## 3 Case-Study Structure  
This section discusses the geometrical and mechanical characteristics of the case study structure. This study considers a masonry wall with geometric characteristics similar to wall D tested in Pavia (IT) [28; 29], belonging to a two-story UnReinforced Masonry (URM) building. The wall dimensions are 600 \(\times\) 643 \(\times\) 25 cm (length, height, and width, respectively) with four openings, two on the first floor with dimensions equal to 94 cm \(\times\) 214 cm (length, height) and two on the second floor with dimensions equal to 94 cm \(\times\) 124 cm (length, height), perfectly aligned with each other, as shown in Figure 3a. Material properties are chosen to match a typical Italian historical masonry building [30]. They are characterized by solid fired-clay bricks with dimensions 25 \(\times\) 12 \(\times\) 5.5 cm\({}^{3}\) (length, thickness, and height), having a mean cubic compressive strength equal to 15 MPa and hydraulic lime mortar with a thickness of 10 mm with a compressive strength of 3.2 MPa. The brick courses are alternated to give discontinuity to the vertical mortar joints, as shown in Figure 3b. The homogenized compressive strength of masonry is assumed to be equal to 3.2 MPa. For the micro-model, the mechanical parameters are divided between those inherent to the brick and those inherent to the mortar, as shown in Tables 1 and 2. The mortar compression strength is assumed equal to the compressive strength of the masonry, while the other parameters are assumed in agreement with the relevant scientific literature. Specifically, these would be the tensile strength of the mortar and the energy parameters of fracture of mortar and brick, which vary for calibration purposes. Figure 3: Wall geometry: (**a**) element dimensions (m), numbering of masonry piers, and (**b**) layout of the numerical model. Equal degrees of freedom in the x direction are located at each floor of the structure to prevent relative displacement between nodes in this direction. This assumption is only an approximation to represent the 2D model; it should be highlighted that the roof in the test was not a rigid diaphragm. The analyses are developed by varying the Vertical Load (VL) representing the dead and live loads of the slabs. Three different VLs are applied to each story, i.e., 10 kN/m, 30 kN/m, and 64 kN/m. The application of the vertical loads involves an increase in the stress at the base of about 5%, 10%, and 20% respectively. The VLs were named as follows: VL-a, VL-b, and VL-c. Figure  
The failure criteria provided in the Italian design code [31; 32] are used in this study. Thus, the in-plane flexural capacity \(M_{u}\) and the shear capacity \(V_{t}\) of the pier are calculated as indicated in Equations (3) and (4). \[M_{u}=\frac{\sigma_{0}l^{2}t}{2}\left(1-\frac{\sigma_{0}}{0.85f_{d}}\right) \tag{3}\]  
where \(M_{u}\) is the in-plane bending moment, \(f_{d}\) design compressive strength of masonry, 0.85 is the stress distribution coefficient, \(\sigma_{0}=N/(l\,t)\) the average compression stress, \(N\) is the vertical action on the pier, and \(l\), \(t\) are the width and thickness of the pier, respectively. \[V_{t}=\frac{1.5\tau_{0}dt}{\beta}\sqrt{1+\frac{\sigma_{0}}{1.5\tau_{0}d}} \tag{4}\]  
where \(V_{t}\) is the shear capacity of the section according to the Turnsek and Cacovic [33] criterion, \(\beta\) is a coefficient taking into account the slenderness of the pier, and \(\tau_{0}d\) is the design shear strength. On the horizontal axis (Figure 4), vertical stress \(\sigma_{\rm c}\) is normalized to the peak compressive strength \(f_{cp}\) (or \(\sigma_{cp}\)) of the masonry, while, on the vertical axis, there is the shear capacity \(V_{b}\) corresponding to the activation of the flexural and shear mechanisms.