# Transient Response of Bridge Piers to Structure Separation under Near-Fault Vertical Earthquake
## 2 Theoretical Model and Vertical Displacement Calculation
### Corresponding Theoretical Solution of Bridge Vertical Separation Stage Displacement  
In the separation process, the beam and the rod do not interact and move at their own characteristic frequencies \(\omega_{nb}\) and \(\omega_{nr}\).  
The displacement response of the main girder and pier can be decomposed into static displacement, rigid displacement, and dynamic deformation, and the main beam does not need to be decomposed into two segments.  
\[\begin{array}{l}\overline{Y}(x,t)=\overline{Y}_{s}(x)+\overline{Y}_{g}(x,t) +\overline{Y}_{d}(x,t)\\ \overline{U}(\xi,t)=\overline{U}_{s}(\xi,t)+\overline{U}_{g}(\xi,t)+\overline{ U}_{d}(\xi,t)\end{array} \tag{18}\]  
The vertical static displacement and rigid displacement of the bridge structure are as follows:  
\[\begin{array}{l}\overline{Y}_{s}(x)=q(-5L^{4}+6L^{2}x^{2}-x^{4})/24E_{b}l_{ b},\ \overline{U}_{s}(\xi)=0\\ \overline{Y}_{g}(x,t)=\overline{U}_{g}(\xi,t)=B(t)\end{array} \tag{19}\]  
The wave equations of girder AB and pier CD can be described as the following:  
\[\begin{array}{l}AB:\frac{\partial^{2}(E_{b}h\partial^{2}\overline{Y}(x,t)/ \partial x^{2})}{\partial x^{2}}+\rho A_{b}\frac{\partial^{2}\overline{Y}(x,t )}{\partial t^{2}}+q=0\\ CD:E_{r}A_{r}\frac{\partial^{2}\overline{U}(\xi,t)}{\partial x^{2}}-\rho A_{r} \frac{\partial^{2}\overline{U}(\xi,t)}{\partial t^{2}}=0\end{array} \tag{20}\]  
The wave mode functions of the main girder and pier are:  
\[\begin{array}{l}\overline{\varphi}_{nb}(x)=A_{3}\sin\overline{k}_{bn}x+B_{ 3}\cos\overline{k}_{bn}x+C_{3}\text{sinh}\overline{k}_{bn}x+D_{3}\cosh \overline{k}_{bn}x\\ \overline{\varphi}_{nr}=E_{2}\sin\overline{k}_{rn}\xi+F_{2}\cos\overline{k}_{ rn}\xi\end{array} \tag{21}\]  
where \(A_{3},B_{3},C_{1},D_{3},E_{2},F_{2}\) are the coefficients.  
The boundary conditions of the characteristic equation are as follows:  
\[\begin{array}{l}\overline{\varphi}_{nb}(-L)=0,\ \overline{\varphi}_{nb}(L)=0,\ \overline{ \varphi}_{nr}(0)=0\\ \overline{\varphi^{\prime\prime}}_{nb}(-L)=0,\ \overline{\varphi^{\prime\prime}}_{nb}(L)=0 \end{array} \tag{22}\]  
The wave functions of the bridge structure can be obtained as follows:  
\[\begin{array}{l}\overline{\varphi}_{nb}(x)=A_{nb}\sin\overline{k}_{bn}(x+L) \\ \overline{\varphi}_{nr}(\xi)=A_{nr}\sin\overline{k}_{rn}\xi\end{array} \tag{23}\]For the main girder and pier, the wavenumber is:  
\[\overline{k}_{bn}=\sqrt{\omega_{bn}/a}=n\pi/2L,\ \ \overline{k}_{m}=\omega_{m}/c=(2n-1)\pi/ 2H \tag{24}\]  
Based on the orthogonality of the wave mode function, the coefficient can be obtained as:  
\[A_{nb}=1/\sqrt{\rho A_{b}L},\ \ A_{nr}=2/\sqrt{\rho A_{r}H} \tag{25}\]  
If the separation contact phenomenon occurs many times, it can be assumed that \(t^{*}=t-t_{2k}\) is the time variable of the k-th collision, \(t^{*}=t-t_{2k+1}\) is the time variable of the k-th separation.  
In the \(k\)-th separation process, the dynamic displacement responses of the main girder and pier are as follows:  
\[\begin{array}{l}q_{nb}(t^{*})=e^{-\zeta_{2}\omega_{b1}t^{*}}(q_{1b}(t^{+}_{2 k+1})\cos\omega_{b1}t^{*}+\frac{\tilde{q}_{n}(t^{-}_{2k+1})+\zeta_{2}\omega_{b1} q_{nb}(0)}{\omega_{b1}}\sin\omega_{b1}t^{*})\\ \qquad\qquad+\frac{1}{\omega_{b1}}\int_{t^{+}_{2k+1}}^{t^{*}}e^{-\zeta_{2} \omega_{bn}t}\tilde{Q}_{bn}(\tau)\sin(\omega_{bd}(t^{*}-\tau))d\tau\\ q_{nr}(t^{*})=e^{-\zeta_{2}\omega_{r1}t^{*}}(q_{1b}(t^{+}_{2k+1})\cos\omega_{b1 }t^{*}+\frac{\tilde{q}_{n}(t^{-}_{2k+1})+\zeta_{2}\omega_{r1}q_{nr}(0)}{\omega _{r1}}\sin\omega_{r1}t^{*})\\ \qquad\qquad+\frac{1}{\omega_{rd}}\int_{t^{+}_{2k+1}}^{t^{*}}e^{-\zeta_{2} \omega_{nr}t}\tilde{Q}_{rn}(\tau)\sin(\omega_{rd}(t^{*}-\tau))d\tau\end{array} \tag{26}\]  
The initial displacement and velocity of the girder and pier structure are considered only by the first mode. By viewing the residual rate and deformation of the last process, the contact separation process is solved continuously.