## 5 Response of Pulse Seismic Waves to Bridge Piers
### Influence of Pulse Ground Motion on the Hydrodynamic Effect of Bridge Piers
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Pier** **Heights**} & \multicolumn{6}{c}{**Stress (MPa)**} \\ \cline{2-10}  & \multicolumn{3}{c}{**Tg = 0.45 s**} & \multicolumn{3}{c}{**Tg = 0.40 s**} & \multicolumn{3}{c}{**Tg = 0.30 s**} \\ \cline{2-10}  & \multirow{2}{*}{**0 m**} & \multirow{2}{*}{**10 m**} & \multirow{2}{*}{**Error** **(\%)**} & \multirow{2}{*}{**0 m**} & \multirow{2}{*}{**10 m**} & \multirow{2}{*}{**Error** **(\%)**} & \multirow{2}{*}{**0 m**} & \multirow{2}{*}{**10 m**} & \multirow{2}{*}{**Error** **(\%)**} \\ \hline
1 & 13.2089 & 12.9822 & \(-1.72\) & 13.3561 & 13.3728 & 0.13 & 15.5058 & 15.0731 & \(-2.79\) \\
3 & 10.7879 & 10.6136 & \(-1.62\) & 10.877 & 10.8807 & 0.03 & 12.7339 & 12.3408 & \(-3.09\) \\
5 & 8.3838 & 8.2299 & \(-1.84\) & 8.4858 & 8.4699 & \(-0.19\) & 9.9335 & 9.6055 & \(-3.30\) \\
6 & 7.1876 & 7.0582 & \(-1.80\) & 7.2298 & 7.2108 & \(-0.26\) & 8.4513 & 8.1749 & \(-3.27\) \\
8 & 4.7929 & 4.6996 & \(-1.95\) & 4.8314 & 4.8205 & \(-0.23\) & 5.7057 & 5.4977 & \(-3.65\) \\
10 & 2.4628 & 2.4125 & \(-2.04\) & 2.4842 & 2.4679 & \(-0.66\) & 2.9353 & 2.8225 & \(-3.84\) \\
12 & 0.0083 & 0.0078 & \(-6.02\) & 0.0086 & 0.0086 & 0.00 & 0.0155 & 0.0135 & \(-12.90\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stress peak values of different water levels under non-pulse ground motions. Figure 11: Peak acceleration trend diagram of different water levels under pulse ground motions. From Figure 12 and Table 4, the peak stress experienced by a bridge pier subjected to pulse ground motion decreases with increasing bridge pier heights under three distinct characteristic periods and water levels. The presence of the water exerts a negligible influence on the stress response. To examine the relationship between pulse ground motion and the hydrodynamic effect in more detail, according to Formula (6), Figures 13 and 14 present a trend of acceleration and stress pulse effects for different water levels under varying characteristic periods of ground motion. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Pier** \\ **Heights** \\ **(m)** \\ \end{tabular} } & \multicolumn{4}{c}{**Tg = 0.45 s**} & \multicolumn{4}{c}{**Tg = 0.40 s**} & \multicolumn{4}{c}{**Tg = 0.30 s**} \\ \cline{2-10}  & \multirow{2}{*}{**0 m**} & \multirow{2}{*}{**10 m**} & \multirow{2}{*}{**Error** **(\%)**} & \multirow{2}{*}{**0 m**} & \multirow{2}{*}{**10 m**} & \multirow{2}{*}{**Error** **(\%)**} & \multirow{2}{*}{**0 m**} & \multirow{2}{*}{**10 m**} & \multirow{2}{*}{**Error** **(\%)**} \\ \hline
1 & 21.2677 & 21.3105 & 0.20 & 21.1582 & 21.1707 & 0.06 & 21.2333 & 21.2878 & 0.26 \\
3 & 20.2414 & 20.208 & \(-\)0.17 & 19.6423 & 19.5614 & \(-\)0.41 & 20.0871 & 20.1301 & 0.21 \\
5 & 16.5325 & 16.472 & \(-\)0.37 & 15.8153 & 15.693 & \(-\)0.77 & 16.3772 & 16.4029 & 0.16 \\
6 & 14.3072 & 14.254 & \(-\)0.37 & 13.664 & 13.5505 & \(-\)0.83 & 14.1611 & 14.1746 & 0.10 \\
8 & 9.5704 & 9.5094 & \(-\)0.64 & 9.1408 & 9.0422 & \(-\)1.08 & 9.5256 & 9.5153 & \(-\)0.11 \\
10 & 4.7636 & 4.7294 & \(-\)0.72 & 4.5588 & 4.5036 & \(-\)1.21 & 4.7181 & 4.7062 & \(-\)0.25 \\
12 & 0.0344 & 0.0330 & \(-\)4.07 & 0.0310 & 0.0270 & \(-\)12.9 & 0.0280 & 0.0310 & 10.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Stress peak values of different water levels under non-pulse ground motions. Figure 12: Peak stress trend diagram of different water levels under pulse ground motions. As shown in Figure 13, the existence of the pulse enhances the acceleration response of the bridge pier. For a characteristic period of Tg = 0.45 s, the pulse effect intensifies as the height of the bridge pier rises. Conversely, for Tg = 0.40 s, the acceleration pulse effect initially increases with height but then begins to decrease as the pier's height continues to rise. In the case of Tg = 0.30 s, the acceleration pulse effect consistently increases with the bridge pier's height.