**3. Construction of Damage Indices**

*Damage Localization Index Based on Macro-Strain*

The reconstructed macro-strain response curve *εcg*(*Lm*, *t*) can obtain *εcg*(*Lm*, *ω*) by Fourier transformed.

$$\Xi\_{\rm c\S}(L\_{m\prime}\omega) = \int\_{-\infty}^{\infty} \psi(L\_{m\prime}t)e^{-j\omega t}dt = \text{Re}(\omega) + jIm(\omega) = |\mathcal{X}(\omega)|e^{j\phi(\omega)}\tag{9}$$

where, the amplitude is |*<sup>X</sup>* (*ω*)| = *Re*2(*ω*) + *Im*2(*ω*). *Re*(*ω*) and *Im*(*ω*) are, respectively, the real and imaginary parts of the Fourier function.

The macro-strain amplitude after Fourier transform was selected to perform a crosscorrelation function calculation among the elements, and the mutual energy density spectrum of the macro-strain amplitude corresponding to the frequency *ω* = <sup>1</sup> *<sup>t</sup>* was obtained, as expressed in Equation (10).

$$m\_{\left(n-1\right)\times n\omega}^{\omega\_{\omega}} = X(\omega\_{n-1}^{\omega\_{\omega}}) \times X(\omega\_{n}^{\omega\_{\omega}}) \tag{10}$$

where *X*(*ωωω <sup>n</sup>* ) and *X*(*ωωω <sup>n</sup>*−1) are the macro-strain response amplitudes at the frequency of the measuring points of elements *n* and *n* − 1, respectively.

The amplitude matrix of mutual energy density spectral between n elements in full frequency domain (or full-time domain) is obtained by calculating the cross-correlation function of the measured data of n long-gauge elements. The cross-correlation of a single element is self-energy density spectrum, and that of different elements is mutual energy density spectrum. The amplitude matrix has multiple forms. Therefore, to better describe its dynamic characteristics, all the types of deformations are uniformly named as the element energy product among the cross-correlated long-gauge elements in the frequency domain. This product is hereinafter referred to as the cross-correlation element energy product, which is equivalent to the amplitude vector of the macro-strain mutual energy density spectrum among the long-gauge elements.

where *n* ∈ Z, and the full-frequency domain is {*ω*<sup>1</sup> , *ω*<sup>2</sup> ··· *ωω*}.

For 32 long-gauge Bragg grating strain sensors pasted on the bridge, the bridge can be divided into 32 long-gauge elements, with each element producing a macro-strain, including macro-static strain and macro-dynamic strain. The data of two types are mixed, and the above-mentioned wavelet transform de-noising and reconstruction technology can be used for separating the macro-static strain and macro-dynamic strain. The long-gauge elements calculated using the cross-correlation function will increase by the square of the

number and is termed the cross-correlation long-gauge elements, that is, there will be 1024 cross-correlation long-gauge elements. Based on the initial element, each element superimposed by the total number of elements is called the cross-correlation long-gauge element group. In order to better display the damage location effect, 1024 cross-correlation element vectors were presented in the form of matrix coordinates of 32 rows and 32 columns, as expressed in Matrix (12). When a certain element is damaged, the coordinate of the corresponding matrix diagonal position changes abruptly.

$$
\begin{bmatrix}
(1,1) & \cdots & (1,32) \\
\vdots & \ddots & \vdots \\
(32,1) & \cdots & (32,32)
\end{bmatrix}
\tag{12}
$$

The above damage localization method is summarized as shown in Figure 3.

**Figure 3.** Damage localization.
