## 5 Probabilistic Seismic Risk Assessment Result
### Logarithmic Standard Deviation  
The logarithmic standard deviations of the different fragility curves are summarized in Table 5. The logarithmic standard deviation, \(\beta_{R}\), obtained from the median model  
Figure 12: Seismic fragility curves for median models: (**a**) UHRS; (**b**) CS1 with \(f_{c}\approx f_{1}/2\); (**c**) CS2 with \(f_{c}=f_{1}\); (**d**) CS3 with \(f_{c}=f_{2}\); (**e**) all fragility curves. Figure 13: Seismic fragility curves for LHS models: (**a**) UHRS; (**b**) CS1 with \(f_{c}\approx f_{1}/2\); (**c**) CS2 with \(f_{c}=f_{1}\); (**d**) CS3 with \(f_{c}=f_{2}\); (**e**) all fragility curves. reflects only the randomness of the seismic input. The logarithmic standard deviation, \(\beta_{C}\), obtained from the LHS model reflects both the randomness of the seismic input and the uncertainty in modeling and limit state criteria. The logarithmic standard deviation \(\beta_{U}\), which represents variability due to uncertainty, was calculated from \(\beta_{C}\) and \(\beta_{R}\) based on the SRSS combination rule, \(\beta_{C}=\sqrt{\beta_{R}^{2}+\beta_{U}^{2}}\). For the UHRS-based seismic input model, the variability of the response spectra with respect to the target spectrum remaining after spectral matching, as observed in Figure 6, was removed from \(\beta_{R}\) based on the SRSS rule [1], where the spectral variability denoted as \(\beta_{PV,R}\) was determined by averaging the logarithmic standard deviations of the spectral acceleration with respect to frequencies. The value of \(\beta_{PV,R}\) was very small relative to the value of \(\beta_{R}\) and had a negligible effect. Consequently, the \(\beta_{R}\) listed for the UHRS-based seismic input model can be viewed as randomness owing to only the horizontal component-to-component directional variability. On the other hand, the three CS-based seismic input models include randomness owing to the variability of the spectral shape, as shown in Figure 8, in addition to the component-to-component directional variability represented in Figure 9.  
The logarithmic standard deviations, \(\beta_{R}\), calculated for the median model are 0.183, 0.273, 0.310, and 0.864 for UHRS, CS1, CS2, and CS3, respectively. CS-based seismic input models generally produce larger \(\beta_{R}\) than UHRS-based seismic inputs. This is because CS reflects an additional source of randomness, that is, spectral shape variability, compared with UHRS. In particular, the value of \(\beta_{R}\) from the response to the UHRS-based seismic input is almost the same as the logarithmic standard deviation of the directional variability factor \(F_{H}\). In addition, for the CS-based seismic input models, \(\beta_{R}\), decreases at a lower control frequency. This is because the dominant frequency of the structure moved from \(f_{c2}\) to \(f_{c1}\) as the inelastic deformation of the structure became more significant, and randomness in the CS-based ground motion suites tended to be minimal in the vicinity of the control frequency. For the CS1-based seismic input, \(\beta_{R}\) is 0.273, which is similar to the logarithmic standard deviation of the spectral acceleration, 0.24 to 0.25, at the control frequency summarized in Table 6. However, \(\beta_{R}\)s for CS2 and CS3 were significantly higher than the corresponding standard deviations in Table 6 because the spectral variability at the dominant frequency increased as the control frequency was located far from the dominant frequency. It is noted that the logarithmic standard deviations for CSs in Table 6 are unique to the individual seismic input model but the logarithmic standard deviation, 0.18, for \(F_{H}\) is a generic characteristic of UHRS. Therefore, it is expected that the \(\beta_{R}\)s for the CS would have decreased somewhat if the spectral variability at the control frequency had been adjusted to 0.18, as for the UHRS. The logarithmic standard deviation, \(\beta_{C}\), calculated for the LHS models includes the influence of both randomness and uncertainty.