## 4 Results and Discussion  
For every combination of the frequency, initial vertical effective stress, and density, the test was repeated several times (mostly five or six times), and the maximum and minimum number of cycles were excluded to eliminate the influence of data scatter. The mean number of cycles for the remaining data was used for the subsequent analysis. Table 3 summarizes the results of the cyclic resistance performed at different loading frequencies, initial vertical effective stresses, and relative densities after consolidation (D\({}_{\rm rc}\)). For a given loading frequency, density state, and initial vertical effective stress, three distinct tests with different levels of the cyclic stress ratio (CSR) were conducted to obtain a cyclic resistance curve (number of cycles to liquefaction versus CSR). Figure 4 shows the undrained cyclic direct simple shear response of loose sand at \(f=0.05\) Hz (Figure 4a-d) and 0.5 Hz (Figure 4e-h) for \(\sigma_{\mathrm{v0}}{{}^{\prime}}=100\) kPa and CSR = 0.18. As loading proceeded, shear strain was accumulated continuously, and initially, the excess pore pressure rose rapidly. Subsequently, the increase in the excess pore pressure became gradual and was accompanied by a decrease in the initial vertical effective stress. Once the liquefaction criterion was satisfied, the excess pore pressure at the end of the test exceeded 90% of the initial vertical effective stress. A comparison of Figure 4d,h shows that the magnitude of initial vertical effective stress decrease was influenced by the loading frequency. At \(f=0.05\) Hz, the value of \(\sigma_{\mathrm{v0}}{{}^{\prime}}\) decreased approximately 45%, from 100 kPa to 55 kPa in the first cycle. However, at \(f=0.5\) Hz, the value of \(\sigma_{\mathrm{v0}}{{}^{\prime}}\) decreased only 35%, from 100 kPa to 65 kPa in the first cycle. The faster decrease in initial vertical effective stress in the first cycle is resulted in a less numbers of cycles to liquefy. The different changes in specimen fabric under different loading frequency contribute to observed differences in cyclic response. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample** & \(\sigma_{\mathrm{v0}}{{}^{\prime}}\) **(kPa)** & \(\mathbf{D_{\mathrm{rc}}}\) **(\%)** & \(f\) **(Hz)** & \(\mathbf{CRR_{15}}\) &
\begin{tabular}{c} **Percentage Increase in** \\ **Cyclic Resistance (\%)** \\ \end{tabular} \\ \hline \multirow{8}{*}{Loose} & \multirow{4}{*}{50} & 39.93–40.01 & 0.05 & 0.208 & 0 \\ \cline{3-6}  & & 40.02–40.08 & 0.1 & 0.215 & 3.4 \\ \cline{3-6}  & & 40.03–40.08 & 0.5 & 0.222 & 6.7 \\ \cline{3-6}  & & 39.92–40.03 & 1 & 0.228 & 9.6 \\ \cline{3-6}  & & 39.91–40.01 & 0.05 & 0.189 & 0 \\ \cline{3-6}  & & 39.94–40.09 & 0.1 & 0.196 & 3.7 \\ \cline{3-6}  & & 39.95–40.06 & 0.5 & 0.204 & 7.9 \\ \cline{3-6}  & & 40.02–40.05 & 1 & 0.212 & 12 \\ \cline{3-6}  & & 40.04–40.10 & 0.05 & 0.177 & 0 \\ \cline{3-6}  & & 40.04–40.08 & 0.1 & 0.189 & 6.8 \\ \cline{3-6}  & & 39.89–40.01 & 0.5 & 0.193 & 9 \\ \cline{3-6}  & & 39.92–40.04 & 1 & 0.204 & 15 \\ \hline \multirow{8}{*}{Dense} & \multirow{4}{*}{100} & 80.01–80.54 & 0.05 & 0.311 & 0 \\ \cline{3-6}  & & 79.98–80.04 & 0.1 & 0.320 & 2.9 \\ \cline{3-6}  & & 79.98–80.06 & 0.5 & 0.371 & 19 \\ \cline{1-1} \cline{2-6}  & & 80.01–80.02 & 0.05 & 0.256 & 0 \\ \cline{1-1} \cline{2-6}  & & 79.95–80.03 & 0.1 & 0.261 & 2 \\ \cline{1-1} \cline{2-6}  & & 79.97–80.01 & 0.5 & 0.284 & 11 \\ \hline \hline \end{tabular} Note: initial vertical effective stress (\(\sigma_{\mathrm{v0}}{{}^{\prime}}\)); relative density after consolidation (D\({}_{\mathrm{rc}}\)); loading frequency (\(f\)); cyclic resistance ratio corresponds to liquefaction at 15 cycles (\(\mathrm{CRR_{15}}\)); the percentage increase in cyclic resistance at 0.1, 0.5 and 1 Hz relative to the cyclic resistance at 0.05 Hz (percentage increase in cyclic resistance).  
\end{table} 
Table 3: Summary of direct cyclic shear tests.