## 2 Wind-Resistant Design Parameters and Modal Analysis
### Modal Analysis  
The finite element model (FEM) made of beams elements can accurately predict the eigenmode shapes and eigenfrequencies of cable-stayed bridges that are highly flexible and lightly damped structures, and these frequencies were used to analyze the bridge's response to wind action. The ratio between the torsional and vertical frequencies of homologues mode shapes is critical: The higher this ratio, the higher the flutter velocity, as previously stated. The purpose of the FEM dynamics analysis is:  
1. To gain further understanding of the inherent dynamic properties of bridge structures and to propose design solutions for improving wind resistance.
2. Provide design frequencies and vibration modes for sectional model and full bridge aeroelastic model testing.
3. Provide design parameters for calculating wind vibration response.  
FEM was used to analyze the most unfavorable construction state and the service state of the bridge. The most unfavorable state was chosen as the longest single-cantilever construction state ('construction state' for short). Table 2 shows the basic frequencies of the two states. Figures 3 and 4 shows the fundamental modes of construction and service state, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**States**} & **Design Reference** & **flutter Checking** & **Static Wind Instability Test** \\  & **Wind Speed (m/s)** & **Wind Speed (m/s)** & **Wind Speed (m/s)** \\ \hline construction & 32.21 & 41.39 & 45.34 \\ service & 39.88 & 58.24 & 63.81 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic wind parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{
\begin{tabular}{c} **Construction** \\ **Mode Shape** \\ \end{tabular} } & \multicolumn{2}{c}{**Frequency (Hz)**} & **Mode Shape** & **Frequency (Hz)** \\ \hline L1 & 0.1869 & V2-S & 0.2809 \\ V2 & 0.2688 & L3-S & 0.3613 \\ V4 & 0.4303 & V5-A & 0.4008 \\ T5 & 0.6030 & T8-S & 0.6268 \\ T14 & 1.4145 & T11-A & 0.9377 \\ \hline \hline \end{tabular} Note: V—vertical bending of deck; L—lateral bending of deck; T— torsional bending of deck; S—symmetric mode shape; A—asymmetric mode shape; number—mode order.  
\end{table}
Table 2: Basic frequencies of construction and service state.  
The lateral, vertical, and torsional mode shapes calculated by FEM are shown in Figure 5, respectively. Since the bridge is asymmetric, compared with those of symmetric bridges [6], the first vertical and lateral mode shapes are still symmetric about the mid-span, whereas the first-order torsional mode is no longer symmetrical, and the maximum displacement of positive symmetric torsional mode and 0 displacement of antisymmetric torsional mode appear at the closure rather than the mid-span, as seen in Figure 4a,b. It is assumed that the most unfavorable position of vertical and lateral is in the mid-span, while the most unfavorable position of torsion is in the closure, which will determine the arrangement of sensors in the full bridge aeroelastic model testing. Due to the asymmetry of the structure, the similarity of vertical and torsional modes of the deck is lower than that of the traditional symmetrical structure. Under the same conditions, such as vertical, torsional frequency and the ratio of them, the asymmetric structure is less prone to vertical torsional-coupled flutter.  
Figure 4: _Cont._  
Figure 3: Fundamental modes of construction state. (**a**) Lateral (0.1869 Hz); (**b**) vertical (0.2688 Hz). (**c**) torsional (0.6030 Hz).