This paper proposes a new probability model for the fully nonstationary ground motion acceleration process, and provides a kind of seismic input for structures for use in seismic research. First, based on the Clough-Penzien power spectrum of the stationary ground motion process, we establish the evolutionary power spectrum of the fully nonstationary ground motion process. Most importantly, the evolutionary power spectrum considers not only the nonstationary intensity, but also the nonstationary frequency. Then, based on the seismic design building code (in China), we identify these parameters in the evolutionary power spectrum model for different site conditions. Meanwhile, we applied the spectral representation-random functions method to generate an ensemble of 152 representative samples, and assigned each representative sample a given probability. This method uses a few basic random variables to express the original seismic ground motion process. Next, we obtain second-order statistics and the seismic response spectrum of the representative sample ensemble, and compare them with the target values. The results show that the general characteristics of all 152 representative samples coincide well with the target values. In this study, to verify the superiority and effectiveness of the proposed method, we used ANASYS software to conduct a dynamic time-history analysis of a concrete framework. Finally, by combining the recent probability density evolution method with the control criteria for the displacement angle, we performed a stochastic dynamic response analysis and reliability calculation for the concrete framework.