## 6 Conclusions  
The primary purpose of this study was to investigate the effect of phase spectrum uncertainty on the nonlinear behavior of structural responses. We proposed to use the modified fBm process to simulate phase spectra with uncertainty features of observed ground motion phases. We also defined a rigorous nonlinear design spectrum, named the yield seismic coefficient demand spectrum, which was obtained through statistical analysis of the nonlinear analysis results using simulated DRSCEMs and taking into account the uncertainty of the earthquake motion phases. Using recorded ground motion observed from the world, a nonlinear regression analysis was conducted to calculate the Hurst index and variance parameters for simulating the modified fBm process. Two regression equations were developed to determine the Hurst index and variance as functions of the earthquake magnitude and epicenter distance, respectively.  
Comparing the results obtained by this study with those of previous studies, the principal contents can be summarized as follows:  
* Modification of the fBm process for modeling Fourier phase provides a good physical interpretation for phase uncertainty and led to a feasible and easy way to simulate design-response spectra compatible earthquake motions (DRSCEM).
* A method for generating DRSCEM was developed in this paper and was useful for generating an input ground motion for a seismic response analysis of building structures with nonlinear behavior at sites of interest.
* We developed regression equations of tow parameters (the Hurst index and the variance) to define the power low of the modified fractional Brownian motion as the function of earthquake magnitude, epicenter distance, and bias index.
* Based on the assumption that even one observed earthquake motion phase has an uncertainty, we proposed a new concept to extract the phase uncertainty from one observed earthquake motion phase. In classical criterion many researchers believe that it is useful to collect many observed earthquake motion phases for evaluating phase uncertainty. Using this new criterion to evaluate uncertainty of phase we succeeded to reduce the effect of phase uncertainty compared to the classical criterion.
* We proposed a method to define a unique yield seismic coefficient demand spectrum for design purpose in China, but the same method can be used in all over the world only by changing the seismic design code that is effective at the concerned region.  
[PERSON] conceived the main idea of this work; [PERSON] and [PERSON] conducted numerical analysis; [PERSON] made the theoretical derivation; [PERSON] wrote the original draft and prepared this article as the first author; [PERSON] and [PERSON] revised the final manuscript.  
This research was funded by JSPS, Grant-in-Aid for Scientific Research (Grant number:18K04334), the National Key Research and Development Program of China (Grant number: 2017 YFC0840200), and the National Natural Science Foundation of China (Grant number: 51578140).  
**Acknowledgments:** The authors would like to acknowledge Japan Meteorological Agency and National Research for Earth and Disaster Resilience, as well as the Peer NGA Strong Motion Database, to provide valuable observed earthquake records.  
**Conflicts of Interest:** The authors declared no conflict of interest concerning the research, authorship, and the publication of this article.