# A Feasibility Study of the Drive-By Method for Damage Detection in Railway Bridges
## 2 Train-Track-Bridge Model  
### Railway Bridge Model  
Each bridge is modeled as a simply supported beam, which is a typical configuration for railway bridges or multi-span viaducts. The bridge deck is modeled with Eulero-Bernoulli beams, and the complete model also includes the ballasted track above the bridge deck. Rails are modeled through finite elements, while sleepers are considered as rigid bodies and modeled with lumped masses (220 kg) and their moments of inertia, the frequency range of interest in the present analysis being up to 70 Hz. Two modules of ballasted track of length 29.4 m each are included before and after the bridge model (see Figure 1), to reproduce the continuity of the track, and to avoid introducing an artificial singularity between a fully rigid track and the bridge. To form a track module, the beam nodes are spaced 0.6 m in the case of the rail, and 4.2 m for the bridge, so that seven rail elements correspond to each bridge beam element. Sleepers are spaced every 0.6 m, and therefore for every section containing rail nodes, there is a node representing a sleeper as a concentrated inertia. The ballast is modeled by means of springs connecting the sleepers to the deck, also in the correspondence of sections in between the two nodes of each bridge beam element. The potential energy of the springs associated with these internal locations is calculated by means of the beam shape function. Figure 2 shows the scheme of the sections where both rails and bridge nodes are present. Nodes 1 and 2 are used to mesh the rails, node 4 to mesh bridge, and node 3 represents the sleeper. Four different structures are simulated, as to evaluate the performance of the drive-by method for different span lengths, flexural stiffnesses, and natural frequencies. Bridge models are named in the following, with numbers from #1 to #4, ordered with increasing span lengths. For all of the models, the Young's modulus is equal to 206,000 MPa, and \(\alpha\) and \(\beta\) Railegh coefficients to 0.1 and 0.2  
Figure 1: Rail model on bridge beam elements (side view). respectively. In this paper, steel properties are considered, even if the proposed methodology can be extended to concrete bridges. Table 1 reports all the other bridge properties, together with the first flexural natural frequency f\({}_{1}\) (I\({}_{22}\) is the area moment of inertia along the horizontal axis and I\({}_{33}\) is the area moment of inertia along the vertical axis). Bridges number one and two are short bridges: the first one is 13.2 m long, with a rather high fist natural frequency (i.e., 21.82 Hz), and the second one is 21.6 m long, with properties based on those of an existing bridge. The third bridge is a medium-long span bridge of length 46.8 m, and the last model has a long span (72 m), with properties derived from an existing bridge. Figure 3 reports the flexural stiffnesses El\({}_{22}\) of the four bridge models, as a function of span length. The dependence is a power coefficient that is equal to 3.5. Table 2 reports the characteristics of the stiffnesses that are adopted for the connections between the sleepers and the bridge (k\({}_{1}\) and k\({}_{2}\), representing ballast), and for the connections between the rail nodes and the sleepers (k\({}_{3}\) representing the stiffness properties of the fastenings). As for the ballast, \begin{table}
\begin{tabular}{l c c c c c} \hline  & **L [m]** & **A [m\({}^{2}\)]** & **\(\varrho\) [kg/m\({}^{3}\)]** & **I\({}_{22}\) [m\({}^{4}\)]** & **I\({}_{33}\) [m\({}^{4}\)]** & **f\({}_{1}\) [Hz]** \\ \hline Bridge \#1 & 13.2 & 0.1567 & 0.9783 \(\times\) 10\({}^{4}\) & 0.2570 \(\times\) 10\({}^{-1}\) & 6.38 & 21.82 \\ Bridge \#2 & 21.6 & 0.4352 & 0.224 \(\times\) 10\({}^{5}\) & 0.1190 & 6.380 & 5.9 \\ Bridge \#3 & 46.8 & 0.4 & 0.2340 \(\times\) 10\({}^{5}\) & 0.2400 \(\times\) 10 & 5.546 & 5.34 \\ Bridge \#4 & 72 & 0.3440 & 0.2129 \(\times\) 10\({}^{5}\) & 0.9481 \(\times\) 10 & 5.546 & 4.86 \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of simulated bridges.