## 6 Evaluation of PGA Values for Specific Limit States by Pushover Analysis  
### Detection of Specific Limit States  
A static non-linear pushover method [10; 37] is used to evaluate seismic behavior and the capacity of the building for three limit state (LS) conditions that have been taken into account according to Eurocode 8, part 3 [44], as follows:  
* Near collapse NC--global capacity of the building is taken to be equal to the ultimate displacement capacity;
* Significant damage SD--global capacity of the building is taken to be equal to \(\frac{3}{4}\) of the ultimate displacement capacity;
* Damage limitation DL--the capacity for global assessment is defined as a yield point of the idealized elasto-perfectly plastic force-displacement relationship of the equivalent SDOF. The pushover analysis has been performed by applying the gradually increasing load up to the structural collapse, using a uniform and modal pattern, according to the N2 method of Eurocode 8 [37]. The analysis allows the determination of capacity curves and collapse load as well as damage monitoring, which continuously increases because of the non-linear deformation. Pushover analysis results in the MDOF capacity curve, which is transformed into the SDOF capacity curve, and a bilinear force-displacement curve is obtained. Peak ground accelerations for three mentioned limit states have been determined as follows. We specify that henceforth, the forthcoming equations have been modified with respect to the homologous ones reported in [33] to consider three damage levels. Given that the displacement in the yield point d\({}_{y}^{*}\) of SDOF is linked with the DL state, the ultimate displacement d\({}_{u}^{*}\) with the NC state and SD state with the displacement equal to \(\frac{3}{4}\) d\({}_{u}^{*}\), the corresponding ductilities are expressed as:  
\[\mu_{\rm DL}=\mu_{y}=1\ ;\ \mu_{\rm SD}=\frac{\frac{3}{4}\mathrm{d}_{u}^{*}}{ \mathrm{d}_{y}^{*}}\ ;\ \mu_{\rm NC}=\mu_{u}=\frac{\mathrm{d}_{u}^{*}}{\mathrm{d}_{y}^{*}} \tag{2}\]  
where \(\mu_{\rm DL}=\mu_{y}\), \(\mu_{\rm SD}\) and \(\mu_{\rm NC}=\mu_{u}\) represents damage limitation, significant damage, and near collapse ductility coefficients. The associated elastic spectral displacements can be calculated as follows:  
\[\mathrm{S}_{\rm de,i}(\mathrm{T}^{*})=\frac{\mathrm{d}_{y}^{*}\overline{ \mathrm{R}}_{\mu}(\mu_{i})}{\left[\overline{\mathrm{R}}_{\mu}(\mu_{i})-1 \right]\frac{\mathrm{T}_{c}}{\mathrm{T}}+1}\,\ \mathrm{i}=\mathrm{DL},\ \mathrm{SD},\ \mathrm{NC} \tag{3}\]  
where \(\overline{\mathrm{R}}_{\mu}\) is a reduction factor depending on the ductility coefficient \(\mu\) of SDOF system [35]. The spectral accelerations are given as:  
\[\mathrm{S}_{\rm ae,i}(\mathrm{T}^{*})=\frac{4\pi}{\mathrm{T}^{*2}}\mathrm{S}_ {\rm de,i}(\mathrm{T}^{*})\,\ \mathrm{i}=\mathrm{DL},\ \mathrm{SD},\ \mathrm{NC} \tag{4}\]  
The periods \(\mathrm{T}_{\rm B}\), \(\mathrm{T}_{\rm C}\), and \(\mathrm{T}_{\rm D}\) divide the elastic response spectrum [37] into four spectral acceleration branches represented with the functions \(\mathrm{f}_{\mathrm{i}}\) (\(\mathrm{i}=1,\ \ldots\,4\)). Therefore, it can be expressed as follows:  
\[\mathrm{S}_{\rm ae}(\mathrm{T})=\mathrm{PGA}\cdot\mathrm{f}_{\mathrm{i}}( \mathrm{T}) \tag{5}\]  
where \(\mathrm{PGA}=\mathrm{a}_{\mathrm{g}}\) is peak ground acceleration and \(\mathrm{T}\) is the period of the structure. Each limit state (DL, SD, and NC) is characterized with the following peak ground accelerations:  
\[\mathrm{PGA}_{\rm DL}=\frac{\mathrm{S}_{\rm ae,DL}(\mathrm{T}^{*})}{\mathrm{ f}_{\mathrm{i}}(\mathrm{T})}\ ;\ \mathrm{PGA}_{\rm SD}=\frac{\mathrm{S}_{\rm ae,SD}(\mathrm{T}^{*})}{\mathrm{f}_{ \mathrm{i}}(\mathrm{T})}\ ;\ \mathrm{PGA}_{\rm c}=\frac{\mathrm{S}_{\rm ae,NC}(\mathrm{T}^{*})}{ \mathrm{f}_{\mathrm{i}}(\mathrm{T})} \tag{6}\]  
Eighteen buildings in the settlement were modelled using 3 MURI software [45] following the equivalent frame model approach. Thus, the structural response is checked alongtwo horizontal axes, in the positive and the negative direction.