## 4 Discussion
### Interference Effects of Wind-Induced Response  
The prototype finite element model was established using the ANSYS 17.0 software platform. Yield stress for the two main types of steel, Q345, and Q235, were 310 and 215 MPa, respectively. Q235 is mainly used in auxiliary materials, whereas Q345 is widely used in the main material. Beam 188 and Mass 21 elements were, respectively, used to simulate the members and the mass of the auxiliary facilities, such as machine houses and lifts. By using the subspace method in the modal analysis, the first three natural frequencies and the vibration modal characteristics were obtained, as shown in Table 8. The first two modal shapes are shown in Figure 17.  
Figure 16: Contour plots of the horizontal wind velocity at the height of the jib of the boom-up crane (h = 57 m): (**a**) \(\alpha\) = 30\({}^{\circ}\); and (**b**) \(\alpha\) = 90\({}^{\circ}\). Figure 17: Modal shapes for the FEM model: (**a**) 1 st mode of boom-down crane; (**b**) 2 nd mode of boom-down crane; (**c**) 1 st mode of boom-up crane; and (**d**) 2 nd mode of boom-up crane. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**No.**} & \multicolumn{2}{c}{**Frequency/Hz**} \\  & **Boom Down** & **Boom Up** \\ \hline
1 & 0.335 & 0.289 \\
2 & 0.527 & 0.527 \\
3 & 0.727 & 0.800 \\
4 & 1.091 & 0.867 \\
5 & 1.434 & 1.214 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The first five natural frequencies of the quayside container crane models. The design wind velocity [12] for both boom-down and boom-up quayside container crane is 20 m/s and 40 m/s (3 s gust wind speed) for the quayside crane. The same value was specified in our research for both isolated and group-arranged cranes. The transient dynamic full-time history analysis can then be conducted using the Newmark-\(\beta\) method. The damping ratio is set as 2% and the collection period is 660 s (the first 60 s is discarded avoiding the impact effect). Both wind load and self-weight were considered. According to the nodal deformation wind vibration coefficient \(\beta_{i}\) is adopted, which can be calculated by the following two formulas:  
\[\beta_{i}=\frac{\overline{r}_{i}+r_{i}}{\overline{r}_{i}}=1+\frac{\mu\sigma_{ i}}{\overline{r}} \tag{16}\]  
\[\sigma_{i}^{2}=\sum_{j=1}^{n}{(R_{j}-\overline{r}_{i})^{2}/(n-1)} \tag{17}\]  
where \(\sigma_{i}\) is the mean square error of the deformation of \(i\)-th node. \(\mu\) is gust loading factor and is taken to be 2.5 [39]. \(\overline{r}_{i}\) is the mean deformation. \(r_{i}\) is the RMS value of fluctuating deformation. \(R_{j}\) is the \(j\)-th sampling point of the time history for the deformation response, which can be obtained by the dynamic analysis of the structure under instantaneous wind loads. \(n\) refers to the number of sampling points. The \(\bar{u}\) and \(\sigma\) of displacements for both isolated boom-up and boom-down cranes are shown in Figures 18 and 19. The maximum \(\bar{u}\) and \(\sigma\) displacements always occur at the far end of the crane jib (node 119 for boom-down and node 174 for boom-up state), the height of which is 52 m and 125 m for boom-down and boom-up states, respectively. And these points are also the peak points of the displacement surface locates, which are 0.072 m and 1.01 m for maximum mean deformation and 0.012 m and 0.27 m for maximum RSM deformation. According to the \(\bar{u}\) and \(\sigma\) deformation surface, 75-105\({}^{\circ}\)are the unfavorable azimuths. The maximum wind vibration coefficient is 1.42 and 1.70 for the boom-down and boom-up cranes, respectively. The response of the nodes with maximum deformation, that is node 119 for the boom-down state and node 174 for the boom-up state, when \(\alpha=90^{\circ}\) and \(S/H=0.484\) of both states are demonstrated in Figure 20. In these cases, the IE is the most obvious. As for the freestanding crane, both peak and mean deformation do not change considerably. It attributed to the similar distribution of wind loads for 1# group-arranged crane and isolated one.