# Effect of Soil Damping on the Soil-Pile-Structure Interaction Analyses in Cohesionless Soils
## 2 The Method
### Soil Models  
The first model is the hyperbolic model, where the modulus degradation curves govern the nonlinear soil behavior, and the ultimate strength is not defined explicitly. The reference strain concept defines the nonlinear soil behavior in the hyperbolic model. Figure 1 shows a loading-unloading-reloading cycle in the hyperbolic model. The second model is the Mohr-Coulomb (MC), in which the shear strength limits the ultimate stress that can be sustained. Figure 1 depicts a loading-unloading-reloading cycle in the MC model. Both models employ [PERSON]'s rule for unloading/reloading behavior. However, the resulting damping ratio could be higher in the MC model, depending on the shear strain. According to Figure 1, the stress-strain behavior in MC for strains up to the yield strain \(\gamma_{\text{m}}\) is the same as in the hyperbolic model. However, the dissipated energy is greater in the MC when the accumulated shear strain is greater than the yield strain. Therefore, the models yield different damping values, particularly at shallow depths where the shear strength of cohesionless soils is significantly lower. Figure 1: Shear stress–strain loops: (**a**) Hardin/Dmrevich hysteretic damping, (**b**) Mohr–Coulomb model with Hardin/Dmrevich hysteretic damping. The definition of hysteretic damping is the ratio of the energy dissipated in one cycle to the maximum stored energy. The damping ratio for the stress-strain loops shown in Figure 1a can be calculated using Equation (1) in the hyperbolic model. \[\mathrm{D_{masing}}=\frac{2}{\pi}\left\{2\frac{1+\frac{\gamma_{c}}{\gamma_{ref}} }{\left(\frac{\gamma_{c}}{\gamma_{ref}}\right)^{2}}\left[\frac{\gamma_{c}}{ \gamma_{ref}}-\ln\left(1+\frac{\gamma_{c}}{\gamma_{ref}}\right)\right]-1\right\} \tag{1}\]  
Equation (1) includes the \(\gamma_{c}\), the shear strain due to the dynamic load, and \(\gamma_{ref}\), the soil property defining the modulus degradation curves. In addition to the parameters used in the hyperbolic model (\(\gamma_{c}\) and \(\gamma_{ref}\)), the MC model includes the strain at which the maximum shear stress is sustained, \(\gamma_{m}\). If the cyclic strain \(\gamma_{c}\) is less than the \(\gamma_{m}\), the resulting damping ratio becomes equal to the one in the hyperbolic model. Once the strain \(\gamma_{c}\) is greater than \(\gamma_{m}\) in the MC model, the damping ratio formulation is as in Equation (2). Equations (1) and (2) were given in the FLAC\({}^{\mathrm{3D}}\) manual [32]. However, in this study, the denominator of the multiplication has been corrected to \(1/(\gamma_{c}/\gamma_{m})\). \[\mathrm{D_{masing}}=\frac{2}{\pi}\left\{2\frac{1+\frac{\gamma_{m}}{\gamma_{ ref}}}{\left(\frac{\gamma_{m}}{\gamma_{ref}}\right)^{2}}\left[\frac{\gamma_{m}}{ \gamma_{ref}}-\ln\left(1+\frac{\gamma_{m}}{\gamma_{ref}}\right)\right]-1 \right\}\frac{1}{\frac{\gamma_{c}}{\gamma_{m}}}+\frac{2}{\pi}\frac{(\gamma_{ c}-\gamma_{m})}{\gamma_{c}} \tag{2}\]  
In this study, the soil's small strain shear moduli (\(G_{max}\)) were determined by Equations (3) and (4) ([PERSON] and [PERSON] [16]), and the bulk modulus is calculated by elastic theory using Poisson's ratio. The maximum shear modulus computed using Equation (3) was applied to the soil domain considering the initial confining stresses. However, a minimum cut-off value of 10 kPa was applied to \(G_{max}\) to prevent the soil domain near the ground surface from having an unrealistically low initial soil modulus. \[G_{max}=21.7\left(K_{2}\right)_{max}\,p_{a}\,\left(\frac{\sigma^{\prime}{}_{m} }{p_{a}}\right)^{0.5} \tag{3}\]  
\[\left(K_{2}\right)_{max}=3.5\,\left(D_{R}\right)^{2/3} \tag{4}\]  
The nonlinear stress-strain behavior of soils can be considered by normalized shear modulus reduction (\(G\)/\(G_{max}\)) curves. Many researchers have studied the nonlinear behavior of cohesionless soils under cyclic loads, and reduction curves have been suggested [14; 15; 16]. The curves suggested by [PERSON] and [PERSON] [16] have been implemented in the verification analyses presented by [PERSON] et al.