Items important to safety can be simply categorized into items that are important in preventing accidents and items that are important in mitigating the consequences of accidents. Items important to mitigating the consequences of accidents are required to function only after the occurrence of an accident, which essentially means that items important in preventing accident, in the first place, are damaged and/or have malfunctioned. Conventional seismic design procedures, however, do not usually distinguish between the roles of these two items explicitly.

The strategy for items important for safety is considered to be developed by combining diversity, physical separation, and functional independence (International Atomic Energy Agency, 2016a). Implementation of physical separation and functional independence is considered to be straightforward, while implementation of diversity to seismic excitation needs to be discussed further. Therefore, implementation of diversity to seismic excitation is discussed in this article. Diversity is provided by different mechanisms to function. In the seismic design of facilities, diversity is considered to be provided through differences in location of items (such as a plan layout or elevation) and by different dynamic characteristics between items (such as structural type, natural period, and damping characteristics). Providing an additional seismic margin, such as differentiation in classes of required seismic margins to each item based on its role, is another means in avoiding cliff-edge effects (International Atomic Energy Agency, 2016a). Typically, conservative parameters are introduced in the analysis of seismic design to deal with uncertainty, which are based on engineering judgment, the results of structural analysis, etc. These conservative parameters result in conservative designs. Quantification of such conservativeness is important to discuss the performance of facility to ground motions greater than the design ground motion (Budnitz et al., 1985; Haselton et al., 2011). In this study, an appropriate combination of seismic margin and diversity is discussed to implement the defense-in-depth concept to seismic design for safety-critical facility based on the risk-informed approach. A method to assign required additional seismic margins to each item is proposed depending on the characteristics of diversity introduced. As mentioned earlier, diversity is important to implement the defense-in-depth concept under seismic excitations. It is, however, not always possible to introduce it, because of the limitation due to the characteristics of item. Additional seismic margin is considered to be effective as a means of supplementing for such cases. Here, additional seismic margin means that seismic margin is required in addition to the seismic margin that is already introduced in the conventional seismic design.

Item B should be designed based on a different concept from that of item A. It is because a role of item B is different from that of item A. Therefore, it has been proposed in this study that the seismic margin m_B(T_B|T_A), which is additionally required for item B, is a function of the obtained spectral acceleration s¯B*TB|TA and is given as follows: (8)mBTB|TA=max1,s¯B*TB|TAsBDTB where S_BD(T_B) is the spectral acceleration at period T_B for the original seismic design obtained using the same concept as that for item A. From Eq. 5, it can be found that the additional seismic margin m_B(T_B|T_A) is almost unity if the difference between T_A and T_B is large enough. This is justified because diversity with respect to dynamic characteristics, such as the natural period, is expected to work effectively. (This will be discussed in the next chapter.). On the other hand, a larger additional margin m_B(T_B|T_A) is required if T_A and T_B are close to each other, i.e., if the diversity in the characteristics of items is not introduced in the seismic design. The proposed method combines the information on regional seismicity, the characteristics of ground motions, and the vulnerability of the facility to determine the additional seismic margin required for items that are important in mitigating the consequences of accidents.

An area source as shown in Figure 3 is used as an example. Point sources are uniformly distributed within a radius of 100 km, whereby their focal depth is 10 km. The facility is assumed to be located on the ground surface above the center of the area source. The probability distribution of the earthquake magnitude is assumed to be in agreement with the Gutenberg–Richter law. The cumulative distribution function for the magnitude F_M(m) is as follows: (9)FMm=exp−b⋅ln10⋅m−exp−b⋅ln10⋅mminexp−b⋅ln10⋅mmax−exp−b⋅ln10⋅mmin where m_max (6.95) and m_min (5.05) are the maximum and minimum magnitudes, respectively. b is assumed to be 0.9. These values are typical for those used for earthquakes without specified source faults in Japan. ν_S30S and z_1500S of Eq. 5 are assumed to be 700 m/s and 100 m, respectively. ν_S30S and z_1500S are the 30 m average shear wave velocity and the depth to shear wave velocity, which is equal to 1,500 m/s at the site, respectively. Seismic hazard curves and uniform hazard response spectra calculated at the facility are shown in Figure 4. The design ground motion for a system is assumed to correspond to the exceedance probability of 10⁻⁴/year.

