## 5 Proposition of YSC Demand Spectra for Design Purposes
### Fluctuation of the YSC Demand Spectra  
Although the mean and mean plus one standard deviation of the YSC demand spectra are very useful as they represent what can be expected beyond the average, it is equally important to know the level of scatter about the mean. A common and effective way to quantify fluctuation is to use the coefficient of variance (CV), which is defined as the ratio of the standard deviation to the mean. Figure 12a shows the CV of a YSC demand spectrum corresponding to all DRSCEMs. Figure 12b shows the mean of YSC demand spectra for all ductility values. The CV, which shows the fluctuation of the YSC demand spectra, is not constant over the entire natural period ranges and depends on the natural period and ductility level selected. In general, fluctuation increases as the level of ductility increases and the natural period of the structural system increases. The CV for a ductility of 1 is almost constant and close to zero for all natural period ranges because it is an elastic response. Additionally,  
Figure 11: YSC demand spectra obtained by simulated 1000 DRSCEMs using modified fBm process, (**a**) Mean, mean plus and minus one standard deviation, 5 and 95 percentile confidence intervals and (**b**) Comparison with results obtained by using DRSCEM simulated by observed ground motion phases. we observe that the CV for all of the ductility values up to the natural period of 3 s are almost in the range of 10%, which is a very peculiar characteristic of the YSC demand spectra because the mean value of the YSC provides the variance of the YSC value within the natural period between 0.1 to and 5.0 s.  
On the other hand, if the data obey a normal distribution at its mean, the statistic shows that over 68% of the data values are within one standard deviation, over 95% of the data values are within two standard deviations and over 99% fall within three standard deviations. Depending on these results, these confidence interval concepts are very useful for defining the design YSC demand spectra by taking into account safety and economic aspects of the structural design. For the purpose to define the design YSC demand spectrum we can use the statistic concept mentioned above. However, even if we use mean + one standard deviation criterion, more than 16% designed structure still drops in the unsafe region. We, therefore, propose to use the smoothed envelope YSC demand spectra shown in Figure 13a by a thick green dash dot line for design purposes. To calculate a smoothed envelope curve of YSC demand spectra we must simulate DRSCEMs as many as possible. If we simulate more than 1000 DRSCEMs and obtain the smoothed envelope curve the ratio that the designed structure drops in the unsafe region is less than 0.1% in all natural period range of structure systems. Figure 12: For ductility values of 1, 1.5, 2, 3, 4, and 6 (**a**) Coefficients of variation and (**b**) Mean of YSC demand spectra. Figure 13: Smoothed envelope curve and the fluctuation of YSC demand spectra obtained by (**a**) 1000 DRSCEMs simulated by the modified fBm process and (**b**) 500 DRSCEMs simulated by observed earthquake motions phases. To verify the efficiency of the proposed method, we also simulated more than 500 DRSCEMs using the observed ground motion phases and investigated the fluctuations of YSC along the range of the concerned natural periods. We found that the maximum exceedance of the fluctuations above the smoothed envelope curve was 1 out of 500 samples, which was a 0.2% exceedance rate. Figure 13b shows the proposed YSC demand spectrum (the thick green dash dot line) and the distribution of the YSC demand spectra calculated using DRSCEMs simulated by taking the Fourier phase from observed earthquake motions. From Figure 13, it is apparent that the YSC demand spectra fluctuate randomly above the smoothed envelope curve by a maximum exceedance of 0.2%, thereby confirming that the smooth envelop curve is suitable for design purposes.