCC, 2019 [36]). Such a document provides the following ranges for the compressive strength: 1.4-2.2 MPa for irregular tuff, 2.0-3.2 MPa for regular tuff, 2.6-4.3 MPa for clay bricks. These values are indicative since they have to be divided by the confidence factor (FC), which is 1, 1.2, or 1.35, according to the knowledge level, and can be increased by coefficients greater than 1 in case of good conditions of masonry (good mortar, presence of transversal connections, presence of well-defined horizontal planes) or presence of strengthening interventions. Thus, the values of compressive strength adopted in the numerical analyses are in agreement with the possible ranges provided by (IBCC, 2019 [35]). Based on these values, the [PERSON]'s modulus was calculated as 1000\(f_{c}\).  
The value of the fracture energy in tension, \(G_{t}=0.012\) N/mm, suggested in [PERSON], 1996 [27], for masonry with strength of about 5 MPa, was firstly considered for both types of masonry substrates. Keeping constant this value of \(G_{t}\), the tensile strength of the masonry was varied, considering 5%, 10%, and 20% of the compressive strength. These cases are indicated in the specimen labels with the letter \"W\", \"N\", and \"S\", respectively.  
Then, higher (10 times \(G_{t}\)) and lower values (0.1 times \(G_{t}\)) were adopted for carrying out parametrical analysis on the effect of \(G_{t}\), and the following theoretical relation for \(G_{t}\) was also considered:  
\[G_{t}=f_{t}^{2}\cdot h/E\ h=\sqrt{2\cdot A}, \tag{17}\]where \(h\) indicates the crack bandwidth (to account for size effect) and depends on the area, \(A\), of the finite element (\(10\) mm \(\times\) 10 mm) used in the model and on the [PERSON]'s modulus, \(E\), and tensile strength, \(f_{t}\), of masonry.  
Moreover, Table 1 reports the theoretical shear strength, \(V_{t,th}\), given by Equation (3) for the un-strengthened walls, the values of the maximum diagonal compression force, \(P\), given by the FE model, and the shear strength, \(V_{t,FEM}\), calculated according to the Equations (7), (10), (13) and (16), based on the value of \(P_{FEM}\). It is worth noting that, in Equation (3), \(b=h/l=1\) and \(\sigma_{\text{od}}=0\) were used.