**1. Introduction**

Transportation is vital to regional economic development, and bridges are key to maintaining a smooth traffic flow. The normal operation of a transportation system relies heavily on the health of bridge structures [1,2]. To ensure the safety of bridge operation, several health monitoring systems have been established. However, existing monitoring systems are complex, expensive, and mostly applicable to long-span bridges. In highway and railway bridges, the vast majority of bridges are medium- and small-span bridges (the middle bridge generally has the total length of porous span between 30 m and 100 m or the single span between 20 m and 40 m, and the small bridge mostly refers to the total length of porous span between 8 m and 30 m or the single span between 5 m and 20 m). The medium- and small-span bridges have small span and large stiffness, so the vehicle–bridge coupling effect is obvious. At the same time, small- and medium-span bridges are prone to cracks and reduce the structural stiffness of bridges. However, because of its complex structure and soft structure, long-span bridges are often the key objects of structural health monitoring and detection, while small- and medium-span bridges lack attention. In recent years, due to the increase in vehicle load, structural deterioration, and structural form that do not conform to the current specification, the tilting erosion and fracture of medium- and small-span bridges have occurred. Throughout the domestic and

**Citation:** Zhang, H.; Zhong, Z.; Duan, J.; Yang, J.; Zheng, Z.; Liu, G. Damage Identification Method for Mediumand Small-Span Bridges Based on Macro-Strain Data under Vehicle–Bridge Coupling. *Materials* **2022**, *15*, 1097. https://doi.org/ 10.3390/ma15031097

Academic Editor: Francesco Fabbrocino

Received: 8 November 2021 Accepted: 24 January 2022 Published: 30 January 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

foreign bridge damage accident cases, the proportion of small- and medium-span bridge damage accidents accounted for the vast majority. Therefore, it is necessary to improve the real-time monitoring of the damage condition of medium- and small-span bridges, and develop a simple damage identification technology suitable for practical applications.

With the development of bridge construction and health monitoring technology, the damage identification method of bridge structure has developed rapidly. So far, many mature theoretical methods have emerged. The classification of monitoring data based on damage identification can be divided into a damage identification method based on static measurement data and a damage identification method based on dynamic monitoring data. The static-based damage identification method is reliable, but it is time-consuming, laborious, and expensive to interrupt the traffic in the detection, which is not conducive to the real-time evaluation of the bridge state. For example, static load test, percussion method, and millimeter wave radar method [3] based on non-destructive damage detection need to interrupt traffic for damage identification, which brings great inconvenience to people's travel. However, the damage identification method based on dynamic response can not interrupt the traffic, and only relying on environmental incentives can achieve the goal of online damage identification for bridges. Different from acceleration sensors, long-gauge strain sensors can not only reflect the overall information of the bridge, but also represent the local information of the structure. The strain index is more sensitive to local damage [4], whereas the displacement index is more effective for overall damage identification [5]. The sensitivity for local damage identification is arranged from low to high, and local damage in structures can be more easily identified using the strain index than the displacement index [6]. With the development of macro-strain measurement technology, the distributed strain sensing technology realizes the regional sensing, which overcomes the shortcomings of the traditional 'point' sensing that is too local, and overcomes the shortcomings of the traditional displacement sensing and acceleration sensing that are too macroscopic. It integrates the local and global information of structures, enables dynamic and static testing, and has advantages such as accurate damage localization and high quantitative accuracy [7–10]. Therefore, in this paper, the measured macro-strain data will be used as the data source for medium- and small-span bridge damage identification.

Based on macro-strain data, the following damage identification methods have been developed in recent years. Li et al. [11,12] employed a macro-strain mode vector with a long gauge as the damage index; it exhibited a high accuracy for damage localization and quantification with a good application effect for long-span flexible bridges. Wu et al. [8] proposed a structural damage identification method based on macro-strain modal vector. The modal parameters were extracted by strain response, and then the two-level strategy for flexible structure damage detection was proposed. Hong [13] proposed a method of only output modal macro-strain extraction and a bridge damage identification method under environmental excitation, and theoretically proved that the modal macro-strain of the strain sensor was uniquely determined by the peak value of the dynamic macro-strain response power spectrum density (PSD). Then, this method was applied to the state assessment of a practical bridge in New Jersey. Xu et al. [14,15] proposed a damage identification method for long-span cable-stayed bridges based on the residual trend of the macro-strain modal and the energy of macro-strain frequency response function. This method is clear in theory and easy to implement. According to the mapping relationship between long strain and displacement, Zhang et al. [10] deduced the macro-strain frequency response function, and proposed two structural strain modal identification methods based on it, and verified the effectiveness of the two methods. Anastasopoulos [16] identified the strain mode and characteristic frequency of the prestressed concrete beam model by macro-strain data to identify the damage and damage location of the structure. There are obvious changes in the amplitude and curvature of the strain mode in the damage area. However, medium- and small-span bridges have high rigidity. When the vehicle-to-bridge mass ratio is relatively high, the vibration effect due to vehicle–bridge coupling cannot be ignored. The structural system will reflect the time-varying vibration characteristics, and the non-stationary characteristics of the vehicle load will make it difficult to accurately obtain the stable strain mode of the bridge [15,17]. Clearly, identifying damages in a time-varying bridge based on the macro-strain modal test method is challenging, and the existing macro-strain theory cannot be directly applied to the damage identification of medium- and small-span bridges [18,19]. Hong et al. [4] established a static damage identification method based on the difference of macro-strain influence line area before and after structural damage by using a vehicle moving load. Li [20] proposed a damage identification method for urban road viaducts based on the working deformation of the macro-strain in the frequency-domain. This method can accurately identify damage locations and provides a good solution to the macro-strain problems encountered in the damage identification of medium- and small-span bridges. The idea is to advance the application of macro-strain-based damage identification to medium- and small-span bridges. Razavi and Hadidi [21] proposed a structural damage identification method based on finite element model correction and wavelet packet transform component energy, and verified the effectiveness and applicability of this method in structural damage identification by taking a two-dimensional steel truss structure as an example. The results show that this method can accurately determine the existence, location, and magnitude of damage, but the method requires the finite element model of the structure without damage. The performance of damage identification using macro-strain data is excellent, and the vehicle–bridge ratio of medium- and small-span bridges is large. When a vehicle passes the bridge, obvious macro-strain data can be measured, which provides a guarantee for subsequent damage identification. However, the complicated damage identification theory makes it challenging for ordinary engineers to grasp. Therefore, how to make the damage identification method based on macro-strain more easily applied in the damage identification of medium- and small-span bridges has become a research focus.

