## 2 Modeling Methodology and Estimation of Input Parameters
### Dynamic Properties of Soil  
The Mohr-Coulomb elasto-plastic soil constitutive model was adopted. This is a representative constitutive model in geotechnical engineering and is widely used in various research works; however, the nonlinear behavior of soil under a strong earthquake cannot be properly simulated with it. For the appropriate simulation of the dynamic behavior of soil-pile-structure system, the nonlinearity of soil under strong earthquake motion should be reasonably modeled. From this perspective, the proper estimation of initial shear modulus and nonlinear variation of shear modulus with shear strain is essential for accurate simulation. In this study, the hysteretic damping model was applied to consider not only the energy dissipation but also the nonlinearity of the soil modulus. As a fitting equation for the hysteretic damping model, the default model that can simulate \(G/G_{max}-\gamma\) curve by using two constants, L\({}_{1}\) and L\({}_{2}\) was adopted. In this model, the \(G/G_{max}-\gamma\) curve can be represented by a cubic equation with zero slope at both low and high strain, as in Equation (1) (Itasca Consulting Group, 2006 [21]).  
\[\mathrm{M_{t}}=\mathrm{s^{2}}(3-2\mathrm{s})-\frac{6\mathrm{s}(1-\mathrm{s})}{ \mathrm{L_{2}}-\mathrm{L_{1}}}\log_{10}\mathrm{e} \tag{1}\]  
where M\({}_{t}\) is the tangent modulus, s = (L\({}_{2}\)-L)/(L\({}_{2}\)-L\({}_{1}\) ), L = \(\log_{10}\) (\(\gamma\)), L\({}_{1}\) and L\({}_{2}\) are the extreme values of L, \(\gamma\) is the shear strain, and e is the void ratio.  
When applying the hysteretic damping model, L\({}_{1}\) and L\({}_{2}\) should be properly estimated according to the soil type and analysis conditions to simulate nonlinear reduction of \(G/G_{max}\) accurately. L\({}_{1}\) and L\({}_{2}\) represent the degradation rate and starting point of degradation of \(G/G_{max}\), respectively. In this study, \(G/G_{max}-\gamma\) curves with various L\({}_{1}\), L\({}_{2}\) were investigated and optimized values for L\({}_{1}\), L\({}_{2}\) were determined by comparison with the \(G/G_{max}-\gamma\) curve obtained by triaxial compression test ([PERSON] et al., 2012 [16]) with Jumunjin sand, which is Korean standard sand. Figure 1 shows the results of calibration of the \(G/G_{max}-\gamma\) curve for Jumunjin sand between the lab test and numerical analysis. As a result, the trace of \(G/G_{max}\) obtained from the numerical analysis matched that obtained from lab test well when L\({}_{1}\), L\({}_{2}\) were determined as 0.5, -3.65, respectively.  
The initial shear modulus generally depends on the soil-confining stress. In this study, the soil initial modulus was obtained by [PERSON] and [PERSON] (1972) [22] as Equation (2). In the equation below, the values of empirical coefficients A, n were determined through regression analysis for the results of bender element tests which were performed by [PERSON] (2009) [23]. In the regression analysis,  
Figure 1: Comparison of measured and calculated \(G/G_{max}\) curve.  
\(\rm G_{max}/F(e)P_{a}\) and \(\rm\sigma_{m}/P_{a}\) for various test cases were plotted, and characteristic values such as A and n was induced.  
\[\rm G_{max}=AF(e)(OCR)^{k}P_{a}^{l-n}(\rm\sigma_{m})^{n} \tag{2}\]  
where \(\rm F(e)=\frac{1}{03+0.7e^{2}}\), \(\rm e\) is the void ratio, \(\rm\sigma_{m}(kPa)\) is the mean principal effective stress (\(\rm\sigma_{m}=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3\)), \(\rm P_{a}(kPa)\) is the atmospheric pressure, k is the over-consolidation ratio exponent (see Table 1), and A and n are empirical coefficients determined as 247.73 and 0.567, respectively.