## 4 Comparative Evaluation of Collapse Probability  
### Methodology to Evaluate Collapse Probability  
Several researchers indicated that the spectral shape of ground motion imposed to a structure, in addition to its intensity, is important to evaluate seismic structural response [38; 39]. In particular, for a given seismic hazard level, the uniform hazard spectrum can be quite different from the shapes of the expected response spectra of real ground motion having a fluctuated spectral amplitude at a single period of interest. These spectral shape characteristics are especially important for the seismic collapse evaluation of a structure in that these differences can be significantly amplified at high amplitudes [40]. A simple procedure for accounting for the spectral shape effects is described on FEMA P695 [17] for non-isolated concrete and/or wood frames with relatively short structural periods. It is questionable to apply this procedure to seismically isolated buildings of which the collapse mechanisms can be developed at the isolation systems rather than the superstructures and, in turn, the fundamental periods are large. In this study, the collapse evaluations of the prototype isolated structures are carried out according to [PERSON] et al. [40] that can directly consider the spectral shape effects in a structure with the relatively large structural period. As shown in Figure 14, the acceleration spectra of an individual ground motion have much different shapes from the uniform hazard spectrum. In this study, the acceleration spectrum used in designing the prototype structures is assumed as the uniform hazard spectrum. The spectral shape effects are represented by a parameter of \(\varepsilon\) that graphically means the difference between an individual ground motion spectrum and a uniform hazard spectrum excluding the variation of earthquake waveforms. Larger \(\varepsilon\) values of a typical individual spectrum usually exist approximately less than 2.0 s. Unlike typical structural systems with relatively short fundamental periods, values of \(\varepsilon\) for isolated buildings would be smaller. A value, \(\varepsilon_{\text{f}}(T_{M})\) at \(T_{M}\) of the \(j\)th ground motion record (\(j\) ranges from 1 to 40) is a coefficient of variance of a random variable of spectral acceleration at a period of \(T_{M}\) and is obtained from:  
\[\varepsilon_{j}(T_{M})=\frac{\ln\Bigl{[}S_{aj}(T_{M})\Bigr{]}-\mu_{\ln\text{ Sat}}(T_{M})}{\sigma_{\ln\text{ Sat}}(T_{M})} \tag{1}\]  
where \(S_{aj}(T_{M})\) is a spectral acceleration at \(T_{M}\) of the \(j\)th ground motion record, and \(\mu_{\ln\text{ Sat}}(T_{M})\) and \(\sigma_{\ln\text{ Sat}}(T_{M})\) are, respectively, the mean and standard deviation of spectral accelerations at \(T_{M}\) of the ensemble of ground motion records. Values of \(\sigma_{\ln\text{ Sat}}(T_{M})\) for the ensembles of earthquake records representing high and low-to-moderate seismicity regions are 0.64 and 0.45, respectively [41; 42]. Values of \(\varepsilon\) dependent upon the site and hazard level need to be calibrated to obtain a representative spectral shape so that a target (or expected) \(\varepsilon\) value, denoted as \(\bar{\varepsilon}_{0}\), is generally introduced. The value of \(\bar{\varepsilon}_{0}\) for the building response assessment can be obtained from the de-aggregation of ground motion hazard (probabilistic seismic hazard analysis) for a specific location, site class, spectral period, and return period. According to the USGS (United States Geological Survey) website [43] providing