# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 4 The George Washington Bridge
### Finite Element Models  
The ANSYS Mechanical APDL numerical models were realized with the geometrical properties found in literature [38; 39; 40]. The suspension cables were modeled as beam elements (BEAM188), with a circular cross-section of radius 0.573 m. An equivalent elastic modulus of 1.07 \(\times\) 10\({}^{11}\) Pa was adopted in order to take in to account the influence of the side spans. The hangers were modeled using LINK180 elements, that transmit only tensile forces between the deck and the main cables. The floor system was modeled by means of equivalent beams having the properties listed in Table 1, wherein the V-shaped up-winds and the lower elements refer to the double-deck configuration. The following material properties were assigned to each element except the suspension cables: [PERSON]'s modulus = 2.1 \(\times\) 10\({}^{11}\) Pa, Poisson's ratio = 0.3 and mass density = 7860 kg/m\({}^{3}\). Point mass elements (MASS21) were attached to the upper chords and to the equivalent longitudinal beams, both lower and upper, in order to model the inertial contribution of the sidewalk slabs, roadway slabs and other secondary elements. According to [PERSON] et al. [39], a total dead load of 417 kN/m and 569 kN/m were obtained for the single-deck and the double-deck configurations, respectively. Lastly, the MATRIX27 elements were attached to each node of the equivalent longitudinal beams to model the aeroelastic forces in terms of aeroelastic stiffness and damping.  
According to the two different methods introduced in the previous section for the double-deck configuration, two different models were realized. In the first model, the pairs of MATRIX27 elements, compiled with the flutter derivatives in Figure 8, were attached only to the upper deck. Whereas, according to the second method, MATRIX27 elements, compiled with the flutter derivatives plotted in Figure 7, were attached to both upper and lower decks. The finite element models are shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Elements**} & \multirow{2}{*}{**Area [m\({}^{2}\)]**} & \multicolumn{2}{c}{**Bending Moments of Inertia [m\({}^{4}\)]**} \\ \cline{3-4}  & & **Horizontal Axis** & **Vertical Axis** \\ \hline Equivalent longitudinal upper beam & 0.4465 & \(I_{y}=0.1867\) & \(I_{z}=38.6601\) \\ Upper chord & 0.0549 & \(I_{y}=0.0049\) & \(I_{z}=0.0060\) \\ Equivalent transversal upper beam & 0.237 & \(I_{x}=0.9498\) & \(I_{z}=3.8102\) \\ V-shaped up-winds & 0.0374 & \(I_{1}=0.0016\) & \(I_{2}=0.0006\) \\ Equivalent longitudinal lower beam & 0.3439 & \(I_{y}=0.1108\) & \(I_{z}=29.3241\) \\ Lower chord & 0.0484 & \(I_{y}=0.0029\) & \(I_{z}=0.0051\) \\ Equivalent transversal lower beam & 0.0816 & \(I_{x}=0.1559\) & \(I_{z}=1.3645\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometrical properties of the truss girder of the double-deck George Washington Bridge.  
Figure 10: ANSYS Mechanical APDL Finite element model. (**a**) Single-deck configuration (MATRIX27 elements are shown in red color); (**b**) Double-deck configuration (MATRIX27 elements are not shown).