## 6 Discussion
### Testing of a 30 m Bridge  
The analysis from the previous section is extended here to a 30 m span bridge (2 nd moment of area \(I\) = 2.2643 m\({}^{4}\) and mass per unit length \(m\) = 21,608 kg/m) traversed by a fleet of 3-axle trucks at a speed of 10 m/s. The first three longitudinal frequencies of the bridge in healthy condition are 3.385 Hz, 13.540 Hz, and 30.465 Hz. For comparison purposes, the damage scenarios have the same relative locations with respect to the total span as in Section 6.1, i.e., damage 'C' is a 15% crack located at 10 m, and damage 'D' is a 5% crack located at 14 m from the first support. Figure 15a,b compare the difference between the forced frequencies extracted from a fleet composed of \(p\) = 1000 trucks and its associated eigenfrequencies for damages 'C' and 'D', respectively. As noted in the methodology section, the proper selection of the locations for frequency estimation is essential. It is clear from Figure 15 that the combination of frequencies used before, i.e., _F\({}_{3}\)_ and _F\({}_{9}\)_, does not work due to the edge effects caused by the axles entering and leaving the bridge. The higher speed has a negative impact on the estimation of the forced frequency by significantly widening the \"blind spots\" around the entrance and exit of vehicle axles. Nevertheless, there is a longer duration of available accelerations with respect to the 15 m bridge, making it possible to capture uncorrupted, instantaneous frequencies for positions of the front axle between 11 and 26 m. Therefore, the frequencies _F\({}_{15}\)_ and _F\({}_{22}\)  
Figure 15: Comparison between transient forced-frequencies due to the crossing of 1000 trucks and eigenfrequencies for scenarios: (**a**) 15% crack at 10 m, and (**b**) 5% crack at 14 m.  
corresponding to positions of the front axle at 15 and 22 m from the start of the bridge, were proposed to establish the eigenfrequency database and the instantaneous frequencies from the dynamic transient analysis. As expected, the first bridge frequency decreased slightly more for damage 'C' than for damage 'D'. For example, the range [\(\mu-3\sigma\), \(\mu+3\sigma\)] within 11-26 m due to the 1000 trucks varies from 3.30 to 3.42 Hz for damage 'C', and from 3.35 to 3.47 Hz for damage 'D'. Figure 16a,b plot the 1000 values'measured' for \(F_{15}\) and \(F_{22}\), and their associated distributions for damages 'C' and 'D', respectively. Figure 16c,d show the'measurements' for the two damages against the theoretical eigenfrequency database. In comparison to the 15 m bridge (Figure 13), there is considerable overlap between theoretical eigenvalue distributions associated with different damage scenarios. The reasons are twofold: (i) the change in the 1 st bridge frequency due to damage is smaller in the 30 m bridge; and (ii) the standard deviations associated with \(F_{15}\) and \(F_{22}\) are larger than those associated with \(F_{3}\) and \(F_{9}\). Regarding (i), even if equivalent damage scenarios cause a similar relative decrement in frequency for both spans, the absolute change in frequency is smaller in the longer span because of its lower first bridge frequency. For instance, damage 'A' in the 15 m bridge causes a reduction of \(-2.81\%\) in \(F_{9}\) with respect to a healthy condition, whereas damage 'C' in the 30 m bridge results in a similar change of \(-2.75\%\) in \(F_{22}\). However, the former is equivalent to a drop of 0.16 Hz, while the latter only represents a reduction of 0.09 Hz. Regarding (ii), the standard deviations in the case of a healthy bridge are 0.0006 Hz, 0.0035 Hz, 0.0145 Hz, and 0.023 Hz, for \(F_{3}(15\) m bridge), \(F_{9}(15\) m bridge), \(F_{15}\) (30 m bridge), and \(F_{22}\) (30 m bridge), respectively.