## 2 Theoretical Analysis of Mass Concrete Strain Monitoring
### Temperature Strain Measurement Based on Vibrating Wire Strain Sensor  
The embedded vibrating wire strain sensor is mainly composed of two tension and compression sheets, protective sleeves, steel strings, thermistors, coils and cables. Thermistors are used to measure temperature data synchronously. The embedded vibrating wire strain sensor is embedded in the concrete to coordinate the deformation with the structure. The deformation is transferred to the steel string through two tension and compression sheets, and the deformation is transformed into the stress change of the steel string, thus changing the vibration frequency of the steel string. The electromagnetic coil excites the steel string and measures its vibration frequency. The frequency signal is transmitted to the reading device through the cable, and the strain inside the structure can be measured.  
Figure 2 is the measurement principle of the strain sensor. When there is no external force in the initial state, the steel string vibrates with stable amplitude according to the initial stress. The working state conforms to the condition of soft and undamped micro vibration. The vibration frequency can be determined by the following formula:  
\[f_{0}=\frac{1}{2L_{0}}\sqrt{\frac{\sigma_{0}}{\rho}} \tag{6}\]  
In the formula,\(f_{0}\) is the initial frequency (Hz); \(L_{0}\) is the initial effective length of steel string (mm); \(\rho\) is the density of the steel string; and \(\sigma_{0}\) is the initial stress of the steel string.  
When an external force \(\Delta P\) is applied, the strain sensor is stretched or compressed accordingly. The stress of the steel string increases or decreases,  
Figure 1: Schematic diagram of a simply supported beam subjected to temperature.  
Figure 2: Schematic diagram of embedded vibrating wire strain sensor.  
also increases or decreases to \(f\). Because the change of the length of the steel string is a minor change compared with its own length, then \(L_{0}+\Delta L\approx L_{0}\).  
\[f=\frac{1}{2L_{0}}\sqrt{\frac{\sigma_{\rm WW}}{\rho}} \tag{7}\]  
Because the mass \(m\), length \(L_{0}\), cross-sectional area \(A_{\rm WW}\) and elastic modulus of the steel string \(E_{\rm WW}\) can be regarded as constants, the relationship between the stress of the steel string and the output frequency can be regarded as:  
\[\sigma_{\rm WW}\propto f^{2} \tag{8}\]  
The strain increment of the steel string is:  
\[\varepsilon_{\rm WW}=\kappa\cdot\left(f^{2}-f_{0}^{2}\right) \tag{9}\]  
The vibrating string strain sensor takes the frequency modulus as a strain output signal. The frequency modulus is \(F=f^{2}\times 10^{-3}\). By combining Formulas (4) and (9), it can be further obtained that the strain value of the structure under temperature change can be obtained by the following formula:  
\[\varepsilon_{2}=\kappa G\cdot\left(F_{1}-F_{0}\right)+\gamma\cdot\left(T_{1}- T_{0}\right) \tag{10}\]  
In the formula, \(G\) is the instrument standard coefficient, and the unit is \(\mu\varepsilon/\)Digit (Digit = \(f^{2}\times 10^{-3}\)); \(\gamma\) is the temperature correction coefficient; and \(T_{0}\) and \(T_{1}\) are the initial temperature and current temperature, respectively [17].