# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 1 Introduction  
Old bridges (conventionally, which are more than 50 years old), require special maintenance, accurate inspections, and a careful assessment of their safety conditions. The conflict between economy and structural performance between the 19 th and the 20 th centuries led the design of long-span bridges to the development of very flexible and slender structures. The use of the elastic deflection theory allowed for very slender decks against static loads and shifted the design trend at that time from rigid trusses to slender edge girders. This evolution ended brutally with the Tacoma Narrows Bridge disaster due to the wind-induced flutter instability on 7 November 1940. From then on, any design of a flexible structure must assure the structure itself to be stable under the dynamic effects of wind loads. In fact, in addition to the already known static divergence due to the steady-state wind loads, flutter stability has become a governing criterion in the design of long-span suspension bridges. The objective of a (linear) flutter analysis is to predict the lowest critical flutter wind velocity and the corresponding flutter frequency. A crucial point in flutter analysis is the definition of motion-dependent aerodynamic loads. The first analytical solutions were developed by [PERSON] [1] and [PERSON] [2] for thin airfoil. [PERSON] defined the self-excited forces as the superposition of circulatory and non-circulatory contributions, the former depending on the oscillation frequency and accounting for flow unsteady effects, and the latter independent of oscillation frequency and including inertial effects due to the moving fluid mass. Subsequently, [PERSON] and [PERSON] exported some features of the [PERSON]'s results extending the formulation to bluff bodies, as bridge cross-sections. In this formulation, the wind loads induced by sectional harmonic motions are described by means of a linear format based on experimentally evaluated parameters, called \"flutter derivatives\", that supply the lack of closed-form analytical formulations [3; 4]. The relation between [PERSON]'s approach based on indicial functions, [PERSON]'s theory based on circulatory function and [PERSON]'s flutter derivatives is well detailed in [5]. In the last decades, [PERSON]'s formulation has been the most widely adopted, and the calculation of flutter derivatives has become an important step for any flutter analysis. Currently, the only method considered reliable for their calculation at the design stage is that of carrying out experimental tests on scale models in wind tunnels. Some notable examples are [6; 7] for the Great Belt East Bridge and [8] for the Akashi Kaikyo Bridge; more recently [9] for the Jianghai Channel Bridge and [10] for the Hardanger Bridge. To overcome costs and difficulties arising from wind tunnel testing, several efforts were made with the aim of developing some simplified methods for the calculation of flutter derivatives. Although these methods are not appropriate for the final design stage, they allow useful and versatile studies in early design phases. Some authors utilized the derivatives of the thin airfoil, as [11] for the Humber bridge. This simplification leads to relatively small errors when the bridges are characterized by streamlined cross-sections. [PERSON] [12] adopted an alternative approach for open-truss stiffened suspension bridges: he estimated the aerodynamic derivatives based on the correlation between the thin airfoil derivatives and those of other bridges having a similar deck configuration. The method was applied to evaluate the flutter stability of the second Tacoma Narrows Bridge, with special focus on the effects of the side grates. Another simplified approach is given by the quasi-steady theory, in which the aerodynamic loads depend on the instantaneous relative velocity between the flow and the cross-section. In this framework the flow-induced forces are described by means of non-linear static relationships involving the wind angle of attack and the displacements of the structure.