The facility is modeled as a system that contains two items, items A and B, as is the case in Section “A Method to Identify the Most Probable Source Characteristics.” The natural period of item A, T_A, is assumed to be 0.02 s. As for item B, three alternative options (items B₀, B_S, and B_T) are assumed as listed in Table 3. It is assumed as an example that the logarithmic standard deviation of the capacity of each item is 0.3, while the conditional probability of failure at the level of design ground motion is 0.01. The most probable spectral acceleration and additional seismic margin required for items that are important in mitigating the consequences of accidents (items B_S and B_T) are obtained based on the proposed method as shown in Figure 5. Seismic fragility curves that show the cumulative distribution function of the capacity as a function of 5% spectral acceleration at the natural period, assumed for items B₀, B_S, and B_T, are shown in Figures 6A,B. The most probable source characteristics and the most probable ground motion parameters that may cause accidents are shown in Table 4. An additional seismic margin of 1.49 for item B_S, as compared to item B₀, is obtained using Eq. 8 for this example, whereas an additional seismic margin is not required for item B_T. If two items have the similar mechanism to resist seismic forces, it is reasonable to assume that the capacities between them are correlated. Therefore, for cases 0 and S, the correlation coefficient ρ between the capacities of A and B is assumed to be 0, 0.3, and 0.6, i.e., for items B₀ and B_S, where ρ = 0 for reference. Independence between items A and B_T is assumed for case T.

Monte Carlo simulations are conducted where the number of samples for the simulation is 10⁸. Samples of hypocenter and magnitude of earthquakes, 5% damped acceleration response spectra, and capacity of items are generated to calculate the fragility curve for failure of the system, i.e., simultaneous malfunction of both items.

Seismic fragility curves for item B_T as a function of 5% damped spectral acceleration at 0.02 s are estimated based on the simulated samples using the maximum likelihood estimation (Shinozuka et al., 2000), as shown in Figure 6A. The logarithmic standard deviation obtained is 0.93, which includes the effects of uncertainties in the shape of acceleration response spectra and the capacity of the item.

Seismic fragility curves of the system representing the cumulative distribution as a function of 5% damped spectral acceleration at 0.02 s, for the occurrence of a simultaneous malfunction of two items, are also obtained using the maximum likelihood estimation (Shinozuka et al., 2000). These are shown in Figure 7. As for case 0, i.e., item B₀, the median capacity of the system is 1.2 times larger than that of item A when ρ = 0, while it is 1.1 times larger when ρ = 0.6. The median capacity decreases as the correlation coefficient ρ increases because of simultaneous damage of two items. As for case S where an additional seismic margin is provided to item B, the median capacity of the system is 1.5 times larger than that of item A when ρ = 0, 0.3, and 0.6. The difference between ρ can be observed for ground motion <2,000 cm/s². As for case T, the case that the natural period of item B is elongated, the median capacity of the system is 2.1 times larger than that of item A, while the logarithmic standard deviation is 0.47, and this is larger than those in case 0 (0.24–0.28) and case S (0.26–0.30). Case T is more effective for larger ground motion levels as compared to cases 0 and S.

The annual failure probability of the system is numerically calculated to discuss the effectiveness of diversity in the natural period of items and additional seismic margins. The annual failure probability of the system, P_fsys, is calculated as follows: (10)Pfsys=∫0∞fSsFSyssds where f_s(s) is the probability density function of the annual maximum 5% damped spectral acceleration at 0.02 s, while F_Sys(s) is the cumulative distribution function of the capacity of the system.

The results are tabulated in Table 5. For case 0, item B₀ is not so much effective to mitigate the consequences of accidents, because the failure probability of the system does not decrease <0.449–0.640 times as compared to that of item A. The failure probability of the system decreases 0.165–0.213 times as compared to that of item A for case S, and it decreases 0.14 times as compared to that of item A for case T. Both cases T and S are effective in mitigating the consequences of accidents, while case 0 is not because of the effects of common cause failure.

It still remains a room for discussion how this framework can be applied to the design of actual safety-critical facility. One of typical examples where the framework can be applied is the case when an emergency operations facility is additionally constructed in the vicinity of the facility. Whether a base-isolated structure is better than an earthquake-resistant structure for the emergency operations facility should be discussed not only by the performance of a single facility but also based on the performance of a group of facilities. The proposed framework can be used to discuss the latter case.