# Seismic Effectiveness of Multiple Seismic Measures on a Continuous Gider Bridge
## 2 Theoretical Pounding Analysis of a Bridge  
The bounding process of bridges is usually described in two stages:  
1. Initially, there is no bounding between the structures, and the bounding stiffness can be regarded as zero. 2. When the ends are bounding, the bounding stiffness will no longer be zero. If seismic measures are adopted on the bridge, the bounding stiffness will be the stiffness of the control measures. In this paper, \(m\) is the mass of the superstructure, \(k\) is the bounding stiffness, and \(c\) is the damping at the time of the structural bounding. The variables \(x(t),\tilde{x}(t),\tilde{x}(t)\) are the displacement, velocity, and acceleration of the structure relative to the ground, respectively, which are all functions of time \(t\). Assuming that the bounding occurs at time \(t=0\) and finishes at time \(t_{0}\), the motion equation of the structure is as follows:  
\[\left\{ \begin{array}{c}m\ddot{x}+\dot{c}\dot{x}+kx=0\\ x(0)=0\\ \dot{x}(0)=v_{0}\end{array} \right., \tag{1}\]  
where \(x(0)\) and \(\dot{x}(0)\) are the structural displacement and velocity at time \(t=0\). The solution of the equation is  
\[x(t)=A\exp(-\xi\overline{\omega}_{\rm n}t)\sin(\overline{\omega}_{\rm d}t), \tag{2}\]  
where \(A=\frac{v_{0}}{\overline{\omega}_{\rm d}^{2}}\), \(\overline{\omega}_{\rm n}=\sqrt{\frac{k}{m}}\), and \(\overline{\omega}_{\rm d}=\overline{\omega}_{\rm n}\sqrt{1-\xi^{2}}\). \(\overline{\omega}_{\rm n}\) is the natural frequency of the structure, \(\overline{\omega}_{\rm d}\) is the natural frequency of the structure with damping, and \(\xi\) is the damping ratio. According to the boundary conditions of the bounding \(\ddot{x}(t_{0})=0\), the bounding time can be calculated as follows:  
\[t_{0}=\frac{\pi-arctan\lambda}{\overline{\omega}_{\rm n}\sqrt{1-\xi^{2}}}, \tag{3}\]  
where \(\lambda=\frac{2\sqrt{1-\xi^{2}}}{1-2\xi^{2}}\). If we substitute \(\ddot{x}(t_{0})=0\) into Equation (1), the equation becomes  
\[\dot{c}\dot{x}(t_{0})+kx(t_{0})=0. \tag{4}\]  
The velocity after the bounding \(\dot{x}(t_{0})\) can be calculated by solving the following equation:  
\[\dot{x}(t_{0})=\frac{v_{0}}{1-2\xi^{2}}\exp\biggl{[}-\frac{\xi}{\sqrt{1-\xi^{ 2}}}(\pi-\arctan\lambda)\biggr{]}\cdot\cos(\pi-\arctan\lambda). \tag{5}\]  
The velocity ratio before and after the bounding is  
\[e=\frac{\dot{x}(t_{0})}{v_{0}}=\frac{1}{1-2\xi^{2}}\exp\biggl{[}-\frac{\xi}{ \sqrt{1-\xi^{2}}}(\pi-\arctan\lambda)\biggr{]}\cdot\cos(\pi-\arctan\lambda). \tag{6}\]  
As the average impact force is \(\overline{F}\), according to the momentum theorem, the relationship between the average bounding force and velocity is as follows:  
\[\overline{F}t_{0}=m\Bigl{(}v_{0}-\dot{x}(t_{0})\Bigr{)}=mv_{0}(1-e). \tag{7}\]As \(A=\frac{v_{0}}{\omega_{\rm d}}\), \(\widetilde{\omega}_{\rm n}=\sqrt{\frac{k}{m}}\), and \(\widetilde{\omega}_{\rm d}=\widetilde{\omega}_{\rm n}\sqrt{1-\xi^{2}}\), the average bounding force can calculated by solving the following equation:  
\[ \begin{array}{l}\overline{F}=\frac{m\cdot v_{0}(1-e)}{l_{0}}\\ =m\frac{v_{0}}{l_{0}}\bigg{\{}1-\frac{1}{1-2\xi^{2}}\exp\left[-\frac{\xi}{ \sqrt{1-\xi^{2}}}(\pi-\arctan\lambda)\right]\cdot\cos(\pi-\arctan\lambda) \bigg{\}}\\ =m\cdot\overline{\omega}_{\rm d}A\frac{\widetilde{\omega}_{\rm n}\sqrt{1-\xi^ {2}}}{n-\arctan\lambda}\bigg{\{}1-\frac{1}{1-2\xi^{2}}\exp\left[-\frac{\xi}{ \sqrt{1-\xi^{2}}}(\pi-\arctan\lambda)\right]\cdot\cos(\pi-\arctan\lambda) \bigg{\}}\\ =m\cdot\overline{\omega}_{\rm n}\sqrt{1-\xi^{2}}A\cdot\frac{\widetilde{\omega} _{\rm n}\sqrt{1-\xi^{2}}}{\pi-\arctan\lambda}\bigg{\{}1-\frac{1}{1-2\xi^{2}} \exp\left[-\frac{\xi}{\sqrt{1-\xi^{2}}}(\pi-\arctan\lambda)\right]\cdot\cos( \pi-\arctan\lambda)\bigg{\}}\\ =Ak\cdot\frac{\big{(}1-\xi^{2}\big{)}}{\pi-\arctan\frac{2\sqrt{1-\xi^{2}}}{1-2 \xi^{2}}}\bigg{\{}1-\frac{1}{1-2\xi^{2}}\exp\left[-\frac{\xi}{\sqrt{1-\xi^{2}} }\bigg{(}\pi-\arctan\frac{2\sqrt{1-\xi^{2}}}{1-2\xi^{2}}\bigg{)}\right]\cdot \cos\left(\pi-\arctan\frac{2\sqrt{1-\xi^{2}}}{1-2\xi^{2}}\right)\bigg{\}}.\end{array}