# Dynamic Interaction Analysis of Long-Span Bridges Under Stochastic Traffic and Wind Loads
## 2 Framework for Stochastic Traffic-Bridge Coupled Vibration Under Combined Wind Action  
### Vehicle-Bridge Coupling Dynamics Equation  
In the vehicle-bridge coupled vibration analysis of large-span bridges, the bridge is typically modeled as a multi-degree-of-freedom system, and the vehicle is coupled to the bridge through a mass model. The displacement vector of the bridge is denoted as \(X_{b}(t)=[x1(t),x2(t),x3(t),\ldots,xn(t)]^{\mathrm{T}}\), the mass of the vehicle is m, its velocity is \(v(t)\), and its position is \(s(t)\). The bridge dynamic equation is  
\[M_{b}\ddot{X}_{b}(t)+C_{b}\dot{X}_{b}(t)+K_{b}X_{b}(t)=F_{v}(t) \tag{1}\]  
where \(M_{b}\), \(C_{b}\), and \(K_{b}\) are the mass, damping, and stiffness matrices of the bridge, respectively, and \(F_{v}(t)\) is the load applied to the bridge by the vehicle.  
The dynamics of a vehicle are usually simplified by means of a mass-spring-damper model, as shown in Figure 2. The mass, velocity, and displacement of the vehicle are \(m_{v},v(t)\), and \(x_{v}(t)\). The vehicle power equation is  
\[m_{v}\ddot{x}_{v}+C_{v}\dot{x}_{v}(t)+K_{v}x_{v}(t)=F_{v}(t) \tag{2}\]  
where \(C_{v}\) is the damping of the vehicle and \(K_{v}\) is the stiffness of the vehicle.  
Pavement roughness significantly influences the dynamic interaction between vehicles and bridges, particularly by inducing vibrations in bridges as vehicles traverse them. This roughness is typically modeled using the power spectral density (PSD) function of the  
Figure 1: The wind–vehicle–bridge coupling vibration analysis model.  
pavement's surface unevenness. It is assumed that the pavement roughness is described by a stochastic process that can be expressed by the following equation:  
\[r(x)=r0+\varepsilon(x) \tag{3}\]  
where \(r(x)\) is the pavement height at location \(x\), \(r0\) is the average pavement height, \(\varepsilon(x)\) is a random perturbation of the pavement unevenness, and \(\varepsilon(x)\) is usually assumed to be a Gaussian white noise process.  
The interaction force between the vehicle and the road surface can be expressed as  
\[F_{road}(t)=k_{road}\cdot r(x)=k_{road}\cdot(r0+\varepsilon(x)) \tag{4}\]  
where \(F_{road}(t)\) is the load due to road roughness and \(k_{road}\) is the pavement stiffness coefficient.  
By integrating the vehicle dynamics equation, the bridge dynamics equation, and accounting for the impact of pavement roughness, the dynamic equation for the vehicle-bridge coupling system can be formulated as follows:  
\[M_{b}\ddot{X}_{b}(t)+C_{b}\dot{X}_{b}(t)+K_{b}X_{b}(t)=\sum_{i=1}^{n}m_{wi}v_{ i}(t)\delta(x-x_{i}(t))+F_{road}(t) \tag{5}\]  
Figure 2: Numerical models of vehicles.  
where \(m_{vi}\) is the mass of vehicle \(i\), \(v_{i}(t)\) is the velocity of vehicle \(i\), and \(x_{i}(t)\) is the position of vehicle \(i\) on the bridge.  
Pavement roughness affects the dynamic coupling between the vehicle and the bridge through the perturbation of its stochastic process \(\varepsilon(x)\). The coupling effect between the vehicle and the bridge due to pavement roughness transmits additional vibrations caused by the road surface unevenness to the bridge. This transmission, in turn, significantly affects the dynamic response of the bridge. Differentiated vehicle types, densities, and speed arrangements were used for the traffic characteristics of the different lanes, and the pavement roughness [38] was assumed to be in good condition. Accurately modeling this coupling is essential for predicting the bridge's vibration behavior under traffic loads. The vehicle-bridge coupling vibration model is shown in Figure 3.