Flutter Instability Calculation for Suspended Bridges Using Geometrically Nonlinear Analyses

Abstract

The paper investigates the modelling and calculation of the flutter instability condition due to the windstructure interaction in the case of suspended bridges. In particular, the paper is focused on FEM (Finite Element Model) analyses on three dimensional models that compute the flutter instability. Geometric non-linear flutter instability analyses were carried out using a research and design software program (TENSO), which enables nonlinear dynamic analysis of wind-structure interaction. All the bridge structure is modelled. In particular, the bridge deck model was simplified by a beam model located in the deck section's center of gravity and two massless rigid links to simulate the connection of the deck to the hangers and cables. The cables are represented by rectilinear cables element. The critical flutter speed was estimated using quasi-static approximation of the unsteady wind loads calculated by aerodynamic coefficients (i.e. lift, drag and moment coefficients) estimated by aerodynamic forces (i.e. lift, drag and moment) directly measured by wind tunnel tests. The executed analyses are by dynamic step-by-step integration of the nonlinear three-dimensional structure with geometric nonlinearities. The global stiffness matrix is updated at each load step by assembly of the stiffness sub-matrices of the elements, updated to account for the strain found at the previous time step, taking into account the geometric nonlinearity of the structure.Firstly, the calculation provides for the static equilibrium of the structure under dead, gravity and construction loads by nonlinear static analysis. It was computed simultaneously using two methods: step-by-step incremental method and a "subsequent interaction" method with variable stiffness matrix (secant method). The secant method is a finite-difference approximation of the Newton-Raphson's modified method for systems of nonlinear algebraic equations. The solution under gravity loads is subsequently used as the initial step of the dynamic wind load analysis. The Newmark-Beta method with Rayleigh damping is used for numerical integration of the dynamic equations. Wind loads on the bridge deck are time dependent and they are simulated by applying the aerodynamic coefficients as a function of the time-dependent angle of attack at a given mean wind speed U. Displacements and rotations of the bridge deck at progressively increasing values of U, are estimated. The velocity at incipient flutter is fixed when the reference deck vibration amplitude exceeds ±5°. Details of the analyses results and calculation are given for an example of suspended foot bridges. Vertical displacements (δ) and rotations of the deck about the longitudinal bridge axis (αx) in normalized format (i.e. upward deck displacements are positive and counterclockwise rotations are positive) are discussed.