## 5 Results and Discussions
Comparisons show that TCU071\(-\)N had more high\(-\)frequency waves than TCU063\(-\)N and HWA027\(-\)N. In addition, amplitudes of FFT for the vertical motion were higher than those for the horizontal motion at different frequency ranges (TCU071\(-\)N at 9-14 Hz, TCU063\(-\)N at 0.2-0.3 Hz, and HWA027\(-\)N at 1.8 Hz). Figure 11: Time histories and FFT of TCU071\(-\)N case. Figure 12: Time histories and FFT of TCU063\(-\)N case. To understand the effects of the frequency content of the vertical motion comprehensively, additional simulations for TCU017\(-\)N and HWA027\(-\)N listed in Table 6 were performed. Figures 14 and 15 compare the amplitude of FFT of input motions for different simulation cases. The observations are summarized as follows:  
1. A comparison of all TCU017\(-\)N cases shows that the decrease in the uplift is caused by the wave in the frequency range of 9-14 Hz in the vertical motion. When the amplitude of FFT in a frequency range of 9-14 Hz in the vertical motion is no longer higher than the amplitude of FFT at the same frequency range in the horizontal motion, the uplift ratio of TCU071\(-\)N cases increases from 0.77 in the 1.00 V case to 1.00 in the 0.33 V case. 2. A comparison of the 2.00 V case of TCU071\(-\)N and all HWA027\(-\)N cases shows that the increase in the uplift is caused by the wave in the frequency range of 1-2 Hz (1.2 Hz for TCU071\(-\)N and 1.8 Hz for HWA027\(-\)N). When the amplitude of FFT at a frequency range of 1-2 Hz in the vertical motion is no longer higher than the amplitude of FFT in the same frequency range in the horizontal motion (0.33 V/0.66 V/1.00 V cases of TCU071\(-\)N and 0.25 V case of HWA027\(-\)N), the uplift ratio decreases. 3. The predominant period, \(T\), of the soil column in the centrifuge test is calculated as follows: \[T=4\times H/V_{s},\] (1) where \(H\) is the height of the soil column in the centrifuge test (which is 10 m), and V\({}_{\rm s}\) is the shear wave velocity of the soil (as listed in Table 2). In this study, for  
Figure 13: Time histories and FFT of HWA027\(-\)N case. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Dr = 30\%** & **Uplift Ratio** & **Peak Input Acc. Ratio** & **Ia Ratio** & **CAV Ratio** \\
**Case** & **(UPVM/UP) (m)** & **(V/N\(-\)S) 1 (g)** & **(V/N\(-\)S) (m/s)** & **(V/N\(-\)S) (g\(-\)s)** \\ \hline TCU071\(-\)N (Chi\(-\)Chi earthquake) & 0.77 (0.836/1.087) & 0.62 (0.44/0.71) & 0.21 (2.49/11.76) & 0.47 (1.80/3.87) \\ TCU063\(-\)N (Chi\(-\)Chi earthquake) & 1.04 (0.796/0.771) & 0.93 (0.13/0.14) & 0.63 (0.75/1.20) & 0.81 (1.21/1.49) \\ HWA027\(-\)N (Hualian earthquake) & 1.13 (0.313/0.278) & 0.69 (0.24/0.35) & 0.30 (0.43/1.41) & 0.58 (0.52/0.90) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Intensity indices and uplift magnitude of selected Dr = 30% cases. \(Dr=30\%\) to 70%, \(T\) ranges from 0.31 s (4 \(\times\) 10 m/130 m/s) to 0.2 s (4 \(\times\) 10 m/200 m/s). The natural frequency of the soil column ranges from 3.22 Hz to 5 Hz. When soil is liquefied, the shear wave velocity of soil will decrease depending on the severity of the liquefaction, causing a decrease in the natural frequency of the soil column. Therefore, a possible explanation for the wave in the frequency range of 1-2 Hz increasing the uplift might be that this wave frequency range is close to the natural frequency of the liquefied soil. When the wave frequency range (9-14 Hz) is much higher than the natural frequency (3.22-5 Hz) of the soil column, these high frequency waves cause transient responses in the excess pore pressure and lower the uplift. [PERSON] and [PERSON] [2] and [PERSON] et al. [6] used different Kobe earthquake motions to study the effects of vertical motion on uplift magnitude. [PERSON] and [PERSON] [2] showed that the vertical motion has a minimal effect on uplift magnitude, but [PERSON] et al. [6] showed that the magnitude of uplift increases with the amplitude of vertical motion.