## 9 Discussion and Conclusions  
The steady-state solutions by modal superposition analyses and direct integration time history analyses of the CTG and STG foundations were computed using several combinations of damping options, which would be generally available in commercial FE software. Frequency-domain steady-state analysis using a frequency-dependent soil-pile interaction was developed, and the effect on hysteretic damping was addressed. Based on the vibration response of the subject foundations, the following conclusions and recommendations can be made from this study.  
1. The steady-state vibration velocity responses computed by SAP2000 and GSTRUDL using modal superposition analysis shows good agreement in both CTG and STG foundations. The cases where there were minor differences are due to the consideration of the incompatible bending mode in the SAP2000 model, which significantly improves local bending behaviour at higher modal frequencies.
2. Modal damping analyses using 4% damping produce consistent vibration performances compared to those with soil radiation damping of 20% (cut-off by EPRI) plus 2% material damping. In other words, the total modal damping by the latter method, which largely depends on the deformation of the soil-foundation interface at the significant modal frequency, would be close to 4%. Modal superposition analyses using 2% and 10% modal damping to some extent overestimate and underestimate the vibration level, respectively. The direct use of damping ratios by the Elastic Half-Space Solution, without cut-off limits, results in an underestimation of the vibration response.
3. The two frequency-point based Rayleigh damping (proportional to mass and stiffness) in the direct integration analysis approximates the replacement of the participating modal damping. The CTG foundation's vibrational responses were almost the same, regardless of the variation of the first modal damping from 4% to 20%, together with 4% damping at 60 Hz. This is primarily due to the fast drop of damping after the first mode and the fast convergence of damping toward 4% until 60 Hz. Furthermore, the 60 Hz vibration response is generally less affected by the first modal damping. After the 60 Hz threshold, the Rayleigh damping increases gradually, but the increase would be negligible for the 60 Hz response.
4. Frequency-dependent vertical and horizontal stiffness gradually increased as the frequency grew. However, frequency-dependent vertical and horizontal damping decreased rapidly as the frequency increased.
5. A full-spectrum steady-state vibration solution compensates for the disadvantage of the modal time history solution, and, thus, increases the reliability of the foundational design. Foundation engineers will be able to estimate the prospective vibration level in a broad frequency range.  
Figure 12: Steady-state vibration analysis considering frequency-dependent interaction: (**a**) hysteretic damping = 1% and (**b**) hysteretic damping = 2%  
**Author Contributions:** Conceptualisation, S.-H.S. Methodology, S.-H.S. Validation, S.S.L. Formal analysis, S.-H.S. Data collection, S.-H.S. Writing--original draft preparation, S.-H.S. Writing--review, S.S.L. Supervision, S.S.L. Project administration, S.S.L. Funding acquisition, S.S.L.  
**Funding:** The research fund of Hanyang University (HY-201700000002431) supported this research and the authors greatly appreciate the financial support.  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study, in the collection, analyses, or interpretation of data, in the writing of the manuscript, or in the decision to publish the results.