## 4 Pushover Analyses  
This section describes the results of the Pushover (PO) analyses for the structure studied. The PO analyses are developed using the horizontal load with a uniform distribution proportional to the masses (positive and negative). The horizontal load is concentrated at the plane levels (at the node).  
Figure 5 shows the PO curves. The vertical axis shows the base shear \(V_{b}\), and the horizontal axis shows the Inter-story Drift Ratio (IDR) or the Roof Drift Ratio (RDR). IDR represents the ratio between the inter-story displacement and the story height, and RDR represents the ratio between the displacement on the top of the structure and its total height.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **VL-a** & & & **VL-b** & & & **VL-c** & \\ \hline
**T (s)** & \(\mathbf{M_{x}}\) **(\%)** & \(\mathbf{M_{y}}\) **(\%)** & **T (sec)** & \(\mathbf{M_{x}}\) **(\%)** & \(\mathbf{M_{y}}\) **(\%)** & **T (sec)** & \(\mathbf{M_{x}}\) **(\%)** & \(\mathbf{M_{y}}\) **(\%)** \\ \hline
0.1328 & 79.57 & 0.00 & 0.1888 & 81.75 & 0.00 & 0.2576 & 83.07 & 0.00 \\
0.0494 & 13.33 & 0.00 & 0.0700 & 14.14 & 0.00 & 0.0947 & 14.47 & 0.00 \\
0.0451 & 0.00 & 87.54 & 0.0634 & 0.00 & 89.93 & 0.0861 & 0.00 & 91.30 \\ \hline \end{tabular}
\end{table}
Table 3: Vibration modes and mass participation ratios for VL-a (at left), VL-b (at centre), and VL-c (at right).  
Figure 5 shows the IDR for the first story (IDR\({}_{\text{S1}}\), left), the IDR for the second story (IDR\({}_{\text{S2}}\), centre), and the RDR (right). Each plot shows the results obtained for VL-a, VL-b, and VL-c.  
The plots show the curves obtained for different VLs. The figure shows that the IDR\({}_{\text{S1}}\) values are higher than the corresponding IDR\({}_{\text{S2}}\) and RDR values. For this reason, the IDRS1 values are considered in the PO graphs below.  
In the range of the vertical loads considered, which are significantly lower than the compressive strength of the masonry, a monotonic increasing trend is observed for the peak value of the PO curve as the vertical load increases.  
Figure 6 reports the PO curve evaluated for VL-a at the first level of the case-study structure and the corresponding cracking pattern at 0.5% 1%, and 2% of IDR\({}_{\text{S1}}\). The first two values are representative of the shear and bending limit states suggested by the Italian technical code [31] and CNR DT 212 [34] for the frame behaviour (piers fixed to spandrels). The Italian technical code suggests a drift limit of 2% for piers that behave as cantilevers. It is used herein as an extreme upper bound of the near-collapse conditions of an unreinforced masonry (URM) wall.  
Figure 7 shows the crack patterns corresponding to VL-a (at left), VL-b (centre), and VL-c (right) for IDR\({}_{\text{S1}}\) equal to 2%. As expected, higher vertical loads correspond to a higher pier capacity, and less damage to the piers corresponds to greater damage to the spandrels.  
The PO curves are described for two reasons: the first is to have a comparison between the resistant capacity of the element (V\({}_{\text{b}}\)) obtained from the PO analysis with that obtained from the cycles of nonlinear dynamic analysis, and the second is to show how the PO curves to vary the static vertical load and then to demonstrate the reliability of the calculation models.  
Figure 5: PO curves obtained for VL-a, VL-b, and VL-c. The three plots report the IDR of the first level (**left**), the IDR of the second level (**centre**), and the RDR (**right**) of the masonry wall.  
Figure 6: Cracking pattern of the masonry wall with VL-a. From left to right, damage scenarios corresponding to (**a**) 0.5%, 1% (**b**), and 2% (**c**) of IDR\({}_{\text{S1}}\).