# Calibration of Finn Model and UBSCAN Model for Simplified Liquefaction Analysis Procedures
## 4 Results and Discussion
### Comparisons of Undrained Cyclic DSS Responses  
[PERSON] and [PERSON] [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\({}_{1}\))\({}_{60\mathrm{cs}}\) = 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, \(r_{u}\), 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 Figures 4 and 5 against curves suggested by [PERSON] et al. [29]. The cycle ratio is defined as the number of cycles (N) divided by the number of cycles to liquefaction (\(N_{liq}\)). When the cycle ratio was close to 1.0 (or 0.8-1.0), the excess pore pressure ratio, \(r_{u}\), of high (N\({}_{1}\))\({}_{60\mathrm{cs}}\) 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. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Procedure** & **(N\({}_{1}\))\({}_{60\mathrm{cs}}\) Range** & **Relationship 1** \\ \hline NCEER & 5–25 & \(K_{G}\)\({}^{p}\) = \(0.06X^{3}\) + \(5.65X^{2}\) + \(18.18X\) + 160 \\ HBF & 5–25 & \(K_{G}\)\({}^{p}\) = \(0.79X^{3}\) = \(8.71X^{2}\) + \(66.85X\) + 350 \\ JRA96 & 5–30 & \(K_{G}\)\({}^{p}\) = \(-0.26X^{3}\) + \(9.02X^{2}\) + \(6.74X\) + 280 \\ T-Y & 5–30 & \(K_{G}\)\({}^{p}\) = \(-0.16X^{3}\) + \(6.29X^{2}\) + \(8.00X\) + 210 \\ \hline \hline \end{tabular} \({}^{1}\)\(X\) = ((N\({}_{1}\))\({}_{60\mathrm{cs}}\) - 5).  
\end{table} 
Table 6: Calibrated \(K_{G}\)\({}^{p}\) of the UBCSAND model. Figure 4: Accumulation of excess pore water pressure ratio and the associated cycle ratio of the Finn model. (**a**) HBF; (**b**) JRA96; (**c**) NCEER; (**d**) T-Y. Figure 3: Undrained cyclic DSS loading response for (N\({}_{1}\))\({}_{60\text{cs}}\) = 10 of the JRA96 procedure with vertical effective consolidation stress \(\sigma_{\text{vx}}\)\({}^{\prime}\) = 101 kPa and initial static shear stress = 0. (a) Finn model; (**b**) UBSCAND model. Laboratory tests show that the CRR is related to the number of uniform loading cycles, \(N\) (related to the earthquake magnitude).