4.2.3. Working Stress Monitoring Error Analysis

To propose a more reliable evaluation method for the working stress of vertical prestressed rebar, the errors of Method 1 and Method 2 proposed were compared. The calculation steps can be shown in Figure 11.

**Figure 11.** Calculation flow chart.

For Method 1, the curve fit degree was good; all groups' R2 were greater than 0.96. This showed that working stress had an excellent functioning relationship with the Δ*Vpp*. The fit relationship was generalized to Equation (16). The measured Δ*Vpp* was substituted into the fit equation. The results were compared with the actual measured working stress. The relative error values of each specimen were calculated as shown in Figure 12.

$$F = AV^3 + BV^2 + CV + D \tag{16}$$

**Figure 12.** The relative error values of the fitted and measured values of Method 1.

From the error analysis of the fitted and measured values under the unloading stage, it was found that the relative error values did not exceed 20% under any working conditions. The relative error values were concentrated below 10% in the high stress section. The measured Δ*Vpp* was substituted into the corresponding fit equation under different working conditions. The percentage of relative error at the turning point of D20-P90-T1 was the highest, 18.21%, and the maximum relative error between the fitted and measured value was 21.73 MPa. The high relative errors were concentrated near the turning point. Therefore, increasing the measurement points near the measured turning points during the calibration in the laboratory could significantly reduce the relative error. The relative error values of all specimens were normalized, and the average relative error values with robustness were used for comparative analysis. The maximum average relative error values

for Group 1, Group 2, and Group 3 at different stress levels were 3.68%, 4.16%, and 2.79%, respectively. The average relative error values for all specimens were less than 5%, close to the results of the REME method for testing strand stresses [28].

For Method 2, the R2 were greater than 0.90. This showed that working stress has a good linear correlation with the *d*Δ*Vpp*. The measured induced voltage peak-to-peak values were substituted into Method 2. The results were compared with the actual working stress. To ensure the consistency of the *d*Δ*Vpp* loading step, only the prestress levels with a design stress above 20% of the yield strength ratio were analyzed. The results are shown in Figure 13.

**Figure 13.** The relative error values of the fitted and measured values of Method 2.

More than 75% of the test points had relative error values below 20%, and the maximum relative error value was 23.85%. The errors of all specimens were normalized and the average relative error value with robustness was used for comparative analysis. The results are shown in Figure 12. In the error analysis, it was found that the maximum average relative error values of each group were 9.77%, 8.20%, and 14.53%, respectively. Therefore, under any working conditions, the average relative error values were less than 15%, better than the 25% average relative error value of the ultrasonic guided wave method [42]. This result showed that using Method 2 to monitor vertical prestressed rebar's working stress loss had good reliability. However, the error was greater than that of the traditional magnetoelastic method [22].

In summary, Method 1 could avoid high error by increasing the measurement points near the turning point. Therefore, the Method 1 test error value can be considered as low and could meet engineering needs. Method 2 avoided the uncertainty of the turning point in the laboratory calibration process, but its error was greater than the traditional magnetoelastic method. Therefore, the cubic polynomial segmental fit (Method 1) was selected to establish the mapping relationship between working stress and the Δ*Vpp*. Then, the working stress monitoring method of prestressed rebar based on magnetic resonance was proposed.

#### **5. Conclusions**

In this paper, the relationship between the sensor induced voltage and the rebar stress was derived based on the electromagnetic induction law, magnetoelastic effect, and magnetic resonance theory. Working stress monitoring experiments with different design stress levels were carried out for rebars with diameters of 16 cm, 18 cm, and 20 cm. The induced voltage peak-to-peak values under working stress variations were collected with a magnetic resonance sensor. The main conclusions were as follows:

(1) The curves of the working stress and the induced voltage peak-to-peak values at different design stress levels showed nonlinear correlation. Due to the hysteresis effect, the induced voltage peak-to-peak values measured in the loading stage differed

from those in the unloading stage. Two characteristic indicators, the Δ*Vpp* and *d*Δ*Vpp*, were proposed for evaluating the working stress. The correlation between the two characteristic indicators and the working stress was analyzed. On this basis, the mapping relationships from the characteristic indicators to the working stress were obtained by nonlinear fitting and linear fitting, respectively.


This paper provided a new method for the working stress monitoring of vertical prestressed rebars.

**Author Contributions:** J.X.: methodology, validation, investigation, and writing—original draft; S.Z.: term and writing—review and editing; H.L.: validation and investigation; L.L.: software, formal analysis, and data curation; Y.S.: methodology and formal analysis. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (52278146, 51978111), the Chongqing Natural Science Foundation of China (cstc2022ycjh-bgzxm0086), and the lnnovation Fund Project of Graduate Education of Chongqing Jiaotong University (2023S0022).

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.
