# Seismic Study of An Isolated Cable-Stayed Bridge under Near-Fault Ground Motions
## 4 Pounding Mitigation Performance of Seismic-Reduction System
### Non-Uniform Excitation  
Seismic waves undergo both temporal and spatial variations during propagation. For small scale structures like typical industrial and residential buildings, the spatial variation in seismic motion is not significant. Neglecting the variation can often suffice for their seismic design requirements. However, for large-scale structures like extensive bridges, the spatial variation in seismic motion becomes a significant factor [31]. The factors contributing to the spatial variation in seismic motion are complex and primarily include the following three aspects: (1) Characteristics of ground motions, where seismic wave velocity is finite and typically ranges from several hundred meters per second to several thousand meters per second; (2) The seismic source (fault) has a finite size and is not a point source, with rupture velocities comparable to the shear wave velocity in the bedrock; (3) Multiple reflections and refractions of seismic waves along their propagation paths. Due to these complex factors, ground motions at different locations can exhibit differences within spatial scales ranging from tens to thousands of kilometers. The wave passage effect, causing the non-uniform excitation, is a multi-support excitation. However, this issue can be addressed by considering the absolute displacement of the structure. This paper will briefly introduce the corresponding equations of motion. When considering multi-support excitation, the equations of motion for the response of the superstructure and its interaction with the foundation can be expressed as Equation (5):  
\[ \begin{bmatrix}M_{ss}&0&0\\ 0&M_{cc}&0\\ 0&0&M_{ff}\end{bmatrix}\begin{bmatrix}\ddot{u}_{s}\\ \ddot{u}_{c}\\ \ddot{u}_{f}\end{bmatrix} + \begin{bmatrix}K_{ss}&K_{sc}&0\\ K_{cs}&K_{cc}&0\\ 0&K_{fc}&K_{ff}\end{bmatrix}\begin{bmatrix}u_{s}\\ u_{c}\\ u_{f}\end{bmatrix} =R \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Earthquake Name** & **Year** & **Station Name** & **Magnitude** & **R\({}_{\text{jb}}\) (km)** & **R\({}_{\text{up}}\) (km)** & **V\({}_{\text{s}}\)30 (m/s)** \\ \hline \begin{tabular}{c} “Imperial Valley-06” \\ \end{tabular} & 1979 & \"Agrarias\" & 6.53 & 0.68 & 0.65 & 242.05 \\ \begin{tabular}{c} “Imperial Valley-06” \\ \end{tabular} & 1979 & \"Brawley Airport\" & 6.53 & 8.54 & 10.42 & 208.71 \\ \begin{tabular}{c} “Imperial Valley-06” \\ \end{tabular} & 1979 & \"EC County Center FF\" & 6.53 & 7.31 & 7.31 & 192.05 \\ \begin{tabular}{c} “Landers” \\ \end{tabular} & 1992 & \"Yermo Fire Station\" & 7.28 & 23.62 & 23.62 & 353.63 \\ \begin{tabular}{c} “Northridge-01” \\ \end{tabular} & 1994 & \"Rinali Receiving Sta\" & 6.69 & 1.2 & 6.5 & 282.25 \\ \begin{tabular}{c} “Chi-Chi, Taiwan\" \\ \end{tabular} & 1999 & \"TCU036\" & 7.62 & 19.83 & 19.83 & 478.07 \\ \begin{tabular}{c} “Chi-Chi, Taiwan\" \\ \end{tabular} & 1999 & \"TCU038\" & 7.62 & 25.42 & 25.42 & 297.86 \\ \begin{tabular}{c} “Chi-Chi, Taiwan\" \\ \end{tabular} & 1999 & \"TCU039\" & 7.62 & 19.89 & 19.89 & 540.66 \\ \
\begin{tabular}{c} “Cape Mendocino” \\ \end{tabular} & 1992 & \"Bunker Hill FAA\" & 7.01 & 8.49 & 12.24 & 566.42 \\
\begin{tabular}{c} “Cape Mendocino” \\ \end{tabular} & 1992 & \"Centreville Beach\_Naval Fac\" & 7.01 & 16.44 & 18.31 & 459.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the selected earthquake motions. Figure 5: Response spectra of near-fault motions. where subscripts s and f denote nodes within the structure and the foundation, and c represents nodes at the interface between the structure and the foundation. The variable \(u\) represents the displacement relative to the free-field displacement (v), and u represents the corresponding relative acceleration. M and K are the mass and stiffness matrices. It is important to note that the mass and stiffness at the interface nodes (i.e., \(M_{\text{cc}}\) and \(K_{\text{cc}}\)) are contributed by both the structure (s) and the foundation (f) and can thus be expressed as Equation (6):  
\[