Simplified Liquefaction Prediction and Assessment Method Considering Waveforms and Durations of Earthquakes

Three series of laboratory liquefaction tests were conducted using 11 types of irregular waves with different waveforms and load cycles, as shown in Fig. 4, together with regular waves, comprising a total of 241 test cases. The laboratory tests performed were undrained cyclic hollow cylindrical torsional shear tests for Series 1 and 2 and constant-volume cyclic simple shear tests for Series 3, as listed in Table 1. All of the tests were carried out using reconstituted specimens of silica sand with a median grain size $D_{50}=0.17\mathrm{mm}$ and a uniformity coefficient $U_c=1.7$. The sizes of the hollow cylindrical specimens were 30 mm in internal diameter, 70 mm in external diameter, and 70 mm in height for Series 1, and 60 mm in internal diameter, 100 mm in external diameter, and 100 mm in height for Series 2. The size of the cylindrical specimen, which was confined in 10 circular Teflon polytetrafluorethylene rings each with a thickness of 4 mm, was 100 mm in diameter and 40 mm in height for Series 3. The samples were prepared through air pluviation by adjusting the fall height to achieve the targeted relative densities, and were frozen and set in the cell with subsequent thawing in Series 1, whereas in Series 2 samples were prepared by dry vibratory compaction. In both series, the samples were saturated so that the pore pressure coefficient B became higher than 0.95 and were isotropically consolidated to an effective stress of ${\sigma }_{m0}^{\prime }=98\mathrm{kPa}$, and then subjected to undrained cyclic torsional shear with given seismic waves. Series 3 samples were prepared by dry vibratory compaction and were $K_o$ consolidated to a vertical effective stress of ${\sigma }_{v0}^{\prime }=98\mathrm{kPa}$, and were then subjected to constant-volume cyclic simple shear with given seismic waves. The 11 types of irregular waves shown in Fig. 4, which had a wide variety of waveforms and durations with the number of seismic waves ranging from 10 to over 300, were collected and/or artificially created on the basis of the seismic records of past earthquakes. The time axes of the loaded seismic waves were calibrated and elongated so as to assure accurate control of the loads during the undrained or constant-volume cyclic shear of the samples (Yamazaki and Emoto 2010).

The example test results are shown in Fig. 5. It is seen that, with seismic loading, the excess pore-water pressure built up, and shear strain developed. The definition of liquefaction used for all test cases under the regular and irregular wave loading is such that liquefaction takes place when the double-amplitude shear strains reach ${\gamma }_{\mathrm{DA}}=7.5\%$ (Tatsuoka et al. 1986). Indeed, the excess pore pressure reached the level of the effective confining pressure (${\sigma }_{m0}^{\prime }=98\mathrm{kPa}$) at around ${\gamma }_{\mathrm{DA}}=7.5\%$ in this case.

The cyclic shear stress ratio, ${\tau }_c/{\sigma }_{v0}^{\prime }$, versus the number of load cycles, $N_c$ leading to liquefaction for the regular wave loading is shown in Fig. 6. The liquefaction resistance for the regular waves was defined by the cyclic stress ratio with $N_c=20$, designated as ${\mathrm{CRR}}_{20}$. For the irregular wave loading, the maximum cyclic stress ratios, ${\tau }_{\max }/{\sigma }_{v0}^{\prime }$, were varied, and their relationships with the developed double-amplitude shear strains, ${\gamma }_{\mathrm{DA}}$, were plotted, as shown in Fig. 7. The liquefaction resistance for the irregular waves was defined by the maximum cyclic stress ratio corresponding to ${\gamma }_{\mathrm{DA}}=7.5\%$, designated as ${\mathrm{CRR}}_{\mathrm{IR}}$. The liquefaction resistances, i.e., cyclic resistance ratios ${\mathrm{CRR}}_{20}$ and ${\mathrm{CRR}}_{\mathrm{IR}}$, obtained for all regular and irregular wave loading cases, are summarized in Table 1, with the relative densities, $D_r$, ranging from 40 to 90%.

