## 4 Results
### Reduction of Torsion Effect for the Structures
#### 4.2.2 Structures with Torsion Sensitivity  
The results of the modal analysis are presented in Tables 13 and 14, before and after the reducing of torsion, respectively.  
In the same manner, in Figure 15, the initial configuration is shown and the final configuration, and behaviour at lateral loads is shown in Figure 16.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Period**} & \multirow{2}{*}{**UX**} & \multirow{2}{*}{**UY**} & \multirow{2}{*}{**RZ**} & \multirow{2}{*}{**Sum UX**} & \multirow{2}{*}{**Sum UY**} & \multirow{2}{*}{**Sum RZ**} \\  & **(s)** & & & & & & \\ \hline
1 & 0.389 & 0.035 & 0 & 0.7815 & 0.035 & 0 & 0.7815 \\
2 & 0.341 & 0.7478 & 0 & 0.0362 & 0.7828 & 0 & 0.8177 \\
3 & 0.314 & 0 & 0.7656 & 0 & 0.7828 & 0.7656 & 0.8177 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Ratios of the modal participating mass before the reducing of the torsion.  
Figure 14: The 1 st mode shape for \(T_{2}\) = 0.376 s: (a) plane view, and (b) 3D view.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Period**} & \multirow{2}{*}{**UX**} & \multirow{2}{*}{**UY**} & \multirow{2}{*}{**RZ**} & \multirow{2}{*}{**Sum UX**} & \multirow{2}{*}{**Sum UY**} & \multirow{2}{*}{**Sum RZ**} \\  & **(s)** & & & & & & \\ \hline
1 & 0.239 & 0.7571 & 0 & 0.0008 & 0.7571 & 0 & 0.0008 \\
2 & 0.229 & 0 & 0.7513 & 0.0008 & 0.7571 & 0.7513 & 0.0008 \\
3 & 0.185 & 0.0008 & 0.0000 & 0.7531 & 0.7579 & 0.7513 & 0.7531 \\ \hline \hline \end{tabular}
\end{table}
Table 14: Ratios of the modal participating mass after the reducing of the torsion.  
In this case, we use the sensitivity concept from the theoretical part of expressions (40) and (41). This equation was applied on this case and indeed the structure has sensitivity to torsion:  
\[\frac{T_{rot}}{T_{tr}}\ =\ 2\pi\sqrt{\frac{kmd^{2}}{K_{\varphi}}}\cdot\frac{1}{2\pi} \sqrt{\frac{K_{min}}{m}}\ =\ \sqrt{\frac{kd^{2}K_{min}}{K_{\varphi}}}\ =\ \sqrt{\frac{\frac{1}{12}\cdot 1534.58\cdot 4}{215.5}}\ =\ 1.54\geq 1, \tag{56}\]  
where \(K_{min}\ =\ 4;\,K_{\varphi}\ =\ 215.5;\,k\ =\ 1/12;\,J\ =\ k\cdot md^{2}\ =\ k \cdot m\Big{(}a^{2}+b^{2}\Big{)};\,d^{2}\ =\ \Big{(}a^{2}+b^{2}\Big{)}\ =\ 27.7^{2}\ +\ 27.7^{2}\ =\ 1534.58.  
Figure 16: The 1 st mode shape for \(T_{1}\) = 0.239 s after the reducing of the torsion: (**a**) plane view and (**b**) 3D view.  
Figure 15: The 1 st mode shape for \(T_{1}\) = 0.389 s before the reducing of the torsion: (**a**) plane view, and (**b**) 3D view.  
To confirm the model analysis, we use the sensitivity concept from the theoretical part at expressions (40) and (41). This equation was applied on this case and indeed the structure has no sensitivity to torsion:  
\[\frac{T_{rot}}{T_{tr}}\ =\ 2\pi\sqrt{\frac{kmd^{2}}{K_{\varphi}}}\cdot\frac{1}{2\pi} \sqrt{\frac{K_{min}}{m}}\ =\ \sqrt{\frac{kd^{2}K_{min}}{K_{\varphi}}}\ =\ \sqrt{\frac{\frac{1}{12}.1534.58\cdot 10.2}{2617.4}}\ =\ 0.71<1, \tag{57}\]  
where \(K_{min}\ =\ 10.2\); \(K_{\varphi}\ =\ 2617.4\); \(k\ =\ 1/12\); \(J\ =\ k\cdot md^{2}\ =\ k\cdot m\Big{(}a^{2}+b^{2}\Big{)}\); \(d^{2}\ =\ \Big{(}a^{2}+b^{2}\Big{)}\ =\ 27.7^{2}\ +\ 27.7^{2}\ =\ 1534.58\).