## 3 Collapse Performance of Seismically Isolated Buildings
### Collapse Capacity Evaluation of the Prototype Isolated Buildings
#### 3.2.1 Results of Incremental Dynamic Analysis
Modeling uncertainty can be determined from Monte-Carlo simulation based-evaluation which is carried out with statistical information on material properties and analytical assumption. Table 5 summarizes the statistical information on modeling uncertainties that are treated as random variables categorized into four structural properties of a superstructure and isolators (denoted as _SSP_ and _ISP_ in the table), inherent damping ratios (denoted as _DP_ in the table), and gravity loads (denoted as _GL_ in the table). This table contains bias factors and the coefficient of variation of each random variable obtained from corresponding references. As shown in Figures 8 and 9, record-to-record uncertainty is an important factor influencing the structural capacities of the prototype isolated buildings. Such record-to-record uncertainty should be eliminated in order to concentrate on the effects of random variables related to the modeling uncertainty. Using the IDA results that are carried out for the prototype isolated buildings, a representative ground motion record is selected as one where the median collapse point is measured. The representative ground motion record of each prototype building can be found in Figures 8 and 9. The four categorized modeling uncertainties are randomly selected from their probabilistic distributions. Generating representative sample sets of modeling uncertainties was comprised using a Latin hypercube sampling method. Since the error rate of Monte Carlo simulation is changed by a sample size, a required minimum sample size was first evaluated. The sampling size was determined by the convergence tests shown in Figure 11. The sampling for the convergence test was continuously performed until a ratio between the mean (or standard deviation) of a sample set and the mean (or standard deviation) of the population is less than a tolerance. Figure 11: Sampling of modeling uncertainty (example values \(f_{y}\) in SSP for 3S\({}_{\text{E}}\)40C): (**a**) Latin hypercube sampling; (**b**) Convergence test for \(f_{y}\) values. Figure 12 shows the IDA curves of the 5L\({}_{\text{E}}\)40C building subjected to the representative ground motion record. The variation of the IDA curves results from the modeling uncertainties due to variations of the four categories (_SSP_, _ISP_, _DP_, and _GL_). The figure clearly shows the dominance of the modeling uncertainty propagation on story drift variance rather than isolation displacement variance. Such behavior is particularly found in the IDA curves of the prototype buildings of which the collapse damage state is determined by the sideway of the superstructure. The variations of seismic capacities \(S_{a}(T_{M})\) induced by the modeling uncertainties are comparatively evaluated in a format of tornado diagrams, as shown in Figure 13. The horizontal axis in the figure indicates the normalized variation ratios \(\hat{\varepsilon}_{SA}\) of \(S_{a}(T_{M})\) values. The \(\hat{\varepsilon}_{SA}\) value is determined by dividing \(\pm 2\) standard deviation dispersion by the median value. Of the four categories chosen for considering the modeling uncertainties in this study, variance of the _ISP_ is the most dominant contributor on the variations of \(\hat{\varepsilon}_{SA}\). A _GL_ variance is positively correlated, while variances of the _SSP_ and _DP_ are negatively correlated with the output random variable, \(\hat{\varepsilon}_{SA}\), of the prototype isolated buildings, which can be confirmed from Figure 12. Table 6 summarizes the lognormal standard deviation parameter, \(\beta_{MDL}\) of the modeling uncertainty derived from the square-root-of-sum-of-squares (SRSS) combination of the standard deviations of the four categories. In consequence, the prototype isolated buildings located at strong seismicity regions present lower probabilistic propagations of the modeling uncertainty in comparison to those located at low-to-moderate earthquake regions.