# Transient Response of Bridge Piers to Structure Separation under Near-Fault Vertical Earthquake
## 2 Theoretical Model and Vertical Displacement Calculation
### Theoretical Solution of Displacement Response of Bridge in Vertical Contact Stage  
The wave equations of girder AB and pier CD can be described as the following:  
\[ \begin{array}{c}OA:\frac{\partial^{2}(E_{1}\partial^{2}Y_{1}(x,t)/\partial x ^{2})}{\partial x^{2}}+\rho A_{b}\frac{\partial^{2}Y_{1}(x,t)}{\partial t^{2} }+q=0\\ OB:\frac{\partial^{2}(E_{1}\partial^{2}Y_{1}(x,t)/\partial x^{2})}{\partial x^{2 }}+\rho A_{b}\frac{\partial^{2}Y_{1}(x,t)}{\partial t^{2}}+q=0\\ CD:E_{r}A_{r}\frac{\partial^{2}U(\xi,t)}{\partial x^{2}\xi}-\rho A_{r}\frac{ \partial^{2}U(\xi,t)}{\partial t^{2}}=0\end{array}  \tag{2}\]  
In these equations, \(Y_{1}(x,t),Y_{2}(x,t)\) is the deflection of the beam; \(U(\xi,t)\) is the axial displacement of pier. The boundary condition of bridge displacement is:  
\[ \begin{array}{c}Y_{1}(-L,t)=0,\ Y_{2}(L,t)=0,\ U(0,t)=0\\ \frac{\partial^{2}Y_{1}(-L,t)}{\partial^{2}x}=0,\ \ \frac{\partial^{2}Y_{2}(L,t)}{ \partial x^{2}}=0\end{array}  \tag{3}\]  
and the displacement continuity condition at the middle of the main girder is:  
\[ \begin{array}{c}Y_{1}(0,t)=Y_{2}(0,t),\ \ \frac{\partial Y_{1}(0,t)}{ \partial x}=\frac{\partial Y_{2}(0,t)}{\partial x},\ \ \ \frac{\partial^{2}Y_{1}(0,t)}{\partial x^{2}}=\frac{\partial^{2}Y_{2}(0,t)}{ \partial x^{2}}\end{array}  \tag{4}\]  
The continuity condition of shear force and displacement between girder, pier and support are:  
\[ \begin{array}{c}U(H,t)-Y_{1}(0,t)=\delta_{1}=\frac{F_{c}}{K_{c}}=-\frac{F_{ c}A_{r}}{K_{c}}\frac{\partial U(H,t)}{\partial\xi}\\ E_{r}A_{r}\frac{\partial U(H,t)}{\partial\xi}=E_{b}I_{b}(\frac{\partial^{3}Y_{1} (0,t)}{\partial x^{3}}-\frac{\partial^{2}Y_{2}(0,t)}{\partial x^{3}})\end{array}  \tag{5}\]  
Vertical displacement field \(Y(x,t)\) of the girder can be divided into static displacement \(Y_{s}\), rigid body displacement \(Y_{g}\) and dynamic deformation \(Y_{d}\). \[ \begin{array}{c}Y(x,t)=Y_{s}(x)+Y_{g}(x,t)+Y_{d}(x,t)\\ U(\xi,t)=U_{s}(\xi)+U_{g}(\xi,t)+U_{d}(\xi,t)\end{array}  \tag{6}\]  
Dynamic displacement satisfies the wave equation, continuity condition, equilibrium differential equation, and force boundary condition. Static displacement of the bridge is as follows:  
\[ \begin{array}{l}Y_{1s}(x)=\big{[}q(-5L^{4}+6L^{2}x^{2}-x^{4})-2F_{c}(2L^{3}-x^{ 3}-3x^{2}L)\big{]}/24E_{b}I_{b}\\ Y_{2s}(x)=\big{[}q(-5L^{4}+6L^{2}x^{2}-x^{4})-2F_{c}(2L^{3}+x^{3}-3x^{2}L)\big{]}/ 24E_{b}I_{b}\\ U_{s}(\xi)=F_{c}\xi/E_{r}A_{r}\end{array}  \tag{7}\]  
where \(F_{c}\) is the axial pressure of the initial girder and the support. The displacement of the rigid body of the bridge is as follows:  
\[Y_{1g}(x,t)=Y_{2g}(x,t)=U_{g}(\xi,t)=B(t) \tag{8}\]  
The dynamic deformation part of the structure can be expanded as an infinite series of wave mode products:  
\[ \begin{array}{l}Y_{1d}(x,t)=\sum\limits_{n=1}^{\infty}\varphi_{nb1}(x)q_{n}( t),\,Y_{2d}(x,t)=\sum\limits_{n=1}^{\infty}\varphi_{nb2}(x)q_{n}(t)\\ U_{d}(\xi,t)=\sum\limits_{n=1}^{\infty}\varphi_{nr}(\xi)q_{n}(t)\end{array}  \tag{9}\]  
The equation includes the bending wave function \(\varphi_{nb1}\),\(\varphi_{nb2}\) of the girder, the longitudinal wave function \(\varphi_{nr}\) of the pier, and the time function \(q_{n}(t)\). The wave model function is solved by the characteristic equations. The characteristic equations of the main girder and pier are, respectively:  
\[ \begin{array}{l}OA:\quad a^{2}\varphi_{nb1}{}^{(4)}(x)-\omega_{n}{}^{2} \varphi_{nb1}(x)=0\\ OB:\quad a^{2}\varphi_{nb2}{}^{(4)}(x)-\omega_{n}{}^{2}\varphi_{nb2}(x)=0\\ CD:\quad c^{2}\varphi_{nr}{}^{(2)}(\xi)+\omega_{n}{}^{2}\varphi_{nr}(\xi)=0 \end{array}  \tag{10}\]  
where \(\omega_{n}\)(\(n\) = 1,2,3, \(\cdots\cdots\)) is the natural frequency of bridge structure, \(a=\sqrt{E_{b}I_{b}/\rho A_{b}}\) is the coefficient related to the beam flexural wave speed, \(c=\sqrt{E_{r}/\rho}\) is the rod phase speed. Flexural wave modes of the main girder and longitudinal wave of the pier can be described as:  
\[