# Railway Bridge Condition Monitoring Using Numerically Calculated Responses from Batches of Trains
## 5 Bridge Health Monitoring
### Preliminary Results with a 2-Axle Half-Car Model  
To illustrate the fleet monitoring concept for a bridge, the simple 2-axle 2-degree-of-freedom half-car and track beam with a smooth profile model shown in Figure 6 is simulated in MATLAB. This track beam is modelled as a simply supported beam using a finite element discretization. Vehicles are simulated crossing this beam and, for each run, moving reference deflections are calculated under each of the two axles. Using the equations described above, fundamental MR-ILs are calculated from the moving reference deflections. Here, a fleet of 20 vehicles is generated using Monte Carlo simulation with different body masses, \(m_{s}\), and body moment of inertias, \(I_{s}\). The vehicle properties are shown in Table 2. The length of the beam is 20 m, and the second moment of area is 0.33 m\({}^{4}\). The Young's modulus is 35 \(\times\) 10\({}^{9}\) N/m\({}^{2}\) and the mass per unit length is 9.6 \(\times\) 10\({}^{3}\) kg/m. The 1 st natural frequency vibration is 4.31 Hz. These vehicle and bridge property values are based on the values gathered from [PERSON] et al. [30] and [PERSON] [44]. The process is repeated for three different speeds, using the same batch of vehicles in each case. The calculated fundamental MR-ILs from 20 vehicle crossings are compared with the 'true' MR-ILs calculated from a simple static analysis, in Figure 7.  
Figure 6: (a) 2-degree-of-freedom half-car and beam model; (b) 2-degree-of-freedom half-car model. Figure 7: True and calculated fundamental MR-ILs for bridge and vehicle at different speeds: (a) 2 m/s; (b) 10 m/s; (c) 20 m/s. For a speed of 2 m/s--effectively pseudo-static--the calculated fundamental MR-IL closely matches the corresponding true MR-IL. The blue dashed curve is the true MR-IL, i.e., the deflection at x due to a unit load at x, calculated using the Unit Load Theorem (a Corollary of the Theorem of Virtual Work). As speed increases, the influence of vehicle/bridge dynamic interaction becomes evident, with a significant divergence from the true MR-IL. A Fast Fourier Transform (FFT) on the MR-IL from the first vehicle at 20 m/s shows a dominant peak at 4.33 Hz, very close to the 1 st fundamental bridge frequency (4.31 Hz). The dynamics of the bridge clearly have an influence on the calculated results. Two bridge states with different levels of global damage are simulated, with the resulting moving reference deflections and MR-ILs. It is acknowledged that the capacity of the bridge is different from its flexural rigidity (stiffness). However, the two are well correlated and changes in flexural rigidity are used here to simulate changes in bridge condition. The overall losses of flexural rigidity are 5% and 10%. Batches of 20 vehicles are generated with the same property values in Table 2 and simulated crossing the bridge in its healthy state and in the two damaged states. Two speeds are used separately: 10 m/s and 20 m/s. The mean calculated MR-ILs of the three bridge states are shown in Figure 8. The mean of the 20 mid-span MR-IL values, denoted \(\overline{IL}\), is chosen as an indicator of bridge damage. It is noted that MR-IL is inversely proportional to the flexural rigidity, so it is inversely proportional to damage. For this reason, _Damage Indicator_ 1 is defined by Equation (24). The area under the mean calculated MR-ILs (denoted \(IL^{*}\)) is used as _Damage Indicator_ 2. \[ \begin{split}\text{{Damage Indicator 1 }(D1)=100\Bigg{\{}1-\frac{\left( \frac{1}{IL}\right)^{D}}{\left(\frac{1}{IL}\right)^{H}}\Bigg{\}}& =100\bigg{\{}1-\frac{\left(\overline{IL}\right)^{H}}{\left( \overline{IL}\right)^{D}}\bigg{\}}\\ \text{{Damage Indicator 2 }(D2)=100\Bigg{\{}1-\frac{\left( \frac{1}{IL}\right)^{D}}{\left(\frac{1}{IL}\right)^{H}}\Bigg{\}}& =100\bigg{\{}1-\frac{\left(\overline{IL}\right)^{H}}{\left( \overline{IL}\right)^{D}}\bigg{\}}\end{split}