# Seismic Response Analysis of Prestressed Concrete
## 6 Time History Analysis of Seismic Response of the PCRF
It can be observed from Figure 8c,d that the phase orbit finally formed a spindle-shaped limit cycle which was centered on the origin in the phase diagram. The energy dissipation in the collisions of the PCRF is continuously supplied by the earthquake. Although the frame vibration in the high-frequency vibration stage is similar to steady-state vibration, it is named pseudo-steady-state vibration, considering that its external excitation is a random earthquake. It can be seen that the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & **Year** & **Record** & **Magnitude** & **Duration(s)** & **PGA (g)** & **PGV (cm/s)** \\ \hline ER19 & 1976 & Gazli, USSR & 6.8 & 16.26 & 0.71 & 71.2 \\ ER20 & 1979 & Imperial Valley-06 & 6.5 & 37.6 & 0.76 & 44.3 \\ ER21 & 1985 & Nahanni, Canada & 6.8 & 20.56 & 1.10 & 43.9 \\ ER22 & 1989 & Loma Prieta & 6.9 & 25 & 0.64 & 55.9 \\ ER23 & 1992 & Cape Mendocino & 7.0 & 29.98 & 1.50 & 119.5 \\ ER24 & 1994 & Northridge-01 & 6.7 & 47.765 & 0.93 & 70.1 \\ ER25 & 2002 & Denali, Alaska & 7.9 & 89.995 & 1.16 & 115.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & \(t_{\text{RI}}\) **(s)** & \(t_{\text{EP}}\) **(s)** & \(\theta_{1,\text{max}}\) **(rad)** & \(t_{\text{RM}}\) **(s)** & \(\dot{\theta}_{1,\text{max}}\)**(rad/s)** & \(t_{\text{VM}}\) **(s)** \\ \hline ER1 & 3.340 & 4.540 & 0.0224 & 5.3358 & 0.4801 & 5.2497 \\ ER2 & 4.020 & 5.030 & 0.0198 & 4.1588 & 0.4512 & 5.176 \\ ER3 & 9.760 & 10.800 & 0.0179 & 11.2637 & 0.4041 & 9.97 \\ ER4 & 5.950 & 10.465 & 0.0118 & 6.4653 & 0.3113 & 6.5353 \\ ER5 & 4.940 & 7.250 & 0.0216 & 8.2107 & 0.4930 & 7.5844 \\ ER6 & 7.040 & 9.170 & 0.0124 & 7.285 & 0.3078 & 7.2108 \\ ER7 & 2.690 & 6.025 & 0.0284 & 5.6652 & 0.6038 & 5.5797 \\ ER8 & 4.425 & 5.120 & 0.0168 & 4.8086 & 0.3823 & 4.6292 \\ ER9 & 4.040 & 6.250 & 0.0139 & 4.2969 & 0.3641 & 4.3665 \\ ER10 & 4.240 & 5.180 & 0.0178 & 5.406 & 0.3988 & 5.4908 \\ ER11 & 32.320 & 36.885 & 0.0193 & 32.4585 & 0.4237 & 32.5385 \\ ER12 & 4.815 & 4.990 & 0.0165 & 5.2269 & 0.4042 & 5.1482 \\ ER13 & 2.720 & 3.280 & 0.0136 & 2.9608 & 0.3003 & 3.4143 \\ ER14 & 5.880 & 14.860 & 0.0210 & 6.0186 & 0.4533 & 6.0991 \\ ER15 & 0.780 & 2.740 & 0.0248 & 4.7134 & 0.5326 & 4.6309 \\ ER16 & 3.320 & 4.200 & 0.0346 & 5.0913 & 0.6778 & 5.1786 \\ ER17 & 26.310 & 47.515 & 0.0122 & 26.563 & 0.3180 & 26.4095 \\ ER18 & 2.795 & 3.420 & 0.0086 & 2.9158 & 0.2555 & 2.9762 \\ ER19 & 2.550 & 8.640 & 0.0211 & 2.6878 & 0.4623 & 2.9058 \\ ER20 & 2.665 & 6.795 & 0.0208 & 7.4363 & 0.4865 & 7.5139 \\ ER21 & 1.320 & 9.160 & 0.0177 & 3.2637 & 0.4117 & 3.1822 \\ ER22 & 7.830 & 10.585 & 0.0130 & 8.3379 & 0.3309 & 8.0201 \\ ER23 & 2.520 & 3.000 & 0.0340 & 3.2384 & 0.6620 & 3.1545 \\ ER24 & 2.890 & 7.775 & 0.0208 & 3.8558 & 0.4960 & 3.930 \\ ER25 & 29.840 & 37.100 & 0.0175 & 30.2572 & 0.3985 & 30.0497 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Time history analysis results of seismic response. In order to investigate the relationship between the key time points \(t_{\text{RI}}\), \(t_{\text{EP}}\), \(t_{\text{RM}}\) and \(t_{\text{VM}}\) of the basic model under different earthquakes, the \(t_{\text{RI}}\) of each response are all set as 1 s. The corresponding unified key time points can be obtained. The unified key time points of far-field and near-field earthquake records are shown in Figures 9 and 10, respectively. Figure 9 illustrates that, under the far-field earthquakes, \(t_{\text{RM}}\) is very close to \(t_{\text{VM}}\) and they generally occur shortly after \(t_{\text{RI}}\) or near \(t_{\text{EP}}\). In most of the cases that \(t_{\text{RM}}\) and \(t_{\text{VM}}\) appeared shortly after \(t_{\text{RI}}\), the frame vibration has entered the pseudo-steady-state vibration stage or a stage with very small amplitudes of responses before \(t_{\text{EP}}\). The amplification effect of PGA on frame response is no longer significant in this situation.