## 2 Study Methods and Tests Materials
### The Model of [PERSON] et al. (1976)  
The model proposed by [PERSON] et al. (1976) is an empirical model derived from a series of experiments that were used to predict changes in the excess PWP ratio (Ru) under cyclic loading. This can be expressed as shown in Equation (1):  
\[\mathrm{R_{u}}=\frac{2}{\pi}\mathrm{sin}^{-1}\bigg{(}\frac{\mathrm{N}}{\mathrm{ N_{L}}}\bigg{)}^{1/2\beta} \tag{1}\]  
where N is the cyclic number, N\({}_{\mathrm{L}}\) is the cyclic number required to reach initial liquefaction, R\({}_{\mathrm{u}}\) is the ratio of excess PWP to the effective confining pressure, and \(\beta\) is the parameter of the curve of the model. In the model developed by [PERSON] et al. (1976), \(\beta\) is the key parameter that influences the curve of the model. [PERSON] (1993) also described \(\beta\) as an empirical parameter that is dependent on the soil type, effective stress, or the rate of cyclic loadings ([PERSON] et al., 2000; [PERSON] et al., 2002). Figure 2 presents curves mapped according to different \(\beta\) values. When \(\beta\) was 0.45-2.0, the curves corresponded more closely to the three stages of the WP excitation of non-plastic sandy soils (i.e., the initial excitation stage, stable excitation stage, and the accelerated excitation stage) ([PERSON] et al., 2000; [PERSON] et al., 2002; [PERSON] et al., 2002). When \(\beta\) was 20-250, the curves corresponded to the PWP excitation behavior of plastic soils described by [PERSON] et al. (2007). The PWP increased sharply in the initial stage and leveled off until the initial liquefaction. Of the various models of PWP excitation under dynamic loading, the mathematical model of PWP developed by [PERSON] et al. (2007) can be used to predict fiber content in sandy soils. In contrast to the model developed by [PERSON] et al., the model proposed by [PERSON] et al. (2007) accounts for factors such as Dr, FC, FL, and CSR and integrates cubic curve functions. [PERSON] et al. (2007); [PERSON] et al. (2007) identified a polynomial relationship between Ru and normalized damage. [PERSON] et al. (2007) introduced a general logarithmic model. [PERSON] et al. (2007) demonstrated that Dr, FC, and CSR must be considered when calculating \(\beta\) values. Overall, although the model developed by [PERSON] et al. (1976) is relatively easy to understand, it may be insufficient for the analysis of multiple parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Soils** & **Fines** &
\begin{tabular}{c} **Undisturbed/** \\ **Reconstituted** \\ **Sample** \\ \end{tabular} & **Set for Testing** & **FC (\%)/PI (Fines)** & **Refs.** \\ \hline \multirow{4}{*}{Liugui sands, Taiwan} & \multirow{2}{*}{Liugui fines} & \multirow{2}{*}{Reconstituted} & \multirow{2}{*}{6} & Same deviate stress & \multirow{2}{*}{[PERSON], et al., 2015 [7]} \\  & & & & 270 kPa, FC = 0, 15, 30, 40, 50, 60\% (PI = 0, 3.1, 5.0, 5.9, 6.7, 7.0) & \\ \hline \multirow{4}{*}{Liugui sands, Taiwan} & \multirow{2}{*}{Liugui fines} & \multirow{2}{*}{Reconstituted} & \multirow{2}{*}{3} & Same relative density, & \multirow{2}{*}{[PERSON], et al., 2015 [7]} \\  & & & & 3.1, 5.0 & \\ \hline Liugui sands, Taiwan & clean & Reconstituted & 3 & FC = 0\%, PI = none & This study \\ \hline Liugui sands, Taiwan & Liugui fines & Reconstituted & 6 & FC = 15, 30\%, PI = 5.7 & This study \\ \hline Liugui sands, Taiwan & Mudstone & Reconstituted & 6 & FC = 15, 30\%, PI = 12.4 & This study \\ \hline Liugui sands, Taiwan & Kaolinite & Reconstituted & 6 & FC = 15, 30\%, PI = 32 & This study \\ \hline \end{tabular}
\end{table}
Table 1: Cont.