## 2 Action Mechanism and Calibration of the Optical-Fiber-Embedded Beam (OFEB)
### Test Principle of the Optical-Fiber-Embedded Beam  
A 0.9 mm diameter single-mode tight-jacketed optical fiber was symmetrically embedded on the upper and lower surfaces of a flexible pipe to construct the OFEB, as shown in Figure 3a. The optical attenuation of the fiber is less than 0.3 dB/km (1550 nm). In addition, it exhibits a long-term 3 N mechanical tensile strength and a compressive strength of 10 N/10 cm, allowing it to adapt to the tension and compression states of the flexible pipe. Moreover, its ambient operating temperature range of \(-40\)\({}^{\circ}\)C to 75 \({}^{\circ}\)C makes it ideal for most engineering environments.  
As shown in Figure 3b, when the OFEB deforms with the subgrade soil, the vertical displacement of the subgrade manifests as a deflection of the OFEB. The deflection of the OFEB can be calculated by integrating the axial strain measured by an optical fiber. The computation is based on the bending deformation theory for a beam. Assuming the flexible pipe beam possesses elastic deformation, the deflection curve of the OFEB may then be  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Indicators** & **Item** & **Indicators** \\ \hline Instrument function & Temperature, strain & Spatial resolution & 5 cm \\ Operating wavelength & 1550 nm & Sensing distance & 5 km \\ Frequency domain & 10-13 GHz & Accuracy & 1 \({}^{\circ}\)C/5 \(\mu\)c \\ Frequency step & 1 MHz (min) & Measuring time & 2 s to a few minutes \\ Sample interval & 2.5 cm (min) & Sensing fiber type & SM/PM fiber \\ Range of strain & \(-\)15,000–15,000 \(\mu\)c & Connector type & FC/APC \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance indicators of the designed instrument.  
Figure 3: (**a**) Setup and (**b**) working principle of the OFEB.  
Figure 2: Schematic of the DPP-BOTDA system. PC: polarization controller; EOM: electro-optic modulator; DC: DC power supply; AFG: arbitrary function generator; EDFA: erbium-doped fiber amplifier module; PD: photo-detector; DAQ: data acquisition.  
determined by measuring the upper and lower surface strains, the beam curvature radius, and the beam deformation angle.  
Figure 4 is a schematic of the deflection of the beam, which is expressed as \(\omega=f(x)\). Assuming plane and small deformation of the beam section and considering the mechanical conditions under which deformation occurs and the definition of the curvature of the beam section, the curvature at any point in the beam can be expressed as follows:  
\[\left\{\begin{array}{l}\frac{1}{\rho}=\left(\varepsilon_{a}-\varepsilon_{b} \right)/D\\ \frac{1}{\rho}=\frac{ds}{d\theta}=\frac{\omega^{\prime\prime}}{\left(1+\omega ^{\prime\prime}\right)^{3/2}}\approx\omega^{\prime\prime}\end{array}\right\} \tag{1}\]  
where at any point of the beam, \(\rho\) is the radius of the curvature; \(\varepsilon_{a}\) and \(\varepsilon_{b}\) are the upper and lower surface strains, respectively; \(\theta\) is the tangent angle; \(s\) is the arc length; \(\omega^{\prime}\) is the slope of the tangent line; \(\omega^{\prime\prime}\) is the change rate of the slope; and \(D\) is the outside diameter of the beam.  
Formula (1) can be used to deduce:  
\[\omega=\int\left(\int\frac{1}{D}(\varepsilon_{a}-\varepsilon_{b})dx\right)dx +C_{1}x+C_{2} \tag{2}\]  
where \(C_{1}\) and \(C_{2}\) are the integration constants.  
According to the boundary conditions on the beam shown in Figure 1,  
\[\left\{\begin{array}{l}x=0,\ w=0\\ x=0,\ w^{\prime}=\theta=0\end{array}\right. \tag{3}\]  
Thus, \(C_{1}=C_{2}=0\).  
The beam deflection can be calculated as follows:  
\[\omega=\int\left(\int\frac{1}{D}(\varepsilon_{a}-\varepsilon_{b})dx\right)dx \tag{4}\]  
where \(\omega\) is the deflection at any point of the beam.  
According to Formula (4), the strains of the upper and lower surface fibers caused by ambient temperature have been eliminated in the process of calculating the subtraction. Thus, it is not necessary to perform temperature compensation for the OFEB.