# Application of Fiber Bragg Grating Sensing Technology for Bol Force Status Monitoring in Roadways
## 2 Structure and Principle of FBG Bolt Force Sensor
### Measuring Principle
The relation between the deflection and strain of the cantilever is as follows [28]:  
\[y=\frac{l^{2}}{h}\varepsilon. \tag{7}\]  
Combining Equations (2)-(7), the strain can be obtained as follows:  
\[\varepsilon=\frac{\frac{3(1-\mu^{2})R_{0}^{4}}{16E_{1}t^{3}}(R_{0}^{4}-r_{0}^ {4}+4R_{0}^{2}r_{0}^{2}\ln(\frac{r_{0}}{R_{0}}))}{\pi(R^{2}-r^{2})\left[\frac {l^{2}}{h}+\frac{(1-\mu^{2})E_{2}bR_{0}^{2}t^{2}}{8\pi E_{1}t^{3}}\left(1-( \frac{r_{0}}{R_{0}})^{2}\frac{1-(\frac{r_{0}}{R_{0}})^{2}+4\ln^{2}(\frac{r_{0 }}{R_{0}})}{1-(\frac{r_{0}}{R_{0}})^{2}}\right)\right]}Q. \tag{8}\]  
When a deflection is generated on the cantilever, the strains of the FBGs bonded on its top and bottom surfaces are equal in size and opposite in direction (\(\varepsilon_{1}=-\varepsilon_{2}=\varepsilon\)). In addition, the two FBGs operate in the same temperature environment (\(\Delta T_{1}=\Delta T_{2}=\Delta T\)). The center wavelength changes of FBG1 and FBG2 are expressed as follows:  
\[\frac{\Delta\lambda_{\text{FBG1}}}{\lambda_{\text{FBG1}}}=(1-P_{e})\varepsilon +(\alpha+\zeta)\Delta T \tag{9}\]  
\[Q=P\pi\Big{(}R^{2}-r^{2}\Big{)} \tag{2}\]0  
The initial center wavelengths of the two FBGs are \(\lambda_{\text{FBG1}}=1530.859\) nm and \(\lambda_{\text{FBG2}}=1537.568\) nm, respectively. The initial center wavelengths are much larger than the wavelength changes caused by strain or temperature (at 1550 nm, the strain and thermal sensitive coefficients are 1.21 pm/\(\mu\varepsilon\) and 10.3 pm/\({}^{\circ}\)C), so the equivalent wavelength \(\lambda_{\text{FBG}}\) (taken as the smaller one out of \(\lambda_{\text{FBG1}}\) and \(\lambda_{\text{FBG2}}\)) can be used to replace two initial center wavelengths [29]. By subtracting Equation (10) from Equation (9), the difference between the wavelength changes of the two FBGs is obtained as follows:  
\[Q=P\pi\Big{(}R^{2}-r^{2}\Big{)} \tag{2}\]1According to Equation (8) and Equation (11), the relation between the two wavelength changes difference and the bolt force can be obtained as follows:  
\[Q=P\pi\Big{(}R^{2}-r^{2}\Big{)} \tag{2}\]2  
\[Q=P\pi\Big{(}R^{2}-r^{2}\Big{)} \tag{2}\]3  
where \(k\) is the sensor force sensitivity. The theoretical sensitivity \(k\) of the sensor is calculated to be a constant of 39.46 pm/kN according to the structural material parameters of the FBG bolt force sensor (Table 1). The modal analysis results of the diaphragm obtained using the ABAQUS finite element analysis software show that the diaphragm has a first-order intrinsic frequency of 3416.0 Hz, and the amplitude is the largest at the center of the diaphragm and decreases gradually from the center to the edge. Therefore, the designed sensor can perform dynamic high-frequency pressure measurement. Moreover, Equation (12) shows that the difference between two wavelength shifts is linear to the force. Compared with a single FBG wavelength shift as a sensing signal, this enhances the measuring sensitivity and effectively avoids the effect of temperature on measuring results due to irrelevance to any temperature change.