Pile foundations in earthquake-prone regions must be analyzed and designed considering the dynamic loads. In the fully nonlinear dynamic analyses, the soil nonlinearity can be considered using the modulus degradation curves in the total-stress approach, and the soil damping is controlled by the unloading/reloading rule. Several researchers have investigated the effect of soil damping on the free-field soil response analyses, but the effect on the soil-pile-structure system response has not been studied thoroughly. In this study, the nonlinear elastic method (hyperbolic model) and the elastoplastic Mohr-Coulomb (MC) models were implemented to investigate the effect of soil damping on the pile and structure response. Dynamic soil-pile-structure interaction analyses were performed by simulating two different centrifuge tests published in the literature, and the analysis results were compared with the centrifuge test results. The analyses with the low-intensity input motions showed that the superstructure accelerations and the bending moments in the single pile were estimated with reasonable accuracy. However, the superstructure accelerations might be underestimated under high-intensity motions, especially in the MC model. Additional analyses were performed under six earthquake records. The constitutive models (MC and hyperbolic) may significantly vary the maximum structural acceleration and pile maximum moments (up to 50%). As a result, the responses of the superstructure and the pile in soil-pile-structure interaction problems are highly dependent on the soil constitutive model that must take the damping into account accurately; in turn, due account must be given to the selection of the constitutive model.