# Preliminary Flutter Stability Assessment of the Double-Deck George Washington Bridge
## 3 Full-Order Flutter Analysis Using ANSYS Mechanical APDL  
The main lack of almost all the FE packages commonly used in civil engineering is the impossibility of including motion-dependent aeroelastic loads into the model and then of performing a complex eigenvalue analysis. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) proposed a method that allows one to perform a flutter analysis using the commercial finite element package ANSYS Mechanical APDL. The method is based on the definition of the aeroelastic loads by means of a particular user-defined element as shortly illustrated below. According to the method developed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018), the motion-dependent effect due to the wind load is considered for each deck element by the element aeroelastic stiffness and damping matrices modeled by the user defined MATRIX27 elements. Following [PERSON]'s  
Figure 4: Flutter derivatives of the Akashi Kaikyo Bridge deck.  
formulation, the self-excited forces are expressed as functions of the flutter derivatives, as recalled in Section 2.  
It is possible to manipulate Equations (2a)-(2c) in order to obtain the equivalent nodal forces acting on the ends of the generic deck element, hence a lumped formulation can be used to derive the element aeroelastic stiffness and damping matrices [30]. Assembling the element matrices into global aeroelastic stiffness (\(K_{ae}\)) and damping (\(C_{ae}\)) matrices, the mathematical model of the system integrated with the effect of aeroelasticity is obtained [31]:  
\[M\ddot{X}+(C-C_{ae})\dot{X}+(K-K_{ae})X=0 \tag{4}\]  
The global aeroelastic stiffness and damping matrices contain flutter derivatives, which depend on the reduced wind velocity \(U_{R}=\frac{U}{JB}\) (\(f=\frac{\omega_{f}}{2\pi t}\), oscillation frequency). Therefore, the system expressed by Equation (4) is parameterized by wind velocity and vibration frequency. By Equation (4), a complex eigenvalue analysis can be carried out to determine the critical wind velocity and vibration frequency. Being the conjugate pairs of complex eigenvalues \(\lambda_{j}=\sigma_{j}\pm i\omega_{j}\), and the conjugate pairs of complex eigenvectors \(\phi_{j}=p_{j}\pm iq_{j}\), the system will be dynamically unstable if the real part of any eigenvalue is positive. Hence, the condition for the onset of flutter instability is stated as follows: at a certain critical wind velocity \(U_{f}\), the system has only one eigenvalue \(\lambda_{f}\) with zero real part, and the imaginary part \(\omega_{j}\) of the complex eigenvalue \(\lambda_{f}\) becomes the flutter frequency. It is necessary to provide the variation of both wind velocity and vibration frequency in the complex eigenvalue analysis, so that a mode-by-mode tracking method can be employed to iteratively search the flutter frequency and the flutter velocity [31].  
The motion-dependent effect due to the wind is taken into account for each deck element by the element aeroelastic stiffness and damping matrices. These matrices should be properly compiled to implement the aeroelastic motion-dependent load by means of the flutter derivatives. User defined MATRIX27 element in ANSYS Mechanical APDL can only model either an aeroelastic stiffness matrix or an aeroelastic damping matrix instead of both of them simultaneously, so a pair of MATRIX27 elements must be attached to each node of a generic bridge deck element as illustrated in Figure 5. The MATRIX27 elements \(e1\) and \(e3\) represent respectively the aeroelastic stiffness and damping of the node \(i\), as the MATRIX27 elements \(e2\) and \(e4\) represent respectively the aeroelastic stiffness and damping of the node \(j\).