# Effect of Soil Damping on the Soil-Pile-Structure Interaction Analyses in Cohesionless Soils
## 2 The Method
### Verification Analyses Using Hyperbolic and MC Constitutive Models  
In this study, two centrifuge tests published in the literature were used for verification purposes. Figure 4 shows the test containers (Gohl [30] and Wilson [31]) in the prototype units. In Gohl [30], a pile with a diameter of 0.57 m was placed inside dry Nevada sand  
Figure 3: The comparison of Masing damping ratio curves for hyperbolic and MC models for (**a**) \(\sigma^{\prime}_{m}=10\) kPa and (**b**) \(\sigma^{\prime}_{m}=100\) kPa. Figure 2: (**a**) Modulus degradation curves of Darendeli [14] (**b**) Masing Damping ratio curves. with a relative density of \(\rm D_{R}=40\%\). The single mass was placed on top of the pile extending to 2.0 m from the ground surface. The diameter and the length of the highly instrumented single pile in Wilson [31] were 0.67 m and 16.7 m, respectively. A 49 Mg mass on the pile created a single degree of freedom system. The free height of the single pile was 3.8 m. The soil in which the piles were embedded was the saturated Nevada sand placed at two different relative densities. The thicknesses of these layers were 9.4 m and 11.3 m, and the relative densities were 55% and 80%, respectively. The damping ratio formulations given in Equations (1) and (2) depend on the reference strain (\(\gamma_{\textit{ref}}\)) and yield strain (\(\gamma_{\textit{m}}\)), where the last of which depends on the shear strength. The friction angles for medium-dense and dense sand were selected as 36\({}^{\circ}\) and 40\({}^{\circ}\) for soil layers in Wilson (1998). The friction angle was assumed to be 34\({}^{\circ}\) for the sand in [PERSON] (1991). The soil parameters selected for the verification analyses are given in Table 1.  
In this study, the numerical model was created in \(\rm FLAC^{3D}\)[32]. The model dimensions in x-y-z directions are 16 m \(\times\) 10 m \(\times\) 12 m and 20 m \(\times\) 51 m \(\times\) 20 m in [PERSON] [30] and [PERSON] [31], respectively. The zone sizes must be fine enough to accurately transmit the waves from the base into the model. Finer zones are used in the vicinity of the pile in the lateral direction, and the zone sizes gradually increase using an aspect ratio of 1.1. In the vertical direction, the dimension of each solid zone is constant. Still, the height of each element must be lower than the limit suggested by [PERSON] and [PERSON] [36], which (\(\Delta\)I) is one-tenth of the wavelength (\(\lambda\)) of the input wave motion (\(\Delta\)I \(<\lambda/10\)). The minimum zone size in the vertical direction is selected (\(\Delta\)I \(<\lambda/10\) = 1.0 m). The bottom boundary of the model was fixed, and the dynamic free-field boundaries were applied to the sides. Due to symmetry, half of the model is presented in Figure 5. \begin{table}
\begin{tabular}{l c c c} \hline  & \multicolumn{2}{c}{**[PERSON], 1998**} & **[PERSON], 1991** \\ \hline
**Layer** & **Layer 1** & **Layer 2** & **Single Layer** \\ \hline Effective unit weight, \(\gamma^{\prime}\) (\(\rm kN/m^{3}\)) & 9.5 & 9.9 & 15.1 \\ Relative density (\%) & 55 & 80 & 40 \\ Friction angle, \(\varphi^{\prime}\) (\({}^{\circ}\)) & 36 & 40 & 34 \\ Dilation angle \(\psi\) (\({}^{\circ}\)) & 4 & 8 & 2 \\ Poisson’s ratio, \(\
u\) & 0.45 & 0.45 & 0.30 \\ \hline \end{tabular}
\end{table} 
Table 1: The soil parameters selected for the verification analyses. Figure 4: (a) Rigid soil container of [PERSON] (1991), and (b) Laminar soil container of [PERSON] (1998). Although the piles and the structure are three-dimensional in nature, they were modeled with the beam elements for simplicity. The structural elements (beam) were rigidly connected to the surrounding soil, where the pile displacement is the same as the deformation of the soil. The rigid connection approach was adopted by several researchers [12, 13, 37]. Since the problem is a laterally loaded pile model under dynamic loading, the key property affecting the response is the flexural stiffness of the pile and the superstructure.