## 4 Results and Discussion
### Shear Strain and Pore Pressure Accumulations  
This increase in the liquefaction resistance could be related to the rates of shear strain and excess pore pressure accumulations. The accumulated shear strain data were automatically generated by the CDSS system as a function of the number of cycles, and they are plotted in Figure 10. Figure 10a shows the accumulated shear strain for loose sand at various loading frequencies for CSR = 0.12. Initially, the accumulated shear strain had a constant low value, implying that elastic strain was predominant. As the number of cycles increased, the accumulated shear strain increased rapidly, indicating a sudden increase in the plastic strain and, subsequently, the shear strain value at liquefaction was reached.  
Figure 8: Effects of loading frequency (**a**) loose sand and (**b**) dense sand on cyclic resistance.  
Figure 9: Percentage increase in cyclic resistance at 0.1, 0.5 and 1 Hz relative to the cyclic resistance at 0.05 Hz.  
At lower frequencies, the plastic shear strain accumulates more rapidly, leading to the sand becoming unstable in fewer loading cycles. Figure 10b shows the accumulation of shear strain in dense sand for CSR = 0.27. It is not identical to that in loose sand. The accumulation of shear strain was initially rapid and, subsequently, it became gradual and continued until the shear strain value at liquefaction was reached. The lower the frequency, the more rapid the accumulation of shear strain and the shorter the time taken to reach the strain value at liquefaction was. The difference in the manner of shear strain accumulation could be attributed to the dilation-dominant behavior of dense sand in this study.  
The effect of loading frequency on excess pore pressure accumulations is evaluated based on the changes in excess pore pressure ratio (r\({}_{\rm u}=\Delta\)u/\(\sigma_{\rm v0}\)\({}^{\prime}\)) during undrained shearing phase. Figure 11 shows the relationship between the excess pore pressure ratio and the number of cycles to liquefaction at different frequencies in CDSS tests when CSR is 0.12 and 0.18 for the loose sand. It can be clearly seen that the excess pore pressure accumulations are significantly influenced by loading frequency. As the loading frequency increases, the excess pore pressure slowly increases at a constant CSR and \(\sigma_{\rm v0}\)\({}^{\prime}\) values. This results in more cycles to reach liquefaction at a higher loading frequency.  
Figure 11: Accumulation of excess pore pressure under different shear values (**a**) CSR = 0.12 and (**b**) CSR = 0.18 effected by loading frequencies for loose sand at \(\sigma_{\rm v0}\)\({}^{\prime}\) = 200 kPa.  
Figure 10: Evolution of accumulated shear strain (**a**) loose sand for CSR = 0.12 and (**b**) dense sand for CSR = 0.27 effected by various loading frequencies at \(\sigma_{\rm v0}\)\({}^{\prime}\) = 100 kPa.  
Since recent studies on the earthquake-induced liquefaction behavior of sand have mainly been conducted at a loading frequency of 0.1 Hz [27; 28; 29; 30], the cyclic resistances from studies performed at 0.1 Hz may not be applicable to all frequencies. As an earthquake contains a range of frequencies, it seems that 0.1 Hz gives an average response to the range of frequencies the authors examined. In practical terms, compared to the frequency of a seismic excitation, the lower loading frequency used in laboratory tests causes an underestimation of the liquefaction resistance of sand, which could make designs costly. Since the effect of the loading frequency on soil liquefaction is not well understood and results from various studies in the literature are contradictory, more than one loading frequency should be used to assess the cyclic resistance of sand in laboratory studies through CDSS tests. This study was conducted on a limited loading frequency range, and sand from only one location was used. Additional studies are required to derive reliable, generalized conclusions.