# Railway Bridge Condition Monitoring Using Numerically Calculated Responses from Batches of Trains
## 5 Bridge Health Monitoring
### Methodology  
The MR-IL derivation process is described for the simplest case of a 2-axle vehicle for which the 1 st and 2 nd axle weights, \(W_{A}\) and \(W_{B}\), are known. The moving reference deflection under Axle A is:  
\[\delta_{A}=W_{A}J^{0}(x)+W_{B}J^{1}(x,\ x-d) \tag{13}\]  
where \(d\) is the axle spacing. Similarly, the deflection under Axle B is:  
\[\delta_{B}=W_{A}J^{1}(x-d,x)+W_{B}J^{0}(x-d) \tag{14}\]  
According to [PERSON]'s Reciprocal Theorem, the deflection at \(x\) due to a unit load at \((x-d)\) equals the deflection at \((x-d)\) due to a load at \(x\), i.e.,  
\[J^{1}(x,\ x-d)=J^{1}(x-d,x) \tag{15}\]  
Hence, the equations above can be manipulated to remove all but the fundamental \(J^{0}\) terms:  
\[W_{A}\delta_{A}-W_{B}\delta_{B}=W_{A}^{2}\ J^{0}(x)-W_{B}^{2}J^{0}(x-d) \tag{16}\]  
Using Equation (16), the fundamental MR-IL, \(J^{0}(x)\), can be found from the moving reference deflections, \(\delta_{A}\) and \(\delta_{B}\), using an approach similar to the Matrix Method proposed by [PERSON] et al. [43]. The process consists of finding the influence line ordinates that best fit the corresponding measured values. An error function is defined as the sum of the squared differences between the weighted measured deflections (left hand side of Equation (16)) and the corresponding theoretical values (right hand side of Equation (16)). Expressing the data in terms of scan number, \(k\), instead of distance, \(x\), gives:  
\[E=\sum_{k=1}^{K}\Bigl{\{}W_{A}\delta_{A,k}-W_{B}\delta_{B,k}-W_{A}^{2}\ J_{k}^{0}+W_{B}^{2}J_{k-c}^{0} \Bigr{\}}^{2} \tag{17}\]  
where \(K\) is the total number of scans and \(c\) is the number of scans corresponding to the distance between the two axles, \(d\). The best fit fundamental MR-IL is that which minimizes \(E\). The minimum error function is found by taking partial derivatives with respect to each influence ordinate, \(J_{k}^{0}\), and setting them to zero. There are \(K\) scans for which at least one axle is on the bridge and for a 2-axle vehicle, \(K-c\) scans corresponding to the bridge length. Hence, there are \(K-c\) unknown influence ordinate values. Setting all partial derivatives to zero results in a system of \(K-c\) simultaneous equations:  
\[[A]\{J^{0}\}=\{b\} \tag{18}\]  
Figure 5: Deflection responses to moving unit load: (**a**) Response at \(x\) to load at \(x\); (**b**) Response at \(x\) to load at \(y\). Here, \(\left\{J^{0}\right\}\) is the fundamental MR-IL vector and \(\left[A\right]\) is a sparse symmetric matrix consisting of axle weight terms:  
\[\left[A\right]=\left[ \begin{array}{ccccccccc}a_{1,1}&0&\cdots&a_{1,1+c}&0& \cdots&&&\\ &\ddots&&&\ddots&&&\vdots\\ &&&a_{i,i}&0&\cdots&a_{i,i+c}&0&\cdots\\ &&&\ddots&&&\ddots&&\ddots&\\ \text{Symmetric}&&&a_{K-2c,K-2c}&0&\ldots&a_{K-2c,K-c}\\ &&&&\ddots&&\vdots&\\ &&&&&&&0\\ &&&&&&&a_{K-c,K-c}\end{array} \right] \tag{19}\]  
where the main diagonal elements are:  
\[a_{i,i}={W_{A}}^{4}+{W_{B}}^{4} \tag{20}\]  
and the upper triangular elements are given by:  
\[a_{i,i+C}=-W_{A}^{2}W_{B}^{2},\;\;\;i+c\leq K-c \tag{21}\]  
The right-hand side of this system of equations consists of a vector, \(\left\{b\right\}\), with components made up of axle weight and measured MR deflection terms:  
\[\left\{b\right\}=\left\{ \begin{array}{c}W_{A}^{3}\delta_{A,1}-W_{A}W_{B}^{2} \delta_{A,c+1}+W_{B}^{3}\delta_{A,c+1}\\ W_{A}^{3}\delta_{A,2}-W_{A}W_{B}^{2}\delta_{A,c+2}+W_{B}^{3}\delta_{B,c+2}\\ \vdots\\ W_{A}^{3}\delta_{A,c}-W_{A}W_{B}^{2}\delta_{A,c+c}+W_{B}^{3}\delta_{B,c+c}\\ W_{A}^{3}\delta_{A,c+1}-W_{A}^{2}W_{B}\delta_{B,c+1}-W_{A}W_{B}^{2}\delta_{A,2 c+1}+W_{B}^{3}\delta_{B,2c+1}\\ \vdots\\ W_{A}^{3}\delta_{A,K-2c}-W_{A}^{2}W_{B}\delta_{B,K-2c}-W_{A}W_{B}^{2}\delta_{ A,K-c}+W_{B}^{3}\delta_{B,K-c}\\ W_{A}^{3}\delta_{A,K-2c+1}-W_{A}^{2}W_{B}\delta_{A,K-2c+1}+W_{B}^{3}\delta_{B,K- c+1}\\ \vdots\\ W_{A}^{3}\delta_{A,K-c}-W_{A}^{2}W_{B}\delta_{A,K-c}+W_{B}^{3}\delta_{B,K}\\ \end{array}