# Seismic Response Analysis of Prestressed Concrete
## 5 Calculation of Collision Energy Dissipation  
It is assumed that energy dissipation occurs only during the collisions when the PCRF is rocking, that is, there is no energy consumption in the process of rotation. Considering that the duration of impact is extremely short, it can be assumed that the PCRF is always in a horizontal state during impacts. Collisions occur at the column feet and the inner faces at the beam-column joint in the case of a one-story single-span PCRF. Regardless of the plastic deformation of the tendons, the elongation of the tendons at the moment of collision is the initial elongation. The forces in the tendons are constant during the collision, which will not affect the momentum moments of the PCRF before and after the collision. The following assumptions are introduced to simplify the analysis and calculation of the collision energy dissipation of the PCRF. It is assumed that the collision forces on each collision surface concentrate on one point which is the postimpact rotation point of the identical rigid body. Relative slips between rigid members on each collision surface are not allowed. Non-collision forces such as gravity and the inertial force of members can be ignored compared to the collision force. The case \(\theta_{1}<0\) is taken as an example for analysis. The collision analysis model of the PCRF is shown in Figure 6 and the frame has returned to a zero-rotation state at this time. It is assumed that the angular velocity \(\dot{\theta}_{1}^{-}\) just before the collision is known of the frame column represented by bar AB. There are five unknowns: impulse \(I_{\rm Ax}\) and \(I_{\rm Ay}\) at collision point A\({}^{\prime}\), \(I_{\rm Dx}\) and \(I_{\rm Dy}\) at collision point D\({}^{\prime}\) and the postimpact angular velocity of the frame column represented by bar C'D'. The impulse \(I_{\text{i}}\) at an arbitrary collision point \(i\) is defined as  
\[I_{\text{i}}=\lim_{ \begin{subarray}{c}\Delta t_{\text{i}}=0\\ \Delta t_{\text{i}}\end{subarray} }\int_{\Delta t_{\text{i}}}F_{\text{ci}}dt \tag{36}\]  
where \(F_{\text{ci}}\) is the impact force and \(\Delta t_{\text{i}}\) is the duration of impact. The ratio of angular velocity after and before the collision is defined as the coefficient of restitution \(r=\dot{\theta}_{1}^{+}/\dot{\theta}_{1}^{-}\). The solution of simultaneous Equations (37)-(41), which are derived in the following, returns the coefficient of restitution \(r\). * Linear momentum conservation along the \(x\)-axis for the whole frame \[I_{\text{Ax}}+I_{\text{Dx}}=\big{[}m_{\text{c}}h(1+\lambda_{13})+m_{\text{b}} \big{(}2h^{\prime}-h_{\text{b}}\lambda_{\text{ L}}\big{)}\big{]}\bigg{(}\dot{ \theta}_{1}^{+}-\dot{\theta}_{1}^{-}\bigg{)}\] (37)
* Linear momentum conservation along the \(y\)-axis for the whole frame \[I_{\text{Ay}}+I_{\text{Dy}}=\big{[}m_{\text{c}}(1+\lambda_{13})+m_{\text{b}} \big{]}b\bigg{(}\dot{\theta}_{1}^{+}+\dot{\theta}_{1}^{-}\bigg{)}\] (38)
* Conservation of moment of momentum about point A\({}^{\prime}\) for the whole frame \[ \begin{array}{rl}-2 LI_{\text{Dy}}=&\big{[}(1+\lambda_{13})\big{(}I_{\text{ c}}^{G}+m_{\text{c}}h^{2}+m_{\text{c}}b^{2}\big{)}-2 Lbm_{\text{c}}-\lambda_{ \text{ L}}I_{\text{b}}^{G}+m_{\text{b}}(2h-h_{\text{b}})(2h^{\prime}-h_{\text{b}} \lambda_{\text{ L}})-m_{\text{b}}bL_{\text{b}}\big{]}\dot{\theta}_{1}^{+}\\ &-\big{[}(1+\lambda_{13})\big{(}I_{\text{c}}^{G}+m_{\text{c}}h^{2}-m_{\text{c} }b^{2}\big{)}+2\lambda_{13}Lbm_{\text{c}}-\lambda_{\text{ L}}I_{\text{b}}^{G}+m_{\text{b}}(2h-h_{\text{b}})(2h^{\prime}-h_{\text{b}} \lambda_{\text{ L}})+m_{\text{b}}bL_{\text{b}}\big{]}\dot{\theta}_{1}^{-}\\ &\bullet\end{array}