## 3 Numerical Modelling and Analysis
### Nonlinear Static Analysis  
Static Pushover Analysis has been widely implemented in determining the nonlinear response of structures and thus their vulnerability [29]. Its application has been recommended in various codes, including [30; 31; 32], among others. It has also been considered an effective tool for evaluating both existing and new buildings. Its correct use gives the users proper information about the expected performance of the building and its components. Mass proportional and force-controlled load patterns are usually recommended in various codes to obtain the capacity curve. The structure's pushover seismic performance was evaluated in the -X direction and Y direction. Figures 7 and 8 depict the matching pushover curves. It was observed that the structure's response in the Y direction was larger than that of the X direction. The markers in the capacity curve indicate instances when significant actions start to take place on the structure. These range from the first crush of the confined concrete to the absolute maximum capacity stage of the curve where the total collapse of the reinforcement materials starts. The bilinearized capacity curve is shown in Figure 9; the corresponding performance points are also indicated on the curve showing the different limit states. For our study, we decided to use capacity curve values from the X direction. Nonlinear static procedures, such as those outlined in ATC 40 [32], FEMA 356 [14; 24; 33; 34] among other  
Figure 8: Pushover curve in the Y-direction. Figure 7: Pushover curve in the X direction. codes, are used to evaluate the seismic performance of buildings by incorporating the nonlinear force-deformation characteristics of individual components and elements due to inelastic material response. These procedures use a pushover curve (Figures 7 and 8), which is a curve of base shear versus top displacement, obtained by subjecting the building model to monotonically increasing lateral forces or increasing displacements until the building collapses [35]. The maximum displacements that are likely to be experienced during a given earthquake are determined using either highly damped or inelastic response spectra. The advantage of these procedures over linear procedures is that they directly consider the effects of nonlinear material response, resulting in calculated internal forces and deformations that are more accurate approximations of those expected during an earthquake [33]. S\({}_{\text{a}}\) represents the spectral acceleration at point S\({}_{\text{d}}\). S\({}_{\text{d}}\) denotes spectral displacement. V signifies the base shear in a pushover analysis at roof displacement, known as \(\Delta_{\text{roof}}\). \(\Delta_{\text{roof}}\) represents the displacement in a pushover curve. There are two fraction terms, PF\({}_{1}\) and \(\alpha_{1}\). \(\alpha_{1}\) symbolizes the modal mass coefficient during a pushover analysis. PF\({}_{1}\), on the other hand, describes the first natural mode's modal participation factor. Performance Point and Capacity Spectrum  
To position the pushover results attained via the N2 method adopted in Seismostruct, the results are plotted against the Acceleration-Displacement Response Spectra (ADRS) format (_S\({}_{a}\)_vs. _S\({}_{d}\)_) as outlined in [36]. This requires the application of various equations to effect the required transformation. \[PF_{1}=\left[\frac{\sum_{i=j}^{N}(w_{i}\varpi_{ij})/g}{\sum_{i=1}^{N}(w_{i} \varpi_{i1}^{2})/g}\right], \tag{2}\]  
\[\alpha_{1}=\frac{\left[(w_{i}\varpi_{ij})/g\right]^{2}}{\left[\sum_{i=j}^{N}w_ {i}/g\right]\left[\sum_{i=1}^{N}(w_{i}\varpi_{i1}^{2})/g\right]}, \tag{3}\]  
\[S_{d}=\frac{V/W}{\alpha_{1}}, \tag{4}\]  
\[S_{a}=\frac{\Delta_{\text{roof}}}{PF_{1}\varpi_{\text{roof},1}}, \tag{5}\]  
\(\varpi_{\text{i}}\) and \(\varpi_{\text{roof}}\) represent the shape of the mode under analysis during a pushover at location i and at the roof, respectively. W\({}_{\text{i}}\) is the weight acting at location i. W is the total weight of the structure.