# Risk and Resilience Assessment of Lisbon's School Buildings Based on Seismic Scenarios
## 4 Seismic Risk and Resilience Assessment Methodology
Therefore, Table 4 presents the minimum objectives, in terms of performance levels, associated with four seismic intensity levels. As can be seen, the minimum performance levels for essential buildings are more demanding than the performance levels applicable to ordinary buildings (e.g., residential buildings). In order to facilitate the visualization of the results, the seismic performance of the school buildings was represented in the form of a performance matrix, as shown in Figure 9, that graphically represents the seismic performance achieved by each structure for four levels of seismic intensity. This representation allows for verification of whether the structural seismic performance meets the minimum requirements for this type of structure (essential performance objective). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Intensity Level of** & **Return Period of** & **Probability of** & **Minimum Seismic** & **Minimum Seismic** \\
**Seismic Action** & **Seismic Action** & **Exceedance** & **Performance for** & **Performance for** \\  & & & **Ordinary Buildings** & **Essential Buildings** \\ \hline Recurrent & 43 years & 50\% in 30 years & Completely operational & - \\ Frequent & 72 years & 50\% in 50 years & Operational & Completely operational \\ Rare & 475 years & 10\% in 50 years & Life safety & Operational \\ Very rare & 975 years & 5\% in 50 years & Near collapse & Life safety \\ Maximum considered & 2475 years & 2\% in 50 years & Imminent collapse & Near collapse \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Definition of performance objectives for current and essential buildings (adapted from [26]). Figure 8: Structural response hazard curve—graphical representation of the procedure for calculating the return periods associated with the exceedance of the structural response limit states EL1 and EL2. The methodology employed herein also yields estimates of losses and recovery times, which are fundamental to assess the risk and resilience of the school network, under different seismic scenarios. Losses are computed based on the return periods that lead to the exceedance of the defined limit states. The expected losses, per year (AEL--Annual Expected Loss), and over 50 years (TEL--50 years life Time Expected Loss) are thus computed by:  
\[\text{AEL (\%)}=\sum\text{P}_{\text{Eli}}\times\text{DF}_{\text{ELi}} \tag{1}\]  
\[\text{TEL (\%)}=\sum(1-(1-\text{P}_{\text{ELi}})^{50})\times\text{DF}_{\text{ELi}} \tag{2}\]  
where \(\text{P}_{\text{ELi}}\) corresponds to the probability that the structure equals or exceeds the \(\text{ELi}\) limit state and \(\text{DF}_{\text{ELi}}\) corresponds to the damage factor (estimated loss ratio as a function of built-up area) associated with the \(\text{ELi}\) limit state. The probability \(\text{P}_{\text{ELi}}\) corresponds to the inverse of the return period associated with limit state \(\text{ELi}\). It should be noted that the computation of the expected loss over a 50-year period assumes that: (i) over this time period, the current state of schools is, at least, maintained by rehabilitation/recovery interventions, guaranteeing that no aggravation on seismic vulnerability occurs; and (ii) the probability of the occurrence of a seismic event is uniform over that time period. The sum of the expected losses, in terms of lost areas, over a 50-year time period in the various CML schools, allows for the computation of a global risk indicator. This corresponds to the expected loss index, computed as:  
\[\text{I}_{\text{v}}(\%)=\Sigma(\text{TEL}_{\text{k}}\cdot\text{A}_{\text{k}})/ \Sigma\text{A}_{\text{k}} \tag{3}\]  
where k varies between 1 and the number of schools and A\({}_{\text{i}}\) is the area of each school. Following a similar approach, the recovery time, \(\text{RT}_{\text{Eli}}\), expressed through the number of interdiction days due to the occurrence of earthquakes, is also estimated.