## 1 Introduction  
The frequency of a seismic excitation, varying mostly from 0 to 15 Hz [1], indicates that loading frequency is an important factor influencing the liquefaction behavior of soil layers. In order to predict the liquefaction induced by earthquakes accurately, the loading frequency effect should be considered in laboratory tests. The effect of the loading frequency for the liquefaction resistance of sand has been studied for a wide range of loading frequencies from 0.05 to 12 Hz, mostly by conducting cyclic triaxial tests on medium saturated sand, and failure criteria based on the axial strain and excess pore pressure ratio have been randomly employed. Nevertheless, conclusions about the effect of the loading frequency on the liquefaction resistance of sand are contradictory. Several investigators have found that the undrained cyclic strength of saturated sand is negligibly small or uninfluenced by the loading frequency [2; 3; 4; 5; 6; 7; 8]. Using limited data, [PERSON] and [PERSON] [9] investigated the effect of frequency in the range 0.17-4 Hz on the liquefaction resistance of saturated Monterey sand by using a Roscoe-type simple shear apparatus. They noted that the effect of frequency variation was small since their data for medium sand with relative density (D\({}_{\rm r}\)) of 50% were scatted within \(\pm\)10% of the mean shear stress, causing failure in 10 cycles, with no special order prevailing. [PERSON] and [PERSON] [10] conducted cyclic triaxial tests on medium saturated sand at frequencies of 0.17 and 1 Hz and found that the sand appeared to have a slightly lower strength at the lower loading frequencies. They also pointed out that the difference in the observed strength was of the same order of magnitude as the scatter in some of the data for a typical series of tests at one frequency on supposedly identical samples. In contrast, some researchers, including [PERSON] and [PERSON] [11] and [PERSON] et al. [12], have examined the effect of the loading frequency on the cyclic strength in the frequency range 0.05-2 Hz and obtained a consistent result: in the cyclic triaxial test, the liquefaction resistance of sand increases significantly with an increase in the loading frequency, regardless of the relative density state. [PERSON] and [PERSON] [13] investigated the influence of the loading frequency on saturated loose (D\({}_{\rm r}\) = 28%) and dense (D\({}_{\rm r}\) = 70%) sand at isotropic consolidation levels of 100 and 200 kPa by conducting cyclic triaxial experiments. They found that varying the frequency from 0.05 to 1 Hz led to an increase in the cyclic strength, with the increase ranging from 12.4%-33.3%. [PERSON] et al. [14] reported that the liquefaction resistance of loose Monterey sand tested at a frequency of 0.017 Hz was greater than that tested at a frequency of 1 Hz by about 12%. Table 1 summarizes previous studies that have investigated the effect of the loading frequency on the liquefaction resistance of sand. The effect of the loading frequency on the liquefaction resistance of saturated sand has been investigated primarily through cyclic triaxial tests. Except for [PERSON] and [PERSON] [9], mentioned above, we could not find any more studies investigating the effect by using cyclic direct simple shear (CDSS)