# Seismic Response Analysis of Prestressed Concrete
## 6 Time History Analysis of Seismic Response of the PCRF  
The seismic response analysis of the PCRF is based on a basic model. The sections of the beam and columns are square. The frame is in the zero-rotation state without initial angular velocity and angular acceleration. The parameters of the basic model are shown in Table 1.  
The term \(\rho_{\rm c}\) is the density of concrete, and the fracture elongation of the tendons is taken as 1%. The set of earthquake motion records recommended by the ATC-63 project [21] is selected as the source of earthquake records. The information of the earthquakes is shown in Table 2. Records ER1 to ER11 are far-field earthquakes, records ER12 to ER18 are pulse near-field earthquakes and the rest are no-pulse near-field earthquakes. The values of the peak ground acceleration (PGA) and peak ground velocity (PGV) of each earthquake record are also shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(L\) (m) & 3 & \(A_{\rm pc}\) (m\({}^{2}\)) & \(5.6\times 10^{-4}\) \\ \(b\) (m) & 0.3 & \(A_{\rm pb}\) (m\({}^{2}\)) & \(3.5\times 10^{-4}\) \\ \(h\) (m) & 1.5 & \(\sigma_{\rm p0}\) (Pa) & \(3\times 10^{8}\) \\ \(h_{\rm b}\) (m) & 0.3 & \(\sigma_{\rm p0}\) (Pa) & \(3\times 10^{8}\) \\ \(\rho_{\rm c}\) (kg/m\({}^{3}\)) & 2400 & \(E_{\rm p}\) (N/m\({}^{2}\)) & \(1.95\times 10^{11}\) \\ \(m_{\rm c}\) (kg) & 2592 & \(m_{\rm b}\) (kg) & 4665.6 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the basic model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Year** & **Record** & **Magnitude** & **Duration(s)** & **PGA (g)** & **PGV (cm/s)** \\ \hline ER1 & 1994 & Northridge & 6.7 & 29.98 & 0.52 & 63 \\ ER2 & 1994 & Northridge & 6.7 & 19.98 & 0.48 & 45 \\ ER3 & 1999 & Duzce, Turkey & 7.1 & 55.89 & 0.82 & 62 \\ ER4 & 1979 & Imperial Valley & 6.5 & 39.03 & 0.38 & 42 \\ ER5 & 1995 & Kobe, Japan & 6.9 & 40.95 & 0.51 & 37 \\ ER6 & 1999 & Kocaeli, Turkey & 7.5 & 27.18 & 0.36 & 59 \\ ER7 & 1989 & Loma Prieta & 6.9 & 39.95 & 0.53 & 35 \\ ER8 & 1989 & Loma Prieta & 6.9 & 39.94 & 0.56 & 45 \\ ER9 & 1987 & Superstition Hills & 6.5 & 22.29 & 0.45 & 36 \\ ER10 & 1992 & Cape Mendocino & 7.0 & 35.98 & 0.55 & 44 \\ ER11 & 1999 & Chi-Chi, Taiwan & 7.6 & 89.995 & 0.44 & 115 \\ ER12 & 1979 & Imperial Valley-06 & 6.5 & 36.815 & 0.46 & 108.9 \\ ER13 & 1972 & Cape Mendocino & 7.0 & 35.98 & 0.63 & 82.1 \\ ER14 & 1992 & Landers & 7.3 & 48.12 & 0.79 & 140.3 \\ ER15 & 1994 & Northridge-01 & 6.7 & 19.9 & 0.87 & 167.3 \\ ER16 & 1994 & Northridge-01 & 6.7 & 39.98 & 0.73 & 122.8 \\ ER17 & 1999 & Chi-Chi, Taiwan & 7.6 & 89.995 & 0.82 & 127.7 \\ ER18 & 1999 & Duzce, Turkey & 7.1 & 25.88 & 0.52 & 79.3 \\ \hline \end{tabular}
\end{table} 
Table 2: Earthquake records. According to the calculation results, the time \(t_{\text{RI}}\) of the rocking motion triggering-off, the time \(t_{\text{EP}}\) of peak ground acceleration (PGA), the maximum rotation \(\theta_{1,\text{max}}\) and its occurrence time \(t_{\text{RM}}\), as well as the maximum angular velocity \(\dot{\theta}_{1,\text{max}}\) and its occurrence time \(t_{\text{VM}}\), can be obtained as shown in Table 3, where the maximum values of rotation and angular velocity are given in absolute values without the consideration of direction. The result of ER15 is illustrated here to explain the characteristics of the seismic response of the PCRF. The time history curve of ER15 is shown in Figure 7 and the calculation results are presented in Figure 8. As shown in Figure 8a,b, \(t_{\text{EP}}\) does not exactly coincide with \(t_{\text{RM}}\) and \(t_{\text{VM}}\). After the appearance of PGA and local maximum values of acceleration, the amplitudes of responses decay rapidly. At a time around 10.5 s, the frame motion approached the high-frequency vibration stage, in which the vibration amplitude is small and the peak of angular velocity is basically stable.