# Fiber Monitoring System Applied to Railway Bridge Structures in a Near-Fault Region
## 2 Measurement Theory
### Strain Distribution of Box Girder  
The main subject of measurement is the box girder. To accurately calculate the strain distribution of a three-dimensional object, its structural geometry must be accounted for [20]. A DFOS system of four sensing fibers is used in this research for accuracy of measurement and reliability.  
First, the coordinate system of the box girder is defined. Take the horizontal plane as the xy plane and the vertical axis as z. The positive x direction will be the direction in which the migration of the bridge propagates. The y direction will hence be perpendicular to the direction of the bridge. Figure 1 shows how the axis is defined.  
Figure 2 is a cross-section of the box girder, and it is divided into four dimensions regarding the y and z axes. There are four fibers parallelly installed on the box girder in the x direction of each dimension. Each of the fibers outputs a set of strain data. The strain values of each fiber are defined as \(\varepsilon 1\), \(\varepsilon 2\), \(\varepsilon 3\), and \(\varepsilon 4\). The vertical distance to the geometric center is c1, c2. The horizontal distance to the geometric center is d1, d2. The distance between the fiber and the geometric center is shown in Figure 2.  
Consider a single box girder as a simply supported beam. First, calculate the axial strain and curvature (Figure 3). Acquire the axial displacement by integrating axial strain. Based on the Euler-Bernoulli beam equation, it is possible to calculate the displacement for curvatures: the vertical displacement and the horizontal displacement.  
Define the variable for calculation.  
Axial strain: \(\varepsilon_{\mathrm{a}}\); Radius of Curvature: \(\rho\).  
Vertical curvature on the left: \(\mathrm{k_{vl}}=\frac{1}{\rho_{\mathrm{vl}}}\), and on the right: \(\mathrm{k_{vr}}=\frac{1}{\rho_{\mathrm{vr}}}\).  
Horizontal curvature on the top: \(\mathrm{k_{hu}}=\frac{1}{\rho_{\mathrm{hu}}}\), and on the bottom: \(\mathrm{k_{hd}}=\frac{1}{\rho_{\mathrm{hd}}}\).  
The strain equation linking the variables is as follows:  
\[\epsilon_{\mathrm{a}}-\mathrm{c_{1}k_{vr}}+\mathrm{d_{1}k_{hu}} =\epsilon_{1}\] \[\epsilon_{\mathrm{a}}-\mathrm{c_{1}k_{vl}}-\mathrm{d_{1}k_{hu}} =\epsilon_{2}\] \[\epsilon_{\mathrm{a}}+\mathrm{c_{2}k_{vl}}-\mathrm{d_{2}k_{hd}} =\epsilon_{3}\] \[\epsilon_{\mathrm{a}}+\mathrm{c_{2}k_{vr}}+\mathrm{d_{2}k_{hd}} =\epsilon_{4}\]  
Figure 3: The directional curvature.  
Figure 2: Cross-section of the box girder.  
This equation is further simplified considering the conditions for approximation.  
Vertical curvature: \(\mathrm{k_{v}}\cong\mathrm{k_{v1}}=\frac{1}{\rho_{\mathrm{v1}}}\cong\mathrm{k_{ vr}}=\frac{1}{\rho_{\mathrm{vr}}}\).  
Horizontal curvature: \(\mathrm{k_{h}}\cong\mathrm{k_{hu}}=\frac{1}{\rho_{\mathrm{pl}}}\cong\mathrm{k_{ hd}}=\frac{1}{\rho_{\mathrm{bd}}}\).  
Hence, the strain equations are simplified and written as a matrix:  
\[\left[\begin{array}{ccc}1&-\mathrm{c_{  
Double integrate the horizontal curvature to obtain the horizontal displacement \(\mathrm{u}_{y}\).  
\[\mathrm{u}_{y}=\iint\mathrm{k}_{\mathrm{h}}\cdot\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{x}\]  
Double integrate the vertical curvature to obtain the vertical displacement \(\mathrm{u}_{z}\).  
\[\mathrm{u}_{z}=\iint\mathrm{k}_{v}\cdot\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{x}\]