## 2 Fully Coupled Traffic-Bridge-Wind (TBW) Simulation
### Modelling of Wind Forces on Bridge  
The wind-induced forces on bridge girders are commonly decomposed into three components, as follows: Static wind forces due to mean wind speed, self-excited forces due to wind-bridge interactions, and buffeting forces due to wind turbulences [23; 24; 25]. Additionally, each wind component can be further discretized as lift force, drag force, and torsional moments. For the bridge pylons, main cables, and suspenders, only drag force from static wind speed is considered [2]. The self-excited forces can be calculated by summing the responses associated with the structural motion in lateral, vertical, and torsional directions. Specifically, the self-excited drag force, lift force, and torsional moments acting on bridge girder per unit length are derived as the convolution integrals between the arbitrary bridge girder motion and the associated impulse functions, as shown below [26]:  
\[D_{se}(t)=D_{sep}(t)+D_{sch}(t)+D_{sca}(t)=\int_{-\infty}^{t}\! \left(f_{Dp}(t-\tau)p(\tau)+f_{Dh}(t-\tau)h(\tau)+\right. \tag{2a}\] \[\left.f_{Da}(t-\tau)\alpha(\tau))d\tau,\right.\] \[L_{se}(t)=L_{sep}(t)+L_{sch}(t)+L_{sea}(t)=\int_{-\infty}^{t}\! \left(f_{Lp}(t-\tau)p(\tau)+f_{Lh}(t-\tau)h(\tau)+\right.\] (2b) \[\left.f_{L\alpha}(t-\tau)\alpha(\tau))d\tau,\right.\] \[M_{se}(t)=M_{sep}(t)+M_{sch}(t)+M_{sea}(t)=\int_{-\infty}^{t}\! \left(f_{Mp}(t-\tau)p(\tau)+f_{Mh}(t-\tau)h(\tau)+\right.\] (2c) \[\left.f_{M\alpha}(t-\tau)\alpha(\tau))d\tau,\right.\]  
where \(f\) = response impulse functions that are derived by experimentally determined flutter derivatives based on the rational approximation approach; the subscripts \(D\), \(L\), and \(M\) = response contribution to the self-excited drag force, lift force, and torsional moment; the subscripts \(p\), \(h\), and \(\alpha\) = impulse response functions with respect to unit impulse displacements in the lateral, vertical, and torsional directions, respectively; and \(p(\tau)\), \(h(\tau)\), and \(\alpha(\tau)\) are the time-dependent lateral, vertical, and rotational displacements of the bridge girder. For each impulse response function, the corresponding frequency function can be represented by rational functions. The frequency independent parameters are obtained through least square fitting of flutter derivatives. Once the frequency response functions are computed, the impulse response functions are then derived using the Fourier transform for each motion. Figure 1: Numerical model of a 3-axle vehicle: (**a**) Elevation view; (**b**) side view. The buffeting forces acting on the bridge girder are due to the unsteady wind turbulence on both horizontal and vertical directions. Similar to the way in which the self-excited forces are constructed, the buffeting forces on the bridge girder per unit span can be expressed as the convolution integral between the wind turbulence and associated impulse response functions, as follows:  
\[D_{b}(t)=D_{bu}(t)+D_{bu}(t)=\int_{-\infty}^{t}(f_{Du}(t-\tau)u(\tau)+f_{Dw}(t -\tau)w(\tau))d\tau, \tag{3a}\] \[L_{b}(t)=L_{bu}(t)+L_{bu}(t)=\int_{-\infty}^{t}(f_{Lu}(t-\tau)u(\tau)+f_{Lw}(t -\tau)w(\tau))d\tau,\] (3b) \[M_{b}(t)=M_{bu}(t)+M_{bw}(t)=\int_{-\infty}^{t}(f_{Mu}(t-\tau)u(\tau)+f_{Mu}(t -\tau)w(\tau))d\tau, \tag{3c}\]  
where \(u\) and \(w\) = turbulent wind in horizontal (normal to bridge longitudinal axis) and vertical direction; \(f_{Du}(t),f_{Dw}(t),f_{Lu}(t),f_{Lw}(t),f_{Mu}(t)\), and \(f_{Mu}(t)\) = the impulse response functions, where the subscripts \(D\), \(L\), and \(M\) = impulse response functions in terms of drag force, list force, and torsional moment and the subscripts \(u\) and \(w\) = the impulse response functions corresponding to wind turbulence in the horizontal and vertical directions. Similar to the way in which the self-excited forces are constructed, the buffeting forces are formulated using frequency dependent aerodynamic admittance functions. It should be noted that both the flutter derivatives and the aerodynamic admittance functions are determined through the wind tunnel experiments.