# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 1 Introduction
A linearization of this model allows a comparison with [PERSON]'s semi-empirical approach and furnishes an expression of the flutter derivatives as functions of the steady-state aerodynamic coefficients of the cross-section [13]. These simplified expressions of the flutter derivatives are widespread for the streamlined deck cross-sections. Some works on the validity of this approach are [14; 15; 16] and some applications for the determination of the critical state can be found in [17; 18; 19]. In a recent work [20], a linear superposition of flat plate aerodynamics was adopted to estimate the flutter derivatives of streamlined multi-box deck sections. In that case, correction factors were introduced into the proposed analytical formulas to better fit the available experimental data. In the present paper, a simplified approach is used based on [PERSON]'s formulation [21], in which the flutter derivatives are expressed in terms of the steady-state aerodynamic coefficients and of the real and imaginary parts of the Theodorsen function. Two-dimensional Computational Fluid Dynamics (CFD) simulations were conducted in ANSYS FLUENT for the determination of the lift and moment coefficients, varying the wind attack angle. In recent years, the CFD approach gained importance with respect to the traditional experimental investigation. Some applications related to the bridge cross-sections are [22; 23]. More recently, [PERSON] et al. [24] pointed out that Reynolds-averaged Navier-Stokes (RANS) approach coupled with the k\(-\)\(\omega\) SST turbulence model, identify the best compromise between accuracy and computational cost. The CFD framework also provides different approximate techniques for the direct calculation of flutter derivatives, some applications are: [25; 26; 27; 28]. Nevertheless, the calculation of the steady-state aerodynamic coefficients requires simpler simulations compared to those required to compute the flutter derivatives, especially in terms of computational effort. In addition, the approximations introduced by a 2D geometrical representation, neglecting several appendages that could affect the aerodynamic behavior, do not justify the use of more refined methods. Concerning flutter analyses, several methods can be found in literature [29; 30], a method developed by [PERSON] and [PERSON] [31], which allows the analysis to be carried out with the FE package ANSYS Mechanical APDL, is adopted in this paper. The software, in fact, provides specific user-defined elements, through which it is possible to implement the motion-dependent aeroelastic forces as expressed in [PERSON]'s formulation. The George Washington suspension bridge was chosen as a case study, partly because of its historical significance. In fact, it was the first bridge whose main span exceeded one kilometer. The bridge was opened to traffic in 1931. It is characterized by a total length of 1450 m and a mid-span of 1067 m. Initially it was composed only of the upper level, whereas the lower deck was constructed from 1958 to 1962 because of the increasing traffic flow. Both configurations will be analyzed in the following: the original one with the single deck and the stiffened one with the two decks (Figure 1). The choice is also motivated by the peculiarity of the current bridge having two superposed decks. A similar configuration can be detected in the Verrazano-Narrows Bridge (USA), also having a great historical relevance, and in the Yangsigang Yangtze River bridge (China), which currently has the third longest span in the world after the Akashi Kaikyo bridge (Japan) and the recent 1915 Canakkale bridge (Turkey), the current World record. As a matter of fact, several investigations on the aerodynamic performances of trussed girders have been made, but only a few for double-deck bridges [32]. Due to the lack of aerodynamic data and studies on the subject, two simplified ways for the application of the aeroelastic forces are followed in the present study with the aim of obtaining approximate predictions.