# Structural Responses of a Superall Building Subjected to a Severe Typhoon at Landfall
## 4 Results and Discussion
### Comparison with Finite Element Results  
A finite element model is established via a beam-slab modeling method in accordance with the structural information of the studied building at its design stage. Twelve-node beam elements and four-node elastic plate elements are employed to model the main body of the structure, three-dimensional link elements are adopted to model the supporting components, and mass elements are used to model the live loads and the nonstructural components. Modal analysis is performed based on the established model via the ETABS software. Relevant results are shown in Figure 16 and Table 6.  
Figure 15: Mode shapes identified via SSI and root-mean-square (RMS) methods against code predictions for the first two swaying modes (i.e., Mode 1 and Mode 3) and the first torsional mode (Mode 2). (**a**) Dir-X; (**b**) Dir-Y.  
Through comparison, the natural frequencies from simulations are found to be collectively lower than the measured values, with the maximum relative difference equal to 13.3%. The discrepancy may be attributable to the following potential reasons. (1) During the simulation process, the live loads for the building were overestimated with respect to real conditions, and the modal mass of the building model was overestimated correspondingly. (2) As the building was constructed, its stiffness may be increased slightly due to the contribution of non-structural components such as doors, windows, and walls, which were not taken into account during the numerical simulation. (3) Because the structure of the studied building is very complex, many uncertainties may exist in the numerical model, which tends to result in simulation errors. It is noted that such discrepancies were also reported in many previous studies [24; 50]. However, more efforts are still required to explore the detailed reasons.  
To examine the working performance of numerical simulation in term of mode shape, the modal assurance criterion (MAC) and the normalized modal difference (NMD) are adopted [24]:  
\[\begin{split} MAC(\{\psi_{i}^{A}\},\{\psi_{i}^{B}\})& =\frac{\left|\{{\psi_{i}^{A}}\}^{T}\{\psi_{i}^{B}\}\right|^{2}}{ \left({{\psi_{i}^{A}}\}^{T}\phi_{i}^{A}\right)\}({\{\psi_{i}^{B}\}}^{T}\phi_{ i}^{B}\})\\ NMD(\{\psi_{i}^{A}\},\{\psi_{i}^{B}\})&=\sqrt{ \frac{1-MAC(\{\psi_{i}^{A}\},\{\psi_{i}^{B}\})}{MAC(\{\psi_{i}^{A}\},\{\psi_{ i}^{B}\})}}\end{split} \tag{19}\]  
in which \(\phi_{i}^{A},\phi_{i}^{B}\) denote the \(i^{\text{th}}\) mode shape vectors from measurement and simulation, respectively. If MAC = 1, it means the two mode shapes are identical, while when MAC = 0, it indicates the two mode shapes are completely different. In contrast, a smaller NMD value indicates a better correlation between the two mode shape vectors.  
Table 7 lists the values of MAC and NMD between the measured and calculated mode shapes. It can be observed that most of the MAC values are greater than 0.88, which indicates that the measured and simulated mode shapes show good agreement. It is also observed numerical simulation tends to provide less accurate prediction of mode shapes for higher order modes.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Mode No.** & **Measurement (Hz)** & **Simulation (Hz)** & **Difference (\%)** & **Mode Type** \\ \hline
1 & 0.183 & 0.164 & 10.4 & 1 st mode in X direction (swap) \\
2 & 0.182 & 0.167 & 8.24 & 1 st mode in Y direction (swap) \\
3 & 0.429 & 0.388 & 9.56 & 1 st mode in Z direction (torsion) \\
4 & 0.683 & 0.592 & 13.3 & 2 nd mode in X direction (swap) \\
5 & 0.656 & 0.612 & 6.71 & 2 nd mode in Y direction (swap) \\ \hline \end{tabular}
\end{table}
Table 6: Results of natural frequencies of Leatop Plaza.  
Figure 16: Mode shapes for the first five orders from simulation via finite element method.