# Seismic Response Analysis of Prestressed Concrete
## 4 Rocking Motion Equation
\begin{array}{ll}I(\theta_{1})=&I_{\rm c}+I_{\rm b}^{\rm c}(\partial_{\theta_{ 1}}\theta_{2})^{2}+I_{\rm c}(\partial_{\theta_{1}}\theta_{3})^{2}+m_{\rm b} \Big{[}(2R_{\rm c}^{\prime})^{2}+4R_{\rm c}^{\prime}R_{\rm b}\cos(\beta-\gamma+ \theta_{1}-\theta_{2})\partial_{\theta_{1}}\theta_{2}+\big{(}R_{\rm b}\partial _{\theta_{1}}\theta_{2}\big{)}^{2}\Big{]}\\ J(\theta_{1})=&\Big{\{}I_{\rm b}^{\rm c}\partial_{\theta_{1}}\theta_{2} \partial_{\theta_{1}\theta_{2}}^{2}\theta_{2}+I_{\rm c}\partial_{\theta_{1}} \theta_{3}\partial_{\theta_{1}\theta_{2}}^{2}\theta_{3}\\ &+m_{\rm b}R_{\rm b}\Big{[}2R_{\rm c}^{\prime}\sin(\beta-\gamma+\theta_{1}- \theta_{2})(\partial_{\theta_{1}}\theta_{2}-1)\partial_{\theta_{1}}\theta_{2} +2R_{\rm c}^{\prime}\cos(\beta-\gamma+\theta_{1}-\theta_{2})\partial_{\theta_ {1}}^{2}\theta_{2}+R_{\rm b}\partial_{\theta_{1}}\theta_{2}\partial_{\theta_{ 1}\theta_{2}}^{2}\Big{]}\Big{\}}\\ G(\theta_{1})=&-\big{[}m_{\rm c}R_{\rm c}\sin(\alpha-\theta_{1})+2m_{\rm b}R_ {\rm c}^{\prime}\cos(\beta+\theta_{1})+m_{\rm b}R_{\rm b}\cos(\gamma+\theta_{2 })\partial_{\theta_{1}}\theta_{2}+m_{\rm c}R_{\rm c}\sin(\alpha-\theta_{3}) \partial_{\theta_{1}}\theta_{3}\big{]}\\ S(\theta_{1})=&-\big{\{}m_{\rm c}R_{\rm c}\big{[}\cos(\alpha-\theta_{1})+\cos( \alpha-\theta_{3})\partial_{\theta_{1}}\theta_{3}\big{]}+m_{\rm b}\big{[}2R_{ \rm c}^{\prime}\sin(\beta+\theta_{1})+R_{\rm b}\sin(\gamma+\theta_{2}) \partial_{\theta_{1}}\theta_{2}\big{]}\big{\}}\\ K(\theta_{1})=&k_{\rm c}\Big{\{}b^{2}(\sin\theta_{1}+\sin\theta_{3}) \partial_{\theta_{1}}\theta_{3})+\sqrt{2}be_{\rm c}[\cos(\theta_{1}/2)+\cos( \theta_{3}/2)\partial_{\theta_{1}}\theta_{3}\big{]}/2\Big{\}}\\ &+k_{\rm b}h_{\rm b}\Big{\{}\Big{[}h_{\rm b}\sin(\theta_{1}-\theta_{2})+e_{ \rm b}\cos\big{(}\frac{\theta_{1}-\theta_{2}}{2}\big{)}\Big{]}\big{(}1- \partial_{\theta_{1}}\theta_{2})+2h_{\rm b}\sin\Big{(}\frac{\theta_{1}+\theta _{2}}{2}-\theta_{2}\Big{)}\Big{(}\frac{1+\partial_{\theta_{1}}\theta_{3}}{2} -\partial_{\theta_{1}}\theta_{2}\Big{)}\\ &+\Big{[}e_{\rm b}0\cos\big{(}\frac{\theta_{1}-\theta_{2}}{2}\big{)}+h_{\rm b }\sin(\theta_{3}-\theta_{2})\Big{]}\big{(}\partial_{\theta_{1}}\theta_{3}- \partial_{\theta_{1}}\theta_{2}\big{)}-h_{\rm b}\sin\Big{(}\frac{\theta_{1}- \theta_{3}}{2}\Big{)}\big{(}1-\partial_{\theta_{1}}\theta_{3}\big{)}\Big{\}} \end{array}  \tag{33}\]  
The simplified and conservative starting condition for the calculation program can be obtained by ignoring the influence of the prestressing force on the triggering-off condition of the rocking motion for the PCRF. With this assumption, the lower bound solution of the minimum ground acceleration \(\ddot{u}_{\rm g,min}\) that initiates the rocking motion of the frame can be derived by substituting the initial conditions \(\ddot{\theta}_{1}=0\), \(\dot{\theta}_{1}=0\) and \(\theta_{1}=0\) into Equation (31):  
\[\lambda_{\rm g}=\frac{\ddot{u}_{\rm g,min}}{\rm g}=\frac{G(0)}{S(0)} \tag{34}\]  
The calculation of \(\lambda_{\rm g}\) can be obtained based on the expression of \(G\) and \(S\) in Equation (32) or Equation (33):  
\[\lambda_{\rm g}=\frac{\ddot{u}_{\rm g,min}}{\rm g}=\frac{G(0)}{S(0)}=\frac{m_{ \rm c}b\Big{(}1+\partial_{\theta_{1}}\theta_{3}\big{|}_{\theta_{1}=0}\Big{)}+m_ {\rm b}\Big{(}2b+L_{\rm b}\partial_{\theta_{1}}\theta_{2}\big{|}_{\theta_{1}=0 }\Big{)}}{m_{\rm c}h\Big{(}1+\partial_{\theta_{1}}\theta_{3}\big{|}_{\theta_{1 }=0}\Big{)}+m_{\rm b}\Big{(}2h^{\prime}+h_{\rm b}\partial_{\theta_{1}}\theta_{ 2}\big{|}_{\theta_{1}=0}\Big{)}} \tag{35}\]