# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 5 Flutter Analyses  
Before introducing the aeroelastic nodal forces by MATRIX27 elements, a modal analysis with no wind was performed: the results are summarized in Figures 11 and 12, where the labels V, T, L stand for vertical, torsional and lateral, respectively; and the labels A and S stand for antisymmetric and symmetric, respectively. Since it is well known that flutter instability usually involves mainly the first modes at lower frequencies [31], only the first six modes were extracted for both configurations. The geometric nonlinearity of the structure was taken into account running the modal analysis in a preloaded configuration accounting for the gravity loads. Once eigenfrequencies and eigenmodes have been evaluated, the flutter analysis was performed by the method described in [31]. The aeroelastic loads were modeled as nodal forces affecting the stiffness and damping matrices of the element to which the nodes belong, so MATRIX27 elements were incorporated into the structural model to perform the damped eigenvalue analyses. velocity is plotted in Figures 13-15, where the labels within parentheses refer to the mode shapes under wind action: for instance, the label VT indicates a mode characterized by both vertical and torsional components, in which the vertical component is predominant. The real and imaginary parts of the eigenvalues represent the logarithm decay rates and damped vibration frequencies of these modes, respectively. As stated above, the flutter condition occurs when the real part of any eigenvalue becomes positive. The results obtained for the 1962 configuration refer to the two methods previously introduced and recalled hereafter:  
1. The first method consists of the application of the aeroelastic forces to the upper deck, where the latter are expressed via the flutter derivatives evaluated using the steady-state aerodynamic coefficients obtained for the whole two-deck cross-section (see Figures 6b, 8 and 9a). 2. The second method consists of the application of the aeroelastic forces to both decks, where the latter are expressed via the flutter derivatives evaluated using the steady-state aerodynamic coefficients obtained for the upper deck (see Figures 6a, 7 and 9b). Figure 14: Variation of complex eigenvalues vs. wind velocity for the double-deck configuration: first method (labels within parentheses refer to mode shapes under wind action). Figure 13: Variation of complex eigenvalues vs. wind velocity for the single-deck configuration (labels within parentheses refer to mode shapes under wind action). The analyses furnished a critical flutter speed of 40 m/s (144 km/h) and a critical flutter frequency of 0.130 Hz for the single-deck configuration. As regards the double-deck configuration, flutter speeds of 107 m/s (385.2 km/h) and 79 m/s (284.4 km/h), and flutter frequencies of 0.203 Hz and 0.195 Hz, were provided by the first and the second method, respectively. As already said, the variations of the imaginary part of the complex eigenvalues in Figures 13-15 describe the change in the oscillation frequencies with the increasing wind speed, whereas the variations of the real part tell us about stability of each single vibration mode, and thus of the bridge deck. Accordingly, the first torsional-vertical anti-symmetric mode (TVA) is responsible for flutter instability in all the analyzed cases. For higher wind speeds, a second flutter mode is attained in the torsional-vertical symmetric mode (TVS); see Figures 13-15. For the double-deck configuration, the first and the second flutter velocities are rather close, being close also the corresponding eigenfrequencies; see Figures 14 and 15. In relation to that, we can say that interaction between modes having similar frequencies could be a triggering factor for instability, thus lowering the critical wind speed. The two different methods adopted for the double-deck configuration furnish a difference of about 26% and 4% in the estimated critical wind speed and flutter frequency, respectively.