# Structural Responses of a Superall Building Subjected to a Severe Typhoon at Landfall
## 4 Results and Discussion
### Identification of Modal Parameters
#### 4.3.3 Mode Shapes  
As discussed in the introduction section, a typical merit of modal identification based on multiple levels of records lies in the ability to determine mode shapes. The following forms are recommended in the national load code of China, i.e., GB50009-2012 [48], for depicting the mode shapes for the first swaying mode \(\phi_{s1}\) and the first torsional mode \(\phi_{\theta 1}\), respectively:  
\[\phi_{s1}=\frac{6\mathrm{z}^{2}H^{2}-4\mathrm{z}^{3}H+\mathrm{z}^{4}}{3H^{4}} \tag{17}\]  
Figure 14: Amplitude-dependence of natural frequencies and damping ratios for the first two swaying modes and the first torsional mode along two measurement directions. (**a**) 1 st swaying mode; (**b**) 1 st torsional mode; (**c**) 2 nd swaying mode.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Mode**} & \multicolumn{2}{c}{**1 st Swaying**} & \multicolumn{2}{c}{**1 st Torsional**} & \multicolumn{2}{c}{**2 nd Swaying**} \\ \hline \multicolumn{2}{c}{**Direction**} & \multicolumn{1}{c}{**Y (S–N)**} & \multicolumn{1}{c}{**X (E–W)**} & \multicolumn{1}{c}{**Y (S–N)**} & \multicolumn{1}{c}{**X (E–W)**} & \multicolumn{1}{c}{**Y (S–N)**} & \multicolumn{1}{c}{**X (E–W)**} \\ \hline \multirow{3}{*}{\(f\) (Hz)} & Approach & 0.185 & 0.186 & 0.437 & 0.437 & 0.673 & 0.699 \\  & Nearest & 0.181 & 0.183 & 0.429 & 0.429 & 0.656 & 0.683 \\  & Leaving & 0.182 & 0.186 & 0.433 & 0.433 & 0.663 & 0.689 \\ \hline \multirow{3}{*}{\(\xi\) (\%)} & Approach & 0.490 & 0.450 & 0.410 & 0.340 & 0.470 & 0.710 \\  & Nearest & 0.840 & 0.900 & 1.03 & 0.910 & 0.710 & 0.890 \\  & Leaving & 0.620 & 0.530 & 0.480 & 0.490 & 0.610 & 0.720 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Values of natural frequency and damping ratio during three selected periods.  
\[\phi_{\theta 1}=\left(\frac{z}{H}\right)^{0.8} \tag{18}\]  
where \(z\) denotes height, and \(H\) stands for building height.  
Figure 15 compares the measured mode shapes with those predicted by the above equations. Besides the SSI method, another method is also utilized to identify mode shapes based on field measurements, i.e., the RMS method, which assumes that the mode shapes can be quantified by the proportional relationship of root-mean-square values for the associated modal component at different testing points. From the figure, results obtained via different methods show good agreement, indicating the credibility of the adopted modal identification methods as well as the empirical predictors recommended in code. Note that the first swaying mode shape deviates from a linear pattern, which is against the basic assumption of linear distribution for the first mode shape involved in the high-frequency force balance (HFFB) wind tunnel tests. Thus, to obtain more accurate testing results, such nonlinear effects should be specially taken into account and compensated [49]. It is also observed that, for the Leatop Plaza building, which adopts a structural system consisting of a steel frame and a reinforced concrete core tube, the first swaying mode shape is convexly curved along height, while the first torsional mode shape is concaved.