## 6 Steady State Time History and Direct Integration Time History Response  
The steady-state time history solutions by modal superposition analysis were executed by both SAP2000 and GTSRUDL. To compute the steady-state modal solution, eigenvalue/eigenvector analysis must be performed before conducting forced vibrational analysis. A total of 20 and 120 modes were considered in the CTG and STG foundation models, respectively, and the higher modes, i.e., more than 120% of excitation frequency, should be included in the modal solution computation. Then, modal combination using either the SRSS (square root of the sum of squares) or CQC (complete quadratic combination) method is performed to compute the total solution. To simplify, only one cycle response was considered using a periodic time history motion.  
* It requires significant computation memory and time.  
\end{table}
Table 1: Analysis methods for the CTG and STG foundation vibration assessment.  
A complete maximum steady-state (or permanent) velocity response at the bearings of CTG and STG foundations are tabulated in the following four tables (Tables 2-5). A separate table is provided for each directional velocity, i.e., vertical and transverse.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Damping Method** & **V60A** & **H60A** & **V60B** & **H60B** & **V60C** & **H60C** & **V60D** & **H60D** \\ \hline Direct soil damping value with & & & & & & & & \\
2\% material damping SAP 2000 & 0.198 & 0.008 & 0.163 & 0.008 & 0.208 & 0.008 & 0.178 & 0.008 \\ Direct soil damping value with & & & & & & & & \\
20\% cutoff by EPRI plus 2\% & 0.224 & 0.053 & 0.185 & 0.028 & 0.229 & 0.043 & 0.188 & 0.020 \\ material damping SAP 2000 & & & & & & & & \\ \hline Modal damping & SAP 2000 & 0.226 & 0.124 & 0.175 & 0.061 & 0.221 & 0.086 & 0.160 & 0.061 \\
2\% & GISTRUDL & 0.272 & 0.147 & 0.241 & 0.069 & 0.310 & 0.127 & 0.262 & 0.081 \\ \hline Modal damping & SAP 2000 & 0.226 & 0.058 & 0.191 & 0.028 & 0.234 & 0.051 & 0.196 & 0.025 \\
4\% & GISTRUDL & 0.267 & 0.074 & 0.213 & 0.033 & 0.300 & 0.066 & 0.236 & 0.033 \\ \hline Modal damping & SAP 2000 & 0.226 & 0.013 & 0.175 & 0.008 & 0.239 & 0.013 & 0.196 & 0.008 \\
10\% & GISTRUDL & 0.244 & 0.015 & 0.173 & 0.008 & 0.262 & 0.015 & 0.203 & 0.010 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Maximum vertical velocity steady-state response at bearings of STG foundation (cm/s).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Damping Method** & **VTX60A** & **VTX60B** \\ \hline Direct soil damping value with 2\% material damping & & & \\ SAP 2000 & 0.132 & 0.155 \\ Direct soil damping value with 20\% cutoff by EPRI 2\% & & & \\ material damping SAP 2000 & 0.178 & 0.302 \\ \hline Modal damping 2\% & SAP 2000 & 0.191 & 0.351 \\ GTSTRUDL & 0.213 & 0.442 \\ \hline Modal damping 4\% & SAP 2000 & 0.185 & 0.333 \\ GTSTRUDL & 0.201 & 0.391 \\ \hline Modal damping 10\% & SAP 2000 & 0.170 & 0.274 \\ GTSTRUDL & 0.170 & 0.290 \\ \hline Rayleigh damping (\(\alpha\) = 5.3174, \(\beta\) = 0.00017) GTSTRUDL & 0.152 & 0.127 \\ Rayleigh damping (\(\alpha\) = 15.4648, \(\beta\) = 0.0001) GTSTRUDL & 0.147 & 0.122 \\ Rayleigh damping (\(\alpha\) = 32.3771, \(\beta\) = 0.0) GTSTRUDL & 0.142 & 0.117 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Maximum vertical velocity steady-state esposne at bearings of the CTG foundation (cm/s).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Damping Method** & **VTX60A** & **VTX60B** \\ \hline Direct soil damping value with 2\% material damping & & & \\ SAP 2000 & 0.127 & 0.112 \\ Direct soil damping value with 20\% cutoff by EPRI 2\% & & & \\ ATGRTUDL & 0.132 & 0.145 \\ \hline Modal damping 2 \% & SAP 2000 & 0.135 & 0.163 \\ GTSTRUDL & 0.127 & 0.165 \\ \hline Modal damping 4\% & SAP 2000 & 0.132 & 0.152 \\ ATGRTUDL & 0.124 & 0.150 \\ \hline Modal damping 10\% & SAP 2000 & 0.124 & 0.122 \\ ATGRTUDL & 0.117 & 0.114 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Maximum transverse velocity steady-state response at bearings of CTG foundation (cm/s).