## 2 Research Methodology  
Damage is seldom localized and quantified considering only the first mode of a structure; it is often necessary to combine this information with data from other modes and dynamic measurements. This paper attempts to overcome this challenge by exploiting the information stored by the time-history of forced frequencies due to a vehicle to estimate the location and severity of the damage induced by a crack. While this paper uses only the frequency associated with the first mode, it is also making the most out of the unique coupling of a specific vehicle fleet with the first frequency of the bridge for the different positions of the vehicle on the bridge. This section presents the basic steps for implementing the methodology, including the practical aspects, as follows:  
1. Build a finite element model of the bridge. For this purpose, it is necessary to know the properties of the healthy bridge, which can be obtained based on original drawings and measurements on site to identify the mass (\([M_{b}]\)), stiffness (\([K_{b}]\)) and damping (\([C_{b}]\)) matrices that are involved in the equation of motion of the bridge (Equation (1)). \[[M_{b}]\{\tilde{u}_{b}\}+[C_{b}]\{\dot{u}_{b}\}+[K_{b}]\{u_{b}\}=\{F_{b}\}\] (1) where \(\{u_{b}\}\), \(\{\dot{u}_{b}\}\) and \(\{\tilde{u}_{b}\}\) are vectors representing the displacement, velocity and acceleration of the Degrees of Freedom (DoFs) of the bridge, respectively; whereas \(\{F_{b}\}\) is the vector representing the time-varying forces applied to the bridge. 2. Build a multi-body dynamic model of the vehicle representative of the truck fleet. The statistical data for axle spacings and static axle weights of the fleet can be collected from a WIM site near the bridge. In the absence of vehicular measurements, the dynamic properties of the vehicle can be gathered from the literature. Equation (2) represents the equation of motion of the vehicle: \[[M_{v}]\{\tilde{u}_{v}\}+[C_{v}]\{\dot{u}_{v}\}+[K_{v}]\{u_{v}\}=\{F_{v}\}\] (2) where \([M_{v}]\), \([K_{v}]\) and \([C_{v}]\) correspond to the mass, stiffness and damping matrices of the vehicle, respectively; \(\{u_{v}\}\), \(\{\dot{u}_{v}\}\) and \(\{\tilde{u}_{v}\}\) are vectors representing the displacement, velocity, and acceleration of the DoFs of the vehicle, respectively; and \(\{F_{v}\}\) is the vector representing the time-varying forces applied to the vehicle. 3. Based on steps (1) and (2), \(n\) mass and stiffness matrices are assembled for the coupled vehicle-bridge system for each position of the vehicle on the bridge. VBI modelling is an efficient approach to estimating the bridge response (i.e., forced vibration) due to a moving vehicle, and has been widely used in past research works [28]. It is important to note here that bridge damping is neglected in the current analysis. Also, eigenvalue analysis does not contemplate the road profile, therefore, the methodology is best suited to bridges with very low roughness coefficient. 4. Using eigenvalue analysis, the time-history of the first eigenvalue of the coupled vehicle-bridge model is calculated for the \(n\) positions of the vehicle on the healthy bridge. The properties of the vehicle are varied randomly based on typical distributions for the fleet configuration under investigation, i.e., based on statistics from a nearby WIM site. Therefore, each point in the time-history of the first eigenfrequency has a statistical distribution associated with it. Equation (3) shows the eigenvalue analysis for a given position \(i\) over the bridge of the vehicle with a set of properties \(j\).