4.1. Finn Model
Table 4 and
Table 5 list the calibrated inputs for all four simplified liquefaction analysis procedures. The friction angle (φ) was φ
peak in
Table 3 with a small modification, and the dilation angle, Ψ (= φ − φ
cv), is the difference between φ and φ
cv. Cohesion (
C) was assigned a value of 1 kPa for numerical simulation stability. The model parameters
C1 and
C3 followed the suggested equations and values of Byrne [
15]. The Finn model was combined with a FLAC built-in hysteretic model (the Hardin hysteretic model) [
27] to model the shear modulus degradation curve of sand [
25]. The shear modulus reduction equation of the Hardin hysteretic model is expressed as follows:
where
G is the current shear modulus under the cyclic loading,
Gmax is the small strain shear modulus (i.e.,
G in
Table 4),
γ is the current shear strain under the cyclic loading, and
γref is the reference shear strain corresponding to
γ when the shear modulus reduction ratio is 0.5 (=
G/
Gmax).
Table 5 shows that for NCEER and HBF procedures, the values of
C_Finn at (N
1)
60cs = 25 could not be determined because the excess pore pressure ratio,
ru, was not able to meet the liquefaction criteria. Values of
C_Finn (=
C1 ×
C2) for (N
1)
60cs between the values listed in
Table 5 could be linearly interpolated.
4.3. Comparisons of Undrained Cyclic DSS Responses
Beaty and Perlea [
28] compared the responses of the Finn model and the UBCSAND model from dynamic analyses of an embankment. However, the element responses between the Finn model and the UBCSAND model were not discussed. In this section, the element responses of undrained cyclic DSS simulations of the Finn model and the UBCSAND model are compared and discussed in terms of (1) stress–strain relationships (shear stress–shear strain, shear stress–vertical effective stress, and excess pore pressure accumulation); and (2) factors affecting the CRR (the number of uniform loading cycles, overburden stress, and static shear stress).
Figure 3 shows the relationships between shear stress–shear strain and shear stress–vertical effective stress. The responses of JRA96 (N
1)
60cs = 10 cases were selected as an illustration. Both models treat the unloading response as elastic to simplify the formulation of the model. The accumulation of the volumetric strain (i.e., the accumulation of excess pore pressure ratio,
ru, or the decrease in the vertical effective stress) of the Finn model occurs every half cycle of loading, whereas the UBCSAND model accumulates during each cycle of loading. The key features that a liquefaction constitutive model attempts to simulate are: (1) banana loop—the plastic shear modulus reduces and increases during a shear stress reversal (from positive shear stress to negative shear stress and vice versa) when the soil reaches the initial liquefaction; and (2) butterfly loop—a significant drop in the vertical effective stress and a subsequent increase in the vertical effective stress when the soil reaches the initial liquefaction.
The stress–strain curve and the stress path in
Figure 3b show that the UBCSAND model could imitate the banana loop and the butterfly loop well because the model tracks the stress ratio history to account for the loading reversal effect on the plastic shear modulus. In addition, the UBCSAND model captured the accumulation of the shear strain during cyclic loading better than the Finn model.
Curves of excess pore water pressure generation are shown in
Figure 4 and
Figure 5 against curves suggested by Seed et al. [
29]. The cycle ratio is defined as the number of cycles (
N) divided by the number of cycles to liquefaction (
Nliq). When the cycle ratio was close to 1.0 (or 0.8~1.0), the excess pore pressure ratio,
ru, of high (N
1)
60cs cases (= 20, 25, 30) of the UBCSAND model was relatively low (
Figure 5) because of the high dilation angle of these cases. Nevertheless, in general, both models were able to capture the general trend of the accumulated excess pore pressure during cyclic loading.
Laboratory tests show that the CRR is related to the number of uniform loading cycles,
N (related to the earthquake magnitude). The relationship between the CRR and
N can be approximated with a power function:
where
a and
b are determined by regression against the experimental data. The normalized CRR (CRR/CRR
N=15) versus
Nliq curves (the weighting curves) are plotted in
Figure 6 against the
b = 0.34 (typical value for clean sand) curve [
30]. The weighting curves of both models follow the trend of the typical clean sand curve, indicating that the effects of earthquake magnitude on the CRR could be adequately modeled.
The effects of the overburden stress and the static shear stress on the CRR were compared with published relationships for cases of the NCEER procedure (
Figure 7 and
Figure 8) because only the NCEER procedure includes these effects in the CRR calculation. The overburden stress effect is represented by
Kσ [
30] and compared with the proposed relationships [
10]:
where
σvc′ is the vertical effective consolidation stress,
Pa is the atmospheric pressure, and
f is the model constant (
DR = 40~60% and
f = 0.8~0.7,
DR = 60~80% and
f = 0.7~0.6). The parameter
m_hfac1 (refer to
Table 2 and
Table 3) of the UBCSAND model is used to include the overburden stress effect. Using the calibrated relationships of
m_hfac1 [
19],
Figure 7 shows that the
Kσ of the UBCSAND model was in good agreement with the proposed curves [
10]. The Finn model could capture the decrease in
Kσ with increasing
σvc′, but the effect of
DR on
Kσ did not coincide well with those of the curves from the NCEER procedure.
The effect of static shear stress on the CRR is represented by
Kα [
31] and compared with experimental data [
32].
Figure 8 shows that the
Kα of the UBCSAND model was in good agreement with experimental data. The Finn model captured the trend of
Kα at the case of (N
1)
60 = 5 only. In summary,
Kσ and
Kα were poorly modeled by the Finn model because the dilatant behavior of sand cannot be modeled appropriately by the simple formulation of the Finn model.