The concept of effective number of waves introduced to characterize the waveforms and durations or number of load cycles of irregular seismic waves is described here. The theme here is how to represent the severity of given seismic waves, with certain maximum amplitudes, yet having distinctly variable waveforms and durations. The key idea is that small waves relative to the maximum waves may not make a significant contribution to the occurrence of liquefaction due to the lesser cyclic plasticity of the soils (e.g., Sassa and Sekiguchi 2001). In other words, seismic waves having intensities above certain proportions of the maximum waves may effectively contribute to the occurrence of liquefaction. The proportion of such waves was defined as $\beta$, and the relationships between $\beta$, specifically the number of half-waves having intensities above $\beta \times {\tau }_{\max }$, and the liquefaction susceptibility defined by the ratio ${\mathrm{CRR}}_{\mathrm{IR}}/{\mathrm{CRR}}_{20}$ was studied. Example relationships are plotted in Fig. 8 for $\beta =10$, 20, 30, 40, 50, 60, 70, 80, and 90% and $D_r=65\%$. The dotted lines represent power-function regression curves whose correlations were analyzed for all of the test cases in Table 1. The results of this statistical analysis are shown in Fig. 9. The correlation coefficient, $R$, was lower for low values of $\beta$. The $R$-value increased with increasing $\beta$, reached a peak and then decreased with increasing $\beta$. For $D_r=40\%$, however, the $R$-value was consistently high, which stemmed from the smaller number of test cases for very soft soils, as given in Table 1. Overall, the results indicate that there exists an optimal value of $\beta$ at around $\beta =60\%$. This means that the number of half-waves having intensities above $0.6\times {\tau }_{\max }$ had the highest correlations with ${\mathrm{CRR}}_{\mathrm{IR}}/{\mathrm{CRR}}_{20}$. Hence, with reference to Fig. 10, the effective number of waves, $N_{\mathrm{ef}}$, was defined as half the number of half-waves above $0.6\times {\tau }_{\max }$ in the time history of the shear stress variation due to irregular seismic waves, as shown in Fig. 10. By definition, the $N_{\mathrm{ef}}$ value becomes equal to $N_c$ in the case of regular seismic waves.

Ishihara and Yasuda (1973, 1975) classified the waveforms of irregular seismic waves as a shock type and vibratory type, depending on the number of waves preceding the maximum whose amplitudes are greater than 60% of the maximum shear stress ${\tau }_{\max }$. This classification, however, has proven insufficient in characterizing the degree of the irregularity because it was based solely on information regarding the preceding seismic waves before ${\tau }_{\max }$ manifested in a given shear stress variation.

A notable feature of the effective number of waves defined above is that it varies not only by the durations or load cycles of the seismic motions, but also by the degree of irregularity of the seismic waveforms through the shear stress variation, such as impulsive or oscillatory waveforms, even for the same durations. As a consequence, the effective number of waves, together with the conventional maximum shear stress, is capable of representing the severity of given seismic loading, simultaneously taking account of the influence of such waveforms and durations of the irregular seismic motions in a simple yet workable way.

The liquefaction resistance for the irregular seismic waves, ${\mathrm{CRR}}_{\mathrm{IR}}$, is plotted against the effective number of waves, $N_{\mathrm{ef}}$, as shown in Fig. 11. The solid lines represent the power regression curves and the dotted lines will be explained later with reference to the proposed equations. It is seen that the ${\mathrm{CRR}}_{\mathrm{IR}}$ value decreases with increasing $N_{\mathrm{ef}}$, and its dependency varies considerably with $D_r$.

For liquefaction prediction and assessment utilizing the effective number of waves concept, and its wider applications, this subsection considers the normalized liquefaction resistance ${\mathrm{CRR}}_{\mathrm{IR}}/{\mathrm{CRR}}_r$ in terms of the liquefaction resistance ${\mathrm{CRR}}_r$ for a reference value of $N_{\mathrm{ef}}$, namely $N_r$, and defines the term ${\mathrm{CRR}}_{\mathrm{IR}}/{\mathrm{CRR}}_r$ as a wave correction coefficient $c_{\alpha }$:

Eq. (4) indicates that under the conditions $N_{\mathrm{ef}}=N_r$, then $c_{\alpha }=1$ ensues. The index $a$ links $c_{\alpha }$ and $N_{\mathrm{ef}}$ through a power function and depends on $D_r$. Regression analysis of the observed data led to the following form:

Here, Eq. (5) is valid for $a\le 0$, thus, $D_r\ge 2/7$, that is, nearly 30%, and under very soft soil conditions with $D_r<2/7$, it follows that $a=0$ and $c_{\alpha }=1$. Eqs. (4) and (5) indicate that $c_{\alpha }$ increases for $N_{\mathrm{ef}}<N_r$ and decreases for $N_{\mathrm{ef}}>N_r$. The $D_r$ value can be assessed from the widely used correlation with the SPT $N$-value (Meyerhof 1957; Ishihara 1996):

Other widely used correlations between $D_r$ and the CPT $q$-value, or between $N$ and $q$ and $Vs$ (Jamiolkowski et al. 1985; Robertson 1990; Ishihara 1996; Andrus and Stokoe 2000; Andrus et al. 2004, among others) can also be employed for calculating $c_{\alpha }$.

With reference to Fig. 11, the dotted lines represent the values predicted from Eqs. (4) and (5). Here, the condition $N_r=5$ was used, and the corresponding ${\mathrm{CRR}}_r$ values were adopted from the measured ones (solid lines) in Fig. 11. There are reasonably good agreements between the two. The form of the proposed Eqs. (4) and (5) is kept constant for any value of $N_r$, which facilitates universal application of the wave correction coefficient, $c_{\alpha }$.

The wave correction coefficients, $c_{\alpha }$, on the basis of Eqs. (4)–(6) are plotted against $D_r$ and $N$ in Fig. 12. It is seen that the $c_{\alpha }$ value changes more markedly with increasing $D_r$ and $N$. The corresponding procedure for the liquefaction prediction and assessment is described subsequently.

Liquefaction charts have been constructed semiempirically following the past case histories of earthquakes. This means that there exist reference earthquakes for which the charts, or more specifically their boundaries for liquefaction and no liquefaction, were best validated and most consistent results obtained. The effective number of waves, $N_{\mathrm{ef}}$, is first assessed for such a reference earthquake and is used as $N_r$. For instance, the liquefaction chart shown in Fig. 2 was best validated against the 1983 Central Japan Sea Earthquake with the corresponding liquefaction and no-liquefaction cases at Akita Port. The effective numbers of waves at various locations at Akita Port during that earthquake were calculated from the site response analysis using the SHAKE code, and the results are shown in Fig. 13. Here, the numbers of soil layers represent the consecutive numbers of soil layers analyzed for various points and depths at this site. It is seen that the $N_{\mathrm{ef}}$ values ranged from 3 to 7, with an average of 5. Accordingly, $N_r$ was set to 5. Then, one can perform liquefaction prediction and assessment that simultaneously considers the influence of the waveforms and durations of earthquakes by plotting the cyclic stress ratio or the equivalent acceleration in Fig. 2, which is divided by the wave correction coefficient, $c_{\alpha }$, as shown in Fig. 14. Here, the open circles denote example plots with $N_r=5$, and the solid circles and triangles represent the corresponding corrected plots for a lower effective number of waves with $N_{\mathrm{ef}}=1$ and for a higher effective number of waves with $N_{\mathrm{ef}}=10$. The symbol ${\alpha }_e^{\prime }$ represents the corrected equivalent acceleration, namely, the corrected cyclic stress ratio as divided by the wave correction coefficient $c_{\alpha }$. The panel on the bottom shows that the plots can vary, crossing the dividing line between liquefaction and no liquefaction.

It is necessary to check the accuracy of the wave correction coefficient, $c_{\alpha }$. The predicted liquefaction resistances using Eqs. (4) and (5), i.e., the dotted lines in Fig. 11, and the measured liquefaction resistances, i.e., the solid lines in Fig. 11, are plotted as solid circles in Fig. 15. Comparison of the predicted and measured liquefaction resistances confirms the accuracy of the wave correction coefficient, $c_{\alpha }$. In Fig. 15, the predictions from a fatigue theory (Annaki and Lee 1977) are also plotted as open circles for comparison. Here, the liquefaction resistances for the irregular waves were predicted using information on the liquefaction resistance curves for the regular waves shown in Fig. 6. Essentially the same degree of accuracy is observed for both theories.