## 4 Comparative Evaluation of Collapse Probability
### Discussion on the Collapse Probability of Prototype Isolated Buildings  
In order to evaluate the seismic performance of a building according to the performance objectives, more specific criteria are required to determine whether or not they are attainable. As a part of carrying out the probability-based evaluation of an isolated building, the target reliability should be established as a quantitative value defined as the conditional probability of the collapse. Since the target reliability depends on the risk category of a building prescribed on a current seismic code, this study sets the required conditional collapse probabilities against the MCE seismic intensity not to exceed 3% for the prototype isolated structures. Note that isolators are usually implemented for essential buildings of the risk category IV falling into important facilities of which the failure could pose a substantial hazard to the community. These target reliabilities should be compared with the calculated probability of collapse presented in Table 7.  
It could be expected in the design phase that the collapse probabilities of the prototype isolated buildings under future MCEs are lower than the required maximum conditional collapse probability of 10% for typical buildings. Values of \(P_{Col,MCE}\) in Table 7 demonstrates that all prototype buildings designed according to the current seismic design procedure well satisfy this expectation. However, the table also shows that the maximum \(P_{Col,MCE}\) value of 7.71% is found at the 10L\({}_{\text{E}}\)40C building, while the minimum value of 0.15% is calculated at the 3S\({}_{\text{E}}\)30B building. Such a huge difference in \(P_{Col,MCE}\) should be minimized to achieve similar reliabilities for every isolated building, regardless of its design conditions. The prototype isolated buildings designed for high seismicity regions, the S\({}_{\text{E}}\) buildings, possess very small values of \(P_{Col,MCE}\) less than 1%, which is still smaller than the acceptable conditional collapse probability of 3% against the MCE earthquake intensity. The fundamental periods, \(T_{M}\), of the S\({}_{\text{E}}\) buildings are considerably larger than 2.5 s, and their collapse mode is dominated by the concentrated displacements on isolators with the undamaged state on their superstructures. For buildings with larger values of \(T_{M}\), the inelastic action of their superstructures is eliminated, and the isolators suffer most of the displacement demands. This simple mechanism causes the decrease in uncertainty propagations to the seismic responses of the S\({}_{\text{E}}\) buildings and, consequently, leads to lower MCE collapse probabilities that those of the L\({}_{\text{E}}\) buildings with similar ACMR and \(\beta_{TOT}\) values. Of the L\({}_{\text{E}}\) buildings designed with smaller seismic loads, the L\({}_{\text{E}}\)40C buildings built at strong windy regions present unacceptably high MCE collapse probabilities that are larger than 3%. The 10L\({}_{\text{E}}\)30B building also develops the higher MCE collapse probability. Excessive stiffening behavior of their isolators in most of the buildings causes ratios of \(T_{\text{I}}/T_{M}\) larger than 0.35, which results in the loss of decoupling effects and the increase in seismic energy dissipation by the superstructure. This more complicated mechanism leading to the increase in uncertainty propagations increases the MCE collapse probabilities of the L\({}_{\text{E}}\) buildings. As mentioned earlier, even if the ACMR values of the L\({}_{\text{E}}\) buildings are quantitatively similar to those of the corresponding S\({}_{\text{E}}\) buildings, the MCE collapse probabilities dependent upon the governing collapse mechanism are quite different. In a probabilistic concept, the most influential parameter including the unacceptable collapse probability of the \(\mathrm{L_{E}}\) buildings is presumed as the total system uncertainty, \(\beta_{TOT}\), which mainly depends on the dominant collapse modes developed by the structural systems.