## 3 Numerical Applications
### Definition of Empirical \"Curvature Variation-Maximum Inter-Story Drift\" Relationslips  
In order to quantify structural damage, a correlation between maximum inter-story drift and maximum curvature difference has been defined. The peak ground acceleration (PGA) values of the accelerograms used for the numerical analyses have been scaled with a scaling factor value between 0.25 and 1.50. The following figures show the different correlations with the related R-squared coefficients: Figure 14 shows the correlation for the 3-story building subjected to three selected natural accelerograms (A1, A5, A6); Figures 15 and 16 show the results for the 5-story regular in plan building without and with infill panels, subjected to three selected natural accelerograms (A1, A5, A6). Figure 17 shows the correlation for the 8-story building subjected to three selected natural accelerograms (A1, A5, A6). \begin{table}
\begin{tabular}{c c} \hline \hline
**Performance Level** & **Inter-Story Drift (\%)** \\ \hline Fully Operational (FO) & 0.2 \\ Operational (DC) & 0.5 \\ Life Safety (LS) & 1.5 \\ Near Collapse (NC) & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance level definition as a function of inter-story drift [50]. Figure 13: Structural damage localization for the numerical models. Results of the 5-story irregular in plan structure: (**a**) curvature difference between the selected time-instants B and A; (**b**) inter-story drift and damage threshold equal to 0.5% (for non-structural components) valid for reinforced concrete framed structures. Figure 16: Structural damage quantification: empirical correlation between the maximum inter-story drift and the maximum curvature difference for the 5-story building with infill panels. Figure 14: Structural damage quantification: empirical correlation between the maximum inter-story drift and the maximum curvature difference for the 3-story building. Figure 15: Structural damage quantification: empirical correlation between the maximum inter-story drift and the maximum curvature difference for the 5-story building. The expected performance of the structure can be associated with the inter-story drift that allows defining, through different performance levels, the expected damage to both the non-structural and structural elements [50]. The following figures show the threshold related to the (LS) performance level (Table 2), where damage is moderate, the structure remains stable, and life safety is generally protected. The following equations show the relationships for the different analyzed structures:  
\[\text{3-story building}\;\;d_{r}(c)=0.180e^{0.68x}\quad 0\leq c\leq 4\% \tag{5}\] \[\text{5-story building}\;\;d_{r}(c)=0.163e^{0.89x}\quad 0\leq c \leq 4\%\] (6) \[\text{5-story infilled building}\;d_{r}(c)=0.064e^{0.69x}\quad 0 \leq c\leq 4\%\] (7) \[\text{8-story building}\;\;d_{r}(c)=0.079e^{4.64x}\quad 0\leq c \leq 2\% \tag{8}\]  
where \(d_{r}(c)\) is the maximum inter-story drift and \(c\) is the maximum curvature difference. The above results show that the relationship between maximum inter-story drift and maximum curvature difference can be defined through the following correlation:  
\[d_{r}(x)=A\cdot e^{B\cdot x} \tag{9}\]  
where \(A\) and \(B\) are parameters depending on the monitored structure. The mode curvature variation is strongly related to the maximum inter-story drift. It can be noted as a quasi-linear behavior between 0% and 0.5% of the maximum inter-story drift. After this range of values, corresponding to a structural linear behavior, it is possible to note a slight nonlinear behavior, with the growth of nonlinear effects occurring within the damaged structure. It is important to highlight the limited dispersion of data. This kind of trend has been found for all of the analyzed numerical models. In conclusion, further studies are necessary to better understand how it could be possible to generalize the empirical relationship mentioned above, with the aim to reduce uncertainties on structural damage localization and quantification.