# Seismic Response Analysis of Prestressed Concrete
## 7 Parameter Analysis of PCRF Seismic Response  
Eight independent dimensionless parameters can be proposed naturally during the derivation of the motion equation of the PCRF, which are the mass ratio of beam to column \(\kappa\), span-height ratio \(\lambda\), column aspect ratio \(\tan\alpha\), height coefficient \(\lambda_{\text{h}}\), initial prestressing force of beam tendon \(\Pi_{\text{ob}}=\sigma_{\text{pb}}A_{\text{bs}}/m_{\text{c}}\)g, initial prestressing force of column tendon \(\Pi_{\text{oc}}=\sigma_{\text{pcb}}A_{\text{cs}}/m_{\text{c}}\)g, linear stiffness of beam tendon \(\Pi_{\text{Ep}}=E_{\text{p}}A_{\text{bs}}/m_{\text{c}}\)g and linear stiffness of column tendon \(\Pi_{\text{Ec}}=E_{\text{p}}A_{\text{cs}}/m_{\text{c}}\)g. These are kept because any reduction of independent dimensionless parameters will inevitably influence the precision of the analysis in consideration of the nonlinearity of the rocking motion for the PCRF. The values of the dimensionless parameters for the basic frame model are \(\lambda=2\), \(\tan\alpha=0.2\), \(\lambda_{\text{h}}=0.2\), \(\kappa=1.8\), \(\Pi_{\text{ob}}=4.13\), \(\Pi_{\text{oc}}=6.61\), \(\Pi_{\text{Ep}}=2684.1\) and \(\Pi_{\text{Ec}}=4294.6\). Earthquake record ER15 is used as a test record and a parameter analysis of the PCRF seismic response is conducted. A frame with a larger \(\kappa\) has a smaller response in the early stage of vibration, as shown in Figure 12. However, the responses of the frame are changed dramatically with the emergence of a large acceleration pulse at 2-3 s. In the meantime, the frame with a larger \(\kappa\) has a significantly larger angle and angular velocity compared with the smaller ones. A number of acceleration pulses appeared around 4 s, which increase the responses of the frame with \(\kappa=1.8\) further. The other frames have a smaller rotation before encountering multiple acceleration pulses and the amplification effect of multiple acceleration pulses on their response is limited. Different frames with different \(\kappa\) got into the pseudo-steady-state vibration eventually, while the frames with smaller \(\kappa\) entered earlier. The excessive increase in the response for the frame with \(\kappa=1.8\) caused by the first large acceleration pulse is counterintuitive. The mechanism of this phenomenon can be explained by the frame with larger \(\kappa\) having greater inertia force and energy after the acceleration pulse, causing a sudden increase in its response. The subsequent multiple acceleration pulses made the system response increase further. This phenomenon can be considered to suggest that the acceleration pulse has amplitude sensitivity to the amplification effect of the response of the PCRF. That is, when the vibration amplitude of the frame is large enough or greater than a threshold, the acceleration pulse has an amplification effect on the system response similar to resonance. The magnification of the system response increases with the enlargement of the system amplitude. For further study of the influence of \(\kappa\) and the influence of earthquake characteristics, the far-field earthquake record of the Northridge earthquake in 1994, named ER2, is selected for analysis. The time history curve of ER2 and similar calculation results are illustrated in Figure 13. It can be seen that the frame with larger \(\kappa\) had a smaller vibration response in the  
Figure 11: Maximum elongations of prestressed tendons. whole vibration process and entered into pseudo-steady-state vibration earlier. Based on the above analysis, cases of different ground motion types should be fully considered in the analysis of PCRFs and time history analysis of various earthquake records is necessary. It can be found in Figure 14 that the responses of frames with different \(\lambda\) are substantially close at the initial stage of vibration and the differences between responses are not changed significantly with the increase in vibration duration.