## 4 Comparative Evaluation of Collapse Probability
### Methodology to Evaluate Collapse Probability
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Design Condition**} & \multirow{2}{*}{**Prototype**} & \multicolumn{3}{c}{**Median Collapse Spectral Acceleration**} & \multicolumn{3}{c}{**Collapse Probability**} \\ \cline{3-10}  & & \(\varepsilon_{0}\) & \(\varepsilon_{0}\) & \(c_{1}\) & \(S_{a,Cd,adj}\) & \(S_{a,MCE}\) & \(\varepsilon_{0}\) & \(\varepsilon_{0}\) & \(\varepsilon_{0}\) \\ \hline \multirow{6}{*}{Buildings at low-to-moderate seismicity region} & 3S\({}_{\text{F}}\)40C & 1.544 & \(-\)0.6246 & 0.0793 & 0.605 & 0.305 & 0.257 & 1.984 & 0.38 \\  & 5S\({}_{\text{F}}\)40C & 1.537 & \(-\)0.8953 & 0.1644 & 0.526 & 0.297 & 0.210 & 1.771 & 0.32 \\  & 10S\({}_{\text{F}}\)40C & 1.479 & \(-\)1.1852 & 0.2665 & 0.453 & 0.245 & 0.229 & 1.851 & 0.36 \\  & 3S\({}_{\text{F}}\)30B & 1.446 & \(-\)1.1723 & 0.1023 & 0.359 & 0.224 & 0.159 & 1.603 & 0.15 \\  & 5S\({}_{\text{F}}\)30B & 1.423 & \(-\)1.2973 & 0.1414 & 0.334 & 0.211 & 0.158 & 1.584 & 0.18 \\  & 10S\({}_{\text{F}}\)30B & 1.370 & \(-\)1.3105 & 0.0574 & 0.292 & 0.187 & 0.154 & 1.560 & 0.19 \\ \hline \multirow{6}{*}{Buildings at low-to-moderate seismicity region} & 3S\({}_{\text{F}}\)40C & 0.808 & \(-\)1.2594 & 0.1803 & 0.328 & 0.180 & 0.380 & 1.824 & 5.69 \\  & 5L\({}_{\text{F}}\)40C & 0.809 & \(-\)1.4851 & 0.2101 & 0.268 & 0.162 & 0.334 & 1.657 & 6.53 \\ \cline{1-1}  & 10L\({}_{\text{F}}\)40C & 0.809 & \(-\)1.5954 & 0.2075 & 0.240 & 0.158 & 0.293 & 1.518 & 7.71 \\ \cline{1-1}  & 3L\({}_{\text{F}}\)30B & 0.818 & \(-\)1.9169 & 0.2590 & 0.182 & 0.109 & 0.217 & 1.667 & 0.93 \\ \cline{1-1}  & 5L\({}_{\text{F}}\)30B & 0.817 & \(-\)1.8140 & 0.1880 & 0.190 & 0.110 & 0.255 & 1.728 & 1.60 \\ \cline{1-1}  & 10L\({}_{\text{F}}\)30B & 0.817 & \(-\)1.8721 & 0.1526 & 0.174 & 0.110 & 0.247 & 1.584 & 3.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of collapse probability evaluation for prototype isolated buildings. Figure 14: Comparison of an observed spectrum (EQ23) with uniform hazard spectrum. (5)  
Compute adjusted collapse margin ratios (ACMR) using the equation:  
\[ACMR=\frac{\hat{S}_{a,col,adj}(T_{M})}{S_{a,MCE}(T_{M})}; \tag{4}\]  
(6)  
Calculate the conditional probability of the collapse caused by MCEs, \(P_{Col,MCE}\) using the equation:  
\[P_{COL,MCE}=\int_{0}^{1}\frac{1}{s\beta_{TOT}\sqrt{2\pi}}\exp\left[\frac{( \ln s-\ln ACMR)^{2}}{2\beta_{TOT}^{2}}\right]ds. \tag{5}\]  
The equation for \(P_{Col,MCE}\) includes the probabilistic propagation of uncertainty sources to the collapse capacities of an isolated building. It is assumed that each uncertainty is independent and lognormally distributed with a median value and lognormal standard deviation. The total uncertainty in the equation, \(\beta_{TOT}\), is calculated by the SRSS combination of a design requirement-related uncertainty, \(\beta_{DR}\), a test-data related uncertainty, \(\beta_{TD}\), a modeling-related uncertainty, \(\beta_{MDL}\), and a record-to-record collapse uncertainty, \(\beta_{RTR}\). Among uncertainty sources, \(\beta_{MDL}\) and \(\beta_{RTR}\) are pre-determined from the incremental dynamic analysis results of the prototype isolated structures. For isolated buildings designed according to a current seismic code, their rigorous design requirements should be followedincluding strict prototype testing procedures, and their excellent seismic performance has been supported by explicit academic investigations against MCE ground motion and seismic renaissance reports. Hence, a value of \(\beta_{DR}\) for the design requirements for isolated buildings can be rated as superior, \(\beta_{DR}\) =1.0. Regarding the test-data related uncertainty, test data related to both superstructure components such as the RC beam, column and beam-column joint, and to isolation systems shall be considered. From a large number of previous research studies related to reinforced concrete components, various experimental verifications are carried out for structural components in RC moment-resisting frames.