## 2 Analytical Programming  
### Formulation of Bridge Dynamic Response  
This section elaborates on how the dynamic responses of bridges are derived analytically. These analytical formulations are important to understanding the key parameters influencing bridge vibration in the case of a moving vehicle. Also, the formulations will be used in this study to verify the adopted FE model. The closed-form formulations of the displacement and acceleration responses of the bridge may be obtained for the vehicle bridge interaction (VBI) problem shown in Figure 1[40].  
Figure 1: Simply supported beam traversed by a moving vehicle.  
The equation of motion for an undamped simply supported beam traversed by a moving load is expressed as follows:  
\[m\ddot{u}+EIu^{\prime m}=P(t) \tag{1}\]  
where \(m\) and \(EI\) are the mass per unit length and flexural stiffness of the bridge, respectively; \(u\) is the displacement response of the bridge, where the dot (\({}^{\bullet}\)) denotes differentiation with respect to time and the dash (\({}^{\prime}\)) denotes differentiation with respect to location; \(P(t)\) is the contact force of the moving load. The deflection of the bridge at the mid-span location can be written in a general form for all modes of vibration for a simply supported beam as follows:  
\[u(\mathrm{x},t)=q_{n}(t)\sin\Bigl{(}\frac{n\pi x}{L}\Bigr{)} \tag{2}\]  
where \(q_{n}\) is the generalized coordinate; \(n\) is the mode number; \(x\) is the location on the bridge; and \(L\) is the total length of the bridge. By differentiating Equation (2) with respect to time and location and substituting the results into Equation (1), then multiplying both sides of the equation by \(\sin(n\pi x/L)\) and integrating with respect to the length \(L\), one may obtain the following:  
\[\ddot{q}_{n}+\omega_{bn}^{2}q_{n}=\frac{-2m_{v}g}{mL}\sin\omega_{dn}t \tag{3}\]  
where \(m_{v}\) and \(v\) are the mass and velocity of the moving load, respectively. \(\omega_{dn}\) is the driving frequency of the moving load of the \(n\)th mode and is expressed such that \(\omega_{dn}=\frac{n\pi v}{L}\), where \(\omega_{bn}\) is the bridge's natural frequency for the \(n\)th mode and can be calculated as follows:  
\[\omega_{bn}=\frac{n^{2}\pi^{2}}{L^{2}}\sqrt{\frac{EI}{m}} \tag{4}\]  
Equation (3) is solved as a differential equation using the zero initial conditions to provide the following equation:  
\[q_{n}(t)=\frac{\Delta_{stn}}{1-S_{n}^{2}}(\sin\omega_{dn}t-S_{n}\sin\omega_{bn }t) \tag{5}\]  
where the constant \(\Delta_{stn}\) is expressed as \(\Delta_{stn}=-\frac{2m_{v}gL^{3}}{EI^{4}\pi^{2}}\) and \(S_{n}\) is called the speed parameter and is presented as \(S_{n}=\frac{\omega_{dn}}{\omega_{bn}}\). The displacement response of the bridge for all modes is obtained by substituting Equation (5) into Equation (2), which is shown in Equation (6). The velocity response is obtained by differentiating the displacement as shown in Equation (7), and the acceleration response is obtained by differentiating the displacement response twice with respect to time, as shown in Equation (8).  
\[u(\mathrm{x},t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\sin\Bigl{(}\frac{n \pi x}{L}\Bigr{)}(\sin\omega_{dn}t-S_{n}\sin\omega_{bn}t) \tag{6}\]  
\[\dot{u}(\mathrm{x},t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\sin\Bigl{(} \frac{n\pi x}{L}\Bigr{)}(\omega_{dn}\cos\omega_{dn}t-S_{n}\omega_{bn}\cos \omega_{bn}t) \tag{7}\]  
\[\ddot{u}(\mathrm{x},t)=\sum_{n}\frac{\Delta_{stn}}{1-S_{n}^{2}}\sin\Bigl{(} \frac{n\pi x}{L}\Bigr{)}\Bigl{(}-\omega_{dn}^{2}\sin\omega_{dn}t+S_{n}\omega_ {bn}^{2}\sin\omega_{bn}t\Bigr{)} \tag{8}\]  
The displacement, velocity, and acceleration responses, considering all modes of vibration of a simply supported beam, may be obtained analytically by the preceding equations. The equations show that bridge vibration is dominated by two frequencies, i.e., the driving frequency and bridge frequency. In this study, the responses obtained from the analytical solution are compared with those of the developed FE model to verify the prepared FE model.