# Risk and Resilience Assessment of Lisbon's School Buildings Based on Seismic Scenarios
## 4 Seismic Risk and Resilience Assessment Methodology
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Limit State** & **Description** & **Structural Response Indicator** \\ \hline \begin{tabular}{c} Completely \\ operational \\ (EL1) \\ \end{tabular} & \begin{tabular}{c} Until this point, continuous service (school operates \\ without any functionality loss) after earthquake is \\ expected, with negligible structural and \\ non-structural damage. \\ \end{tabular} & \begin{tabular}{c} Accounts for 70\% of the spectral displacement \\ associated with the operational limit state (EL2). \\ \end{tabular} \\ \hline \begin{tabular}{c} Operational \\ (EL2) \\ \end{tabular} & \begin{tabular}{c} Structure is safe for occupancy and most operations \\ can resume immediately after earthquake. Repair is \\ required to restore some nonessential services. \\ Damage is light. \\ \end{tabular} & \begin{tabular}{c} Spectral displacement associated with the elastic \\ limit of the capacity curve. \\ \end{tabular} \\ \hline \begin{tabular}{c} Life safety \\ (EL3) \\ \end{tabular} & \begin{tabular}{c} Life safety is generally achieved. Structure is \\ damaged to a moderate level but remains stable. \\ Some building systems or contents may be protected \\ from damage. Extensive repair operations are \\ necessary to rehabilitate the structure and restore \\ full functionality. In some cases, rehabilitation may \\ not be economically viable. \\ \end{tabular} & \begin{tabular}{c} Spectral displacement associated with 3/4 of the \\ spectral displacement value associated with the near \\ collapse limit state (EL4). \\ Note: in the case of shear failures in columns, this \\ limit state corresponds to the point at which such \\ brittle failure occurs. \\ \end{tabular} \\ \hline \begin{tabular}{c} Near collapse(EL4) \\ \end{tabular} & \
\begin{tabular}{c} Although structural collapse is prevented, \\ non-structural elements may fail. Structural damage \\ is severe. Repair operations, if viable, are costly and \\ generally long (depending on the allocated \\ resources). \\ \end{tabular} &
\begin{tabular}{c} Displacement associated with the point at which the base shear force (or spectral acceleration) decreases by 20\% relatively to the maximum base shear force (or the maximum spectral acceleration). \\ Note: In the case of shear failures in columns, this \\ limit state is associated with a spectral displacement equal to 4/3 of the spectral displacement associated \\ with the life safety (EL3) limit state. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definition of seismic response limit states. Figure 7: Illustrative capacity curve in the acceleration–displacement (Sa–Sd) format and identification of damage and performance levels, as well as response limit states. hazard curve in terms of structural displacement, represented in Figure 8. This curve relates the spectral displacement of the structure with the seismic intensity, defined herein through its return period. The structural response hazard curve is then used to determine the intensity of the seismic action, i.e., the return period, leading to the exceedance of the limit states (EL) represented in Figure 7 and described in Table 3. Figure 8 illustrates graphically the computation of the return periods T\({}_{\text{EL1}}\) and T\({}_{\text{EL2}}\) associated with the exceedance of EL1 and EL2, respectively. The return periods of the seismic action associated with the exceedance of the defined structural response limit states are then compared against well-known thresholds, such as the ones proposed in VISION 2000 [26]. Schools are considered in this study as part of the third (\(\gamma_{\text{III}}\)) class of importance, according to NP EN 1998-1:2010, which corresponds to an essential performance objective. This classification takes into account the type of occupation of the schools and their importance in the post-earthquake response, namely their use in the allocation of key post-event services.