# Calibration of Finn Model and UBSCAN Model for Simplified Liquefaction Analysis Procedures
## 3 Methodology of Model Calibration  
In this study, the constitutive models are calibrated using a single-element simulation of a cyclic undrained direct simple shear (DSS) test for four simplified liquefaction analysis procedures, including NCEER [10], HBF [11], JRA96 [12], and T-Y [13] procedures. The initial stress condition is the \(K_{o}\) (coefficient of earth pressure at rest) condition. The liquefaction criteria specify that the excess pore pressure ratio, \(r_{u}\) (= excess pore pressure/vertical effective consolidation stress), is equal to or greater than 95% at 15 cycles of uniform shear loading under one atmosphere (101 kPa) vertical effective consolidation stress (\(\sigma_{\text{nc}}\)\({}^{\prime}\)). Given that the laboratory test results for these simplified liquefaction analysis procedures were lacking, the Finn model and the UBCSAND model were calibrated by fitting the liquefaction triggering curves of these simplified procedures (Figure 2). For simplicity and comparison purposes, the constitutive models were both calibrated at selected SPT N-values ((N\({}_{1}\))\({}_{60\text{cs}}\) = 5, 10, 15, 20, 25, 30). (N\({}_{1}\))\({}_{60\text{cs}}\) is the equivalent clean-sand SPT-N value at the energy ratio of 60% and under one atmosphere overburden stress. The primary parameters of the Finn model with [PERSON] formulation are listed in Table 1. In this study, only \(C_{2}\) was adjusted to fit the liquefaction triggering curves. \(C_{1}\) and \(C_{3}\) were values adopted from those proposed by [PERSON] [15], and other parameters were adopted  
Figure 2: Liquefaction triggering curves of four simplified liquefaction analysis procedures. equations suggested by [PERSON] and [PERSON] (1993). Because the Finn model cannot fully simulate the shear modulus degradation under cyclic loadings, it was combined with FLAC's built-in hysteretic model during the calibration to achieve reasonable predictions of shear strain and volumetric strain. The primary parameters of the UBCSAND model are listed in Table 2. The plastic shear modulus, \(G^{p}\), is the key factor affecting the accumulation of the excess pore pressure and the CRR. The calibration process was simplified by adjusting the paraemeters related to plastic shear modulus (\(K_{G}\)\({}^{p}\) and \(np\)) and maintaining other parameters same with the values provided by [PERSON] and [PERSON] (1993). The UBCSAND model can model the shear modulus degradation behavior, and no supplementary hysteretic model is needed. Relationships of the input parameters provided by [PERSON] and [PERSON] (1993) are listed in Table 3 and described as follows. The small strain shear modulus (\(G_{max}\)) is estimated using equations in the literature (25; 26):  
\[G_{max}=21.7\times(K_{2})_{max}\times P_{a}\times(\sigma^{\prime}/P_{a})^{0.5}, \tag{9}\]  
\[(K_{2})_{max}=3.5\times(D_{R})^{2/3}=20\times((\mathrm{N}_{1})_{60\mathrm{cs}}) ^{1/3}, \tag{10}\]  
where \((K_{2})_{max}\) is a modulus parameter, \(P_{a}\) is atmospheric pressure, \(\sigma^{\prime}\) is the mean effective stress in the plane of loading, and \(D_{R}\) is the relative density of sand. Then, \(G_{max}\) and \(K_{G}\)\({}^{e}\) can be expressed as follows:  
\[G_{max}=21.7\times 20\times((\mathrm{N}_{1})_{60\mathrm{cs}})^{1/3}\times P_{a} \times(\sigma^{\prime}/P_{a})^{0.5}, \tag{11}\]  
\[K_{G}\)\({}^{e}=21.7\times 20\times((\mathrm{N}_{1})_{60\mathrm{cs}})^{1/3}\) and \(ne=0.5\), \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter 1** & **Meaning** & **Parameter 1** & **Meaning** \\ \hline _friction_ & friction angle, \(\varphi\) & _ff\_c2_ 2 & \(C_{2}\) of Equation (1) \\ _cohesion_ & cohesion, \(C\) & & \\ _dilation_ & dilation angle, \(\Psi\) & & \\ _shear_ & shear modulus, \(G\) & _ff\_c3_ 2 & \(C_{3}\), threshold shear strain—the shear \\ _bulk_ & bulk modulus, \(B\) & & \\ _ff\_c1_ 2 & \(C_{1}\) of Equation (1) & & \\ \hline \hline \end{tabular} 1
[FOOTNOTE:1]Footnote 1: footnotetext: Parameter title in the FLAC program. 2  
\end{table}