**3. Methods**

For eliminating the time-lag effect, the premise is obtaining the phase difference between the temperature and the temperature-induced strain. To scientifically and accurately acquire the phase difference, a method based on Fourier fitting [23] is proposed.

Since any continuous periodic signal can be composed of a set of appropriate sinusoids, the Fourier transform was firstly performed on the temperature and the temperature-induced strain [24]. Specifically, the original data was fitted to obtain the phase difference of various orders [25]. Then, the phase difference between the two signals was weighted and summed, so the total phase difference was obtained. Furthermore, the flow chart of the Fourier series expansion fitting method is shown in Figure 4.

According to Figure 5, the specific processes of the method are as follows.

**Figure 5.** The flow chart of the Fourier series expansion.

## *3.1. Temperature-Induced Response Separation*

The temperature history data and structural strain data of the same day are represented as *ftemp* and *f*sr, respectively. The wavelet decomposition and reconstruction method is used to separate the measured strain data. Consequently, the temperature-induced strain *fsr*,*tem* and live-load strain *fsr*,*load* are extracted. The principle and process of the method are described by Zhao et al. [20].

## *3.2. Fourier Frequency Decomposition*

The Fourier frequency decomposition in the time domain of *fsr*,*tem* and *ftemp* is used to obtain the frequency components of the data signal. The specific steps are as follows.
