## 3 Numerical Study
### Numerical Study II: Comparison of Responses with Reduced Degrees of Freedom
For the spatial evaluation from 15 elements onward (for some cases even starting from 3 elements), maximum convergency is observed for top-floor displacements and accelerations. This is tabulated in Table 6. Similar observation can be made for Table 7. The tabulated and compared quantities are the maximum and the RMS of displacements/accelerations at the top. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No. of Elements** & \(\mathbf{Disp_{max}}\) & \(\mathbf{Disp_{RMS}}\) & \(\mathbf{Acc_{max}}\) & \(\mathbf{Acc_{RMS}}\) \\ \hline
1 & 0.1565 & 0.1081 & 0.2323 & 0.0210 \\
2 & 0.2056 & 0.1452 & 0.0745 & 0.0278 \\
3 & 0.2364 & 0.1572 & 0.1576 & 0.0524 \\
15 & 0.2238 & 0.1588 & 0.1319 & 0.0429 \\
30 & 0.2265 & 0.1593 & 0.1386 & 0.0463 \\
60 & 0.2234 & 0.1591 & 0.1307 & 0.0446 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Spatial reduction summary for no turbulence, damping ratio \(\zeta=0.01\), continuous, \(0^{\circ}\) angle of wind. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No. of Elements** & \(\mathbf{Disp_{max}}\) & \(\mathbf{Disp_{rms}}\) & \(\mathbf{Acc_{max}}\) & \(\mathbf{Acc_{rms}}\) \\ \hline
1 & 0.5884 & 0.2233 & 0.4389 & 0.1855 \\
2 & 0.6804 & 0.2922 & 0.5340 & 0.2292 \\
3 & 0.8332 & 0.3266 & 0.8698 & 0.3220 \\
15 & 0.6785 & 0.3099 & 0.5926 & 0.2549 \\
30 & 0.8567 & 0.3201 & 0.9044 & 0.3015 \\
60 & 0.7511 & 0.3061 & 0.6936 & 0.2497 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Spatial reduction summary for with turbulence, damping \(\zeta=0.01\), continuous, \(0^{\circ}\) angle of wind. Figure 9: Top-floor response (acceleration and displacement) time history with no turbulence in the wind. Two damping ratio \(\zeta\) are plotted. This part of this study focuses on the evaluation of an extensive parametric study, showing a clear summary of the comprehensive data available. The values on the vertical (y) axis are normalized with the values of the highest level of spatial resolution. The effects of  
* Wind flow conditions
* Angle of attack
* Presence of natural incoming turbulence
* Structural model  
are plotted for the top-floor displacement in all the above cases in Figures 10-13. It can be observed that for the cases with and without turbulence, the quantity of interest shows similar characteristics. This may be due to the normalization with respect to the highest resolution. As commented earlier, the resolution of 15 spatial elements is accurate in representing the maximum and RMS of displacement at the top. The amount of structural damping plays an insignificant role compared to the spatial resolution of modeling. A similar study was conducted on the model results and is plotted in Figures  
Figure 11: Spatial reduction—Maximum top-floor displacement as a ratio of the finest resolution model against the number of elements in the beam model with no turbulence in the incoming wind. Four damping ratios \(\zeta\) are plotted. The title of each sub-figure is on top. Figure 12: Spatial reduction—RMS of top-floor displacement as a ratio of the finest resolution model against the number of elements in the beam model with turbulence in the incoming wind. Four damping ratios \(\zeta\) are plotted. The title of each sub-figure is on top. Figure 14: Modal reduction—Maximum top-floor displacement as a ratio of the finest resolution model against the number of modes considered with turbulence in the incoming wind. Four damping ratios \(\zeta\) are plotted. The title of each sub-figure is on top. Figure 13: Spatial reduction—RMS of top-floor displacement as a ratio of the finest resolution model against the number of elements in the beam model with no turbulence in the incoming wind. Four damping ratios \(\zeta\) are plotted. The title of each sub-figure is on top. Figure 16: Modal reduction—RMS of top-floor displacement as a ratio of the finest resolution model against the number of modes considered with turbulence in the incoming wind. Four damping ratios \(\zeta\) are plotted. The title of each sub-figure is on top.