# Accounting for Resilience in the Selection of R Factors for a RC Unsymmetrical Building
## 3 Design Concept and Structural Details of the Building
### Vulnerability Assessment  
As per the HAZUS methodology [55], all of the buildings experience four damage states, i.e., slight, moderate, extreme and collapse. By creating fragility curves, it was possible to determine the vulnerability in terms of the likelihood of exceeding each damage condition in each situation. In this study, the fragility curves were plotted against cumulative probability exceedance of each damage state and spectral displacement. Using HAZUS methodology [55], the damage probability of exceedance is given by the following expression:  
\[P\left(ds\diagup S_{d}\right)=\phi\left[\frac{1}{\beta_{ds}}\ln\left(S_{d} \diagup S_{d,ds}\right)\right] \tag{1}\]  
The median (\(S_{d,ds}\)) at each damage state with respect to spectral displacement was found from the proposed equations [56]. Lognormal standard deviation (\(\beta_{ds}\)) values that describe the variability (dispersions) of fragility curves are developed for each damage state (i.e., slight, moderate, extensive and complete). The total variability of each structural damage state (\(\beta_{ds}\)) was taken from the HAZUS MR4 technical manual. The spectral displacement (\(S_{d}\)) on the \(x\)-axis and the probability of exceedance on the \(y\)-axis were used to illustrate the vulnerability in terms of the fragility curve. The probability of exceeding the stated damage level or state is shown by the \(y\)-axis, which ranges from 0% to 100%. For each example of the construction, fragility curves were created (Figures 12-15).  
Figure 12: Fragility curve for unsymmetrical building Case I (R = 3).  
From the fragility curves, the probability of the building exceeding each damage state was evaluated. The vertical line that intersects the fragility curves of various damage states was shown with regard to the spectral displacement at each design level. The likelihood percentage of exceeding for that particular damage condition is provided by the intersection point.  
Figure 14: Fragility curve for unsymmetrical building Case III (R = 5).  
Figure 13: Fragility curve for unsymmetrical building Case II (R = 4).  
Figure 15: Fragility curve for unsymmetrical building Case IV (R = 6).  
From Figures 12-15, it was observed that at the DBE design level, the building with R = 5 and 6 (Cases III and IV) shows 45 to 55% certainty of experiencing extreme damage. The building in all cases shows less than a 20% probability of experiencing a collapse damage level. This shows that the building has much less collapse probability at the DBE design level. At a higher design level (MCE), it was noted that all of the building cases (Case I to IV) have more than 90% certainty to experience moderate damage and 70% certainty to experience extreme damage states. Since the collapse probability of the building ranges from 45 to 57% at higher R factors such as R = 5 and 6, it shows that the building is more prone to the collapse damage level, which affects the recovery planning of a post-disaster event. Under bidirectional loading, at the MCE design level, the building Cases I and II have more than 75-82% certainty of experiencing an extreme damage state, whereas Cases III and IV have 88-92% certainty of experiencing an extreme damage state. The slight increase in the damage probability was observed at a higher R factor with bidirectional loading.