## 2 Fully Coupled Traffic-Bridge-Wind (TBW) Simulation
### Governing Equations of Motion for the TBW Coupled System  
The governing equations of motions for the coupled traffic-bridge-wind system are given as follows below:  
\[\left[\begin{array}{cccc}M_{b}&0&\dots&0\\ 0&M_{v1}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&M_{tm}\\ \end{array}\right]\left(\begin{array}{c}\ddot{q}_{b}\\ \ddot{q}_{v1}\\ \vdots\\ \ddot{q}_{vm}\\ \end{array}\right)+\left[\begin{array}{cccc}C_{b}&0&\dots&0\\ 0&C_{v1}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&C_{tm}\\ \end{array}\right]\left(\begin{array}{c}\dot{q}_{b}\\ \dot{q}_{v1}\\ \vdots\\ \ddot{q}_{vm}\\ \end{array}\right)\] \[+\left[\begin{array}{cccc}K_{b}&0&\dots&0\\ 0&K_{v1}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&K_{tm}\\ \end{array}\right]\left(\begin{array}{c}q_{b}\\ q_{v1}\\ \vdots\\ q_{vm}\\ \end{array}\right)=\left\{\begin{array}{c}(\sum_{i=1}^{n}F_{vv}^{G})+F_{b}^{R }+F_{b}^{C}+F_{b}^{Se}+F_{b}^{Bu}\\ F_{v1}^{R}+F_{v1}^{C}+F_{v1}^{W}\\ \vdots\\ F_{vn}^{R}+F_{vn}^{C}+F_{vn}^{W}\\ \end{array}\right\}, \tag{7}\]  
where \(n\) = the total number of vehicles in the traffic flow; \(q\) = the displacement vector; \(M\), \(F\), \(C\), and \(F\) = structural mass, stiffness, damping matrices, and force vector; subscripts \(b\) and \(v_{i}\) (\(i\) =1,2,... \(n\)) indicate the parameters for bridge and \(i\)th vehicle, respectively; and the superscripts \(G\), \(R\), \(C\), \(W\), \(Se\), and \(Bu\) = excitation loads from gravity, road surface roughness, coupling interaction forces, static wind, self-excited and buffering forces, respectively. The coupling interaction force between the bridge and the \(i\)th vehicle are given by:  
\[\left\{\begin{array}{c}F_{b}^{C}=\sum_{i=1}^{n}C_{kv}\Big{[}\dot{q}_{b}\Big{]} +\sum_{i=1}^{n}K_{kv}\big{[}q_{b}\big{]}+\sum_{i=1}^{n}C_{b,\ v_{i}}\Big{[} \dot{q}_{v_{i}}\Big{]}+\sum_{i=1}^{n}K_{b,\ v_{i}}\big{[}q_{v_{i}}\Big{]}\\ F_{v_{i}}^{C}=C_{v_{i}b}\Big{[}\dot{q}_{v_{i}}\Big{]}+K_{v_{i}b}\big{[}q_{v_{i}} \Big{]}\\ \end{array}\right., \tag{8}\]  
where \(K_{kv}\) and \(C_{kv}\) = stiffness and damping contribution to the bridge caused by the coupling effects between bridge and \(i\)th vehicle in the traffic flow and \(K_{b,v_{i}}(K_{v_{i}b})\) and \(C_{b,v_{i}}(C_{v_{i}b})\) are the coupled stiffness and damping matrices between the bridge and the vibration of \(i\)th vehicle, respectively.  
As the complicated dynamic coupling effects among the bridge, traffic, and wind cannot be modeled appropriately using the existing finite element (FE) software, an analytical framework of the coupled TBW dynamic system was established using the computer programming language MATLAB and commercial FE software ANSYS. The whole simulation process is composed of three steps. Firstly, the numerical models for the bridge and vehicles are established to obtain the initial coefficient matrices in ANSYS. Secondly, the stochastic wind field and traffic flow are simulated to calculate the wind forces on the bridge/vehicle as well as the coupled bridge-vehicle forces. Thirdly, the governing equations of the coupled TBW system are developed to perform the dynamic analysis in MATLAB. In this step, the geometric nonlinear effect of axial forces on the bridge girder and pylons and the cable tension can be taken into account. After the coupled governing equations are solved, the stress vector can be further computed based on the deflection-strain relationship and the stress-strain relationship, as given by the following:  
\[[S]=[E][B]\{q_{b}\}, \tag{9}\]  
where \([S]\) = the predicted stress vector; \([E]\) = the stress-strain relationship matrix, which is assumed to be constant; and \([B]\) = the stress-deflection relationship matrix deduced from the element shape functions.