## 5 AAF Numerical Application
### Numerically Equivalent Model of the DDTG  
A 3D finite element model (3D model for short) of the case bridge was established with the commercial software ANSYS, as shown in Figure 13a. It was developed using APDL code, and all structural components, connections and boundary conditions of the bridge were properly simulated. The beam188 is a kind of bar element, which allows the users to apply a real section of the truss component. The shell181 element is suitable for the shell structure with medium thickness. Therefore, in the 3D model, the truss bars are modelled by 3D beam188 elements with real sections, and the bridge decks are modelled by shell181 elements. For the combined roadway and steel decks, the different girder types are shown in Figure 14a,b. The main simplification considered herein to reduce the computational effort of the buffeting analyses, especially for the partial members of the DDTG, is to put forward an equivalent \"fish-bone\" model to replace the 3D model while retaining the global mechanical behaviour of the DDTG. The fish-bone model was also established in ANSYS, as shown in Figure 13b. The fish-bone model was adopted for the nonlinear transient analyses. The beam4 element has only mechanical characteristics of the section, which is appropriate for simulating the virtual girder of the fish-bone model. In the model, the girder is modelled by beam4 elements with equivalent static and dynamic mechanical characteristics, as shown in Tables 5 and 6, connected with cables by rigid elements. The cables are modelled by the tension-only element, link10 elements, to account for the geometric nonlinearity according to the sag. The towers are modelled by beam4 elements, along with their mechanical properties, in all models. Figure 12: Configuration of the case bridge (unit: m). Figure 13: _Cont._  
To ensure the accuracy of the fish-bone model's dynamic properties, the natural frequencies of the 3D and fish-bone models were carefully checked and compared. A modal analysis with the Block Lanczos method was applied to compare the primary natural modes and frequencies of the 3D model and fish-bone model, as listed in Table 7. According to the frequency result comparisons, the fish-bone model includes most vibration modes and mechanical properties of the 3D model, which proves that the equivalent fish-bone model can replace the 3D model in buffeting analysis. Due to the paper limitations, only the 1 st symmetric lateral, vertical and torsional and antisymmetric vertical mode \begin{table}
\begin{tabular}{c c c c} \hline
**Section Type** & **Ixx** & **Iyy** & **Izz** \\ \hline Composite girder & 687.8 & 168.6 & 679.3 \\ Steel girder & 273.3 & 108.6 & 502.3 \\ \hline \end{tabular}
\end{table} 
Table 6: Inertia moment of the fish-bone model girder (m4). Figure 14: Segment model of different girder types. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Weight per metre of composite girder & 87,029 kg/m \\ Weight per metre of steel girder & 48,873 kg/m \\ Additional highway concrete deck load & 9102.04 kg/m \\ Additional highway steel deck load & 5469.39 kg/m \\ Additional railway load & 28,440 kg/m \\ Material damping ratio & 0.005 \\ \hline \end{tabular}
\end{table} 
Table 5: Static and dynamic parameters of the equivalent fish-bone model of the girder. Figure 13: The 3D and equivalent “fish-bone models” of the case bridge. shape comparisons are shown in Figure 15, which indicates that the two models have good consistency in the dominant mode shape of the buffeting analysis. The natural frequencies of the lateral and vertical deflections are always low, which illustrates that the stiffness of the girder in both directions is small and that deformation may occur more easily in these directions than in other directions in the buffeting analysis. The results of the modal analysis indicate that the response of the case bridge under a turbulent wind load can be assumed to be dominated by lateral and vertical deflections. \begin{table}