4.2.1. Relationship between Working Stress and the ΔVpp

Due to the measurement under different working conditions, the changing trend between the stress of prestressed rebar and the Δ*Vpp* was basically the same. Therefore, a representative Δ*Vpp*-*σ* curve was selected from three diameters for further analysis. Because the design stress of 90% yield strength included the stress process of 50% and 70% yield strength design conditions, this paper selected specimens D16-P90-T1, D18-P90-T1, and D20-P90-T1 for discussion. In working stress monitoring, the working stress was unknown and needed to be evaluated based on the measured Δ*Vpp*. Therefore, the Δ*Vpp* was used as the abscissa and the stress converted by tension was used as the ordinate, which was recorded as Method 1. The *σ*-Δ*Vpp* curves of D16-P90-T1, D18-P90-T1, and D20-P90-T1 are shown in Figure 6.

**Figure 6.** The Δ*Vpp*-*σ* curves and fitted curves of three specimens: (**a**) D16-P90-T1; (**b**) D18-P90-T1; (**c**) D20-P90-T1.

As shown in Figure 6, during the unloading stage, the Δ*Vpp* decreased first and then increased with the decrease of rebar working stress. Therefore, when the increase of the Δ*Vpp* was observed, it could be considered that the working stress of the rebar had dropped to a low stress level relative to the design prestress. However, all three specimens had a Δ*Vpp* corresponding to two different rebar prestress levels, and the mapping relationship between the Δ*Vpp* and working stress could not be established. Therefore, the corresponding relationship between working stress and the Δ*Vpp* variation under each stress level was discussed in sections.

The whole unloading stage bounded by the turning point can be divided into two sections: the high stress section and low stress section. Since the importance of the two sections was the same, it was necessary to evaluate the fit effect as a whole. The Taylor expansion of Equation (13) was carried out, the higher order term after the third order was ignored, and Equation (14) was obtained. The first, second, and third orders of the corresponding relationship between the Δ*Vpp* and rebar working stress were discussed separately, as shown in Equation (15). The turning points of the corresponding curves of each specimen in Figure 6 were 119.37 MPa, 157.19 MPa, and 119.37 MPa, respectively. Taking the turning point as the dividing line, the three specimens were fitted to obtain the corresponding linear, quadratic, and cubic fit curves. Therefore, the goodness of fit (R2) was used to evaluate the fit effect. The R<sup>2</sup> of each specimen was calculated based on two segmented data.

$$
\sigma \approx \lg(0) + \lg'(0)(u) + \frac{\lg''(0)}{2!}(u)^2 + \frac{\lg'''(0)}{3!}(u)^3 \tag{14}
$$

$$\begin{array}{l}\sigma \approx a(\mu) + b\\\sigma \approx a(\mu)^2 + b(\mu) + c\\\sigma \approx a(\mu)^3 + b(\mu)^2 + c(\mu) + d\end{array} \tag{15}$$

The R2 is shown in Figure 7, demonstrating that as the order of fit increased, the R2 approached one. The R<sup>2</sup> of cubic polynomial fit was higher than that of quadratic polynomial fit and linear fit, and its R2 reached 0.98 on average. The cubic polynomial R2 of the specimen D20-P90-T2 was as high as 0.99781, which was close to 1. In addition, when the order increased from three to four, there was little room for improvement in the R2. Considered comprehensively, the cubic polynomial was selected for piecewise fit to explore the correlation between working stress and the Δ*Vpp*. To verify the feasibility of using cubic polynomial fit to determine the correlation between Δ*Vpp* and working stress, the Δ*Vpp* data of each specimen were fitted by cubic polynomial, and the R<sup>2</sup> was shown as follows.

**Figure 7.** The goodness of fit R2 of different fit methods of three specimens.

As shown in Figure 8, the R2 of D18-P50-T2 was at least 0.96928. The R2 of each specimen was more significant than 0.96, indicating a high degree of compliance with the cubic polynomial fit of the line between *σ*-Δ*Vpp*. Therefore, the working stress of the rebar could be determined from the *σ*-Δ*Vpp* curve.

**Figure 8.** The cubic polynomial R2 of the relationship between the Δ*Vpp* and working stress change under different working conditions.
