*4.2. Phase Subtraction*

The separated temperature-induced strain and temperature data was fitted subject to the Fourier decomposition. Furthermore, the temperature fitting curves of the 1st order, 4th order, and 8th order decomposition on 20 April 2017 are shown in Figure 7a. With the increase of the Fourier expansion order, the consistency was better, as shown in Figure 7b. When the data was expanded to the 6th order, the accuracy requirement was met.

**Figure 7.** *Cont*.

**Figure 7.** The Fourier curve fitting results (20 April 2017).

The direct consequence of the time-lag phenomenon is the decline in correlation between the temperature and the temperature response. The hysteresis loop area directly reflects the degree of the time-lag phenomenon. Specifically, the larger the area, the more significant the lag effect. Therefore, the hysteresis loop area and the correlation coefficient can be used as indicators to verify the effectiveness of subtracting the time-lag phenomena by the Fourier series expansion method.

With the Fourier fitting method, the phase difference of the measured temperature and strain data on 22 July 2017 (summer) was calculated. The lag time of the S8 strain measuring point and T4 temperature measuring point can be calculated through Equation (9). As a result, the lag time was approximately 176 min in the summer season. Then, a translation of the temperature data by the lag time achieves the goal of subtracting the time-lag phenomenon. The temperature and strain correlation before and after the subtraction is shown in Figure 8. Using the same method to subtract the time-lag phenomenon in winter, Figure 9 is obtained. Meanwhile, through Equation (9), the lag time was calculated to be around 129 min in the winter season. As mentioned before, the displacement and temperature data for steel bridges possess a lag time of approximately 45 min in the research by Guo et al. [21]. As a result, the lag time in concrete structures is longer than that in steel structures.

(**a**) Before the phase difference is translated

(**b**) After the phase difference is translated

**Figure 8.** Temperature vs. strain plots in summer (22 July 2017).

**Figure 9.** Temperature vs. strain plots in winter (27 November 2017).

By calculating the hysteresis areas and lag time of raw data and phase difference eliminated data, Table 1 was obtained. From Table 1, the hysteresis curve and lag times in winter with those in summer, respectively, were compared. Consequently, after subtracting, the hysteresis loop was notably reduced and the correlation coefficient was significantly improved. Therefore, the Fourier fitting method was verified to be effective.


**Table 1.** Comparison of indicators before and after phase difference elimination.

With the proposed method, the time-lag phenomenon can be reduced. Furthermore, the long-term stable relationship between temperature load and the structural corresponding response can be more clearly reflected. Specifically, the same temperature and strain measurement points of four days in the four seasons of 2017 were selected to draw the temperature–strain curve. Then, the subtracted data curve was plotted in Figure 10. It is clear to see, although the specific distribution difference of strain varying with temperature exists in different seasons, the hysteresis loops in four seasons possess similar slopes overall. In particular, by eliminating the phase difference, the reduced curve can more clearly reflect the corresponding linear relationship characteristics of temperature-induced effects and temperature in different seasons. This feature reflects the specific mapping relationship between temperature and structure response and it is related to structure physical properties, such as structural geometry and material properties.

In summary, the wavelet decomposition and reconstruction method is useful and handy to extract temperature-induced strain. The lag time in a concrete structure is longer than that in a steel structure. Moreover, according to the comparison of hysteresis loop areas and correlation coefficient before and after subtracting the time-lag effect, the effectiveness was verified.

**Figure 10.** Temperature–strain curve and the subtracted data curve in different seasons.
