# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 2 Motion Related Wind Load
The expressions of the flutter derivatives are the following:  
\[KH_{1}^{\#}(K)=-C_{l}^{\prime}\ \widetilde{F}(K), \tag{3a}\] \[KH_{2}^{\#}(K)=C_{l}^{\prime}\ \Bigg{[}\bigg{(}\frac{1}{4}-a\bigg{)} \widetilde{F}(K)+\frac{\widetilde{G}(K)}{K}\Bigg{]}+\frac{C_{l}^{\prime}}{4},\] (3b) \[K^{2}H_{3}^{\#}(K)=C_{l}^{\prime}\ \bigg{[}\widetilde{F}(K)-K\widetilde{G}(K) \bigg{(}\frac{1}{4}-a\bigg{)}\bigg{]}+C_{l}^{\prime}\ K^{2}\frac{a}{4},\] (3c) \[K^{2}H_{4}^{\#}(K)=C_{l}^{\prime}\ K\widetilde{G}(K)+\frac{C_{l}^{\prime}\ K^{2}}{4},\] (3d) \[KA_{1}^{\#}(K)=C_{m}^{\prime}\widetilde{F}(K),\] (3e) \[L_{SE}=\frac{1}{2}\rho U^{2}B\left[KH_{1}\frac{\dot{h}}{U}+KH_{2}\frac{B\; \dot{\alpha}}{U}+K^{2}H_{3}\alpha+K^{2}H_{4}\frac{h}{B}+KH_{5}\frac{\dot{p}} {U}+K^{2}H_{6}\frac{p}{B}\;\right], \tag{2a}\]0 (3f) \[L_{SE}=\frac{1}{2}\rho U^{2}B\left[KH_{1}\frac{\dot{h}}{U}+KH_{2}\frac{B\; \dot{\alpha}}{U}+K^{2}H_{3}\alpha+K^{2}H_{4}\frac{h}{B}+KH_{5}\frac{\dot{p}} {U}+K^{2}H_{6}\frac{p}{B}\;\right], \tag{2a}\]1 (3g) \[L_{SE}=\frac{1}{2}\rho U^{2}B\left[KH_{1}\frac{\dot{h}}{U}+KH_{2}\frac{B\; \dot{\alpha}}{U}+K^{2}H_{3}\alpha+K^{2}H_{4}\frac{h}{B}+KH_{5}\frac{\dot{p}} {U}+K^{2}H_{6}\frac{p}{B}\;\right], \tag{2a}\]2  
where \(A_{i}^{\#}(K)\) and \(H_{i}^{\#}(K)\) are the flutter derivatives for the thin airfoil as expressed by Fung (\(K=\omega B/U\)), \(\widetilde{F}(K)\) and \(\widetilde{G}(K)\) are respectively the real and the imaginary parts of the Theodorsen function, \(C_{l}^{\prime}\) and \(C_{m}^{\prime}\) are the derivatives of the lift and moment coefficients with respect to the wind angle of attack and \(a\) is the distance between the shear center and the centroid of the airfoil, normalized with respect to the chord \(B\). For low values of reduced frequency, \(\widetilde{F}(K)\) tends to one, \(\widetilde{G}(K)\) tends to zero and the Fung formulation tends to the quasi-steady one. These formulas were tested for two bridges of which both steady-state aerodynamic coefficients and flutter derivatives are available: the Great Belt East in Denmark, with a streamlined deck ([PERSON], 1992; [PERSON], 1992), and the Akashi Kaikyo in Japan, with a truss girder ([PERSON], 1992; [PERSON], 1992). The derivatives of the aerodynamic coefficients, expressed in \([rad^{-1}]\), are: \(C_{l}^{\prime}=-4.37\) and \(C_{m}^{\prime}=1.17\) for the Great Belt Bridge; and \(C_{l}^{\prime}=-1.19\) and \(C_{m}^{\prime}=0.3\) for the Akashi Kaikyo Bridge. The curves in Figures 3 and 4 show a comparison between the flutter derivatives: obtained experimentally (solid crossed line), calculated by the Fung formulas (solid line), calculated by the quasi-steady formulation (dashed line) ([PERSON], 1992), and predicted by the Theodorsen theory for the thin airfoil (dash dotted line). From the comparisons in Figures 3 and 4, it is possible to notice that the Fung formulation leads generally to a good alignment with the experimental curves: the average trend is always respected except for \(H_{4}\), that in some cases can be neglected ([PERSON], 1992). A remarkable superposition is obtained for \(A_{1}\) and \(A_{3}\) of both bridges, and also for \(H_{3}\) in the case of the Great Belt Bridge. Good predictions are also obtained for \(A_{2}\) derivative. As expected, for a truss-stiffened girder, [PERSON] formulation for thin airfoil becomes unable to reliably predict the aerodynamic behavior. Even though the quasi-steady formulation also shows a good agreement, the Fung formulation was chosen over the quasi-steady approach because this latter is better suited for low values of reduced frequency and high values of reduced wind velocities ([PERSON], 1992). In fact, the bridge chosen as a case study is relatively stiff and characterized by high values of torsional and vertical eigenfrequencies, therefore flutter instability is expected to occur at a relatively low reduced velocity.