Keywords

1 Introduction

The frequency (f) of the actual earthquake motions is random and varies from a few seconds to a few hertz. On the other hand, due to the limitation of the devices, laboratory tests were performed at lower frequencies (usually 0.1 Hz). Hence, f plays an important effect in laboratory testing results.

The effects of cyclic loading frequency on the liquefaction resistance of sand are summarized in Table 1. Some early studies found that increasing f had little impact on sand liquefaction resistance, but some more recent studies found that liquefaction resistance increased as f increased. Interestingly, some findings indicated that liquefaction resistance decreased with increased f or remained unchanged with a small f and increased with f. Therefore, the f effect on sand's liquefaction resistance is an interesting topic that needs further investigation.

A literature review of the effect on liquefaction resistance of the uniformity coefficient (Cu) also is shown in Table 2, which cannot be drawn to a consistent conclusion.

Table 1. The literature review of the effect of f on liquefaction resistance [1]
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Table 2. The literature review of the effect of Cu on liquefaction resistance
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This study used two types of Silica sand with the same mean diameter (D50 = 0.64) but varying Cu and Cc, classified into SP sand and SW sand, to investigate the coupling effect of Cu and f on the liquefaction resistance of the sand.

2 Material and Testing Method

2.1 Materials

Silica sand with a specific gravity of 2.6 was used as a base sand, and several grain sizes of this sand were separated using the dry sieving technique. Two gradation curves for the poorly-graded sand (SP) and the well-graded sand (SW) were established by mixing various grain size groups with different proportions. The material properties of sand and particle-size distribution curves are shown in Table 3 and Fig. 1.

Table 3. Material properties of Silica sand
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Fig. 1.
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Particle-size distribution curve of SP and SW sand

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2.2 CDSS System and Sample Preparation

The testing program was conducted based on the CDSS system manufactured by Geocomp Corporation [1]. SP and SW silica sands in a dry state are poured into a cylinder-shaped steel-reinforced membrane by the dry deposition method to prepare a cylindrical sample with an initial relative density (Dr) of 60% (medium state). The typical samples have an initial height of 25 mm and an initial diameter of 63.5 mm.

All samples are consolidated to confining stress of 100 kPa (i.e.,σ’ = 100 kPa). Then the samples were loaded under harmonic form loading under constant volume conditions at cyclic stress ratios, CSR of 0.1 with a wide range of loading frequency, f (0.03, 0.05, 0.1, 0.2, and 0.5 Hz). The CDSS tests can be terminated when the double amplitude shear strain (DA) has reached 7.5%.

3 Result and Discussion

3.1 CDSS Testing Result

Table 4 summarizes the CDSS test results and the typical cyclic response of SP sand at the medium state with CSR of 0.1 and f of 0.03 Hz is illustrated in Fig. 2. While the test was performed until liquefying, the shear strain oscillated almost symmetrical at zero. The excess pore pressure rapidly increased in the first few cycles, then developed uniformly, leading to the effective vertical stress decreasing to zero at liquefy state.

Table 4. CDSS test results
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Fig. 2.
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Undrained response of Poorly-graded sand (SP) in CDSS test with CSR of 0.1 and f of 0.03 Hz.

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3.2 Effect of F on Ncyc

Figure 3 illustrates the relationship between Ncyc and f for CSR of 0.1. For SP sand, at CSR = 0.1, Ncyc remains unchanged (Ncyc = 70) when frequency increases from 0.03 Hz to 0.1 Hz and rises from 70 to 111 when f increases from 0.1 Hz to 0.5 Hz. In general, SP sand's liquefaction resistance is affected by high f. By contrast, for the SW sand, the effect of f on Ncyc can be neglected. Moreover, the Ncyc of SW sand is higher than that of SP at f less than 0.2 Hz, which leads to the fact that the liquefaction resistance of SW sand is higher than that of SP sand.

Fig. 3.
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Relationship between Ncyc and f

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For practical use, the normalized number of cycles (NNcyc) is proposed by Eq. (1).

$${NN}_{cyc}={N}_{cyc}^{f=iHz}/{N}_{cyc}^{f=0.1Hz}$$
(1)

where \({N}_{cyc}^{f=iHz}\) is Ncyc at f of i Hz, and \({N}_{cyc}^{f=0.1Hz}\) is Ncyc at f of 0.1 Hz.

Figure 4 displays a function of NNcyc and f, and the relationship can be easily used in practice as the reference index.

Fig. 4.
figure 4

Relationship between NNcyc and f

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4 Conclusion

The note presented a laboratory testing analysis on the effect of f on Ncyc of Silica sand with the same mean diameter and different particle size distribution (SP and SW). The CDSS test results showed that when f increased from 0.03 Hz to 0.1 Hz, Ncyc remained unchanged, while it increased when f increased from 0.1 Hz to 0.5 Hz. It can be stated that SP sand's liquefaction resistance is affected by the high f. In contrast, f had little effect on the liquefaction resistance of SW sand. Furthermore, SP sand was more resistant to liquefaction than SW sand. According to this study, Cu and f should be included in the sand's liquefaction resistance analysis.