As a time–frequency analysis method, wavelet analysis has the characteristics of multiresolution and has a strong ability to represent the local characteristics of a signal, particularly when dealing with non-stationary signals. It can realize an effective decomposition and noise reduction of signal data [22]. Yu [23] conducted a modal analysis of a cracked bridge based on the structural dynamic equation of motion, studied the modal characteristics of the damaged bridge, and realized damage localization through the wavelet coefficients of the wavelet transform. Liu [24] employed the curvature of the displacement response and the cross-correlation function of the vehicle-excited bridge response as damage indices and applied the time–frequency analysis method of lifting wavelet transforms to identify bridge damage under moving loads. This method does not require damage-free structural response information, and the damage can simply be identified through the vibration response data of the bridge under different moving loads. Although it can effectively identify the damage location, the degree of damage cannot be easily quantified. Macro-strain damage identification methods are mostly based on empirical mode decomposition analysis, while wavelet transform for noise reduction and time–frequency analysis are rarely used.

Based on the above research, this paper reports a damage localization index in the form of an amplitude vector matrix of the mutual energy density spectrum based on macrostrain, constructed using wavelet transform de-noising and reconstruction technology and cross-correlation function. With the macro-dynamic strain and macro-static strain experimental data, the accuracy of the proposed damage localization index was verified in terms of the damage degree, damage location, and vehicle-to-bridge mass ratio. The experimental results showed that the proposed indicators could achieve precise damage localization for medium- and small-span bridge systems with different damage degrees under the action of vehicle–bridge coupling.

#### **2. Theoretical Basis**

#### *2.1. Damage Identification Based on Vehicle–Bridge Coupling Theory*

For a small-span bridge with a relatively high vehicle-to-bridge mass ratio, the moving vehicle-mounted action can be approximately considered vehicle–bridge coupling, and the vehicle–bridge system has time-varying characteristics [25–27]. This is shown in Figure 1. *P* is the force actions on the bridge; *t* is the vehicle travel time in the bridge; *v* is the speed of the vehicle; *M*<sup>1</sup> is the vehicle mass; *y*(*x*, *t*) is the vertical dynamic displacement of the bridge; *x* is the distance traveled by the vehicle on the bridge; *l* is the length of the bridge.

**Figure 1.** Vehicle–bridge time-varying system.

When the vehicle–bridge coupling effect is considered, the macro-strain generated by the moving vehicle-mounted action includes two categories. One is the macro-strain generated by the vehicle weight, i.e., the static strain. The other is the macro-dynamic strain due to the vehicle–bridge coupling. Therefore, based on the measured macro-strain data and wavelet transform, this study reconstructed and separated the macro-dynamic and static strains, and then used the reconstructed macro-static and dynamic strain data as the research object to perform an impact analysis on damage localization [19,28].

Based on the modal superposition method and D'Alembert's principle, a vehicle– bridge coupling model was established [29], as shown in Figure 2. *Ms* is the mass of the car body; *Mt*<sup>1</sup> represents the total mass of suspension device of vehicle front axle and tires; *Mt*<sup>2</sup> represents the total mass of rear axle suspension device and tires. *ks*<sup>1</sup> and *ks*<sup>2</sup> represent the spring stiffness of the front and rear axle suspension system of vehicle, respectively; *kt*<sup>1</sup> and *kt*<sup>2</sup> are, respectively, the stiffness of the front and rear axle tires of the vehicle. *cs*<sup>1</sup> and *ct*<sup>1</sup> are the damping of vehicle front axle suspension system and tires, respectively. On the contrary, *cs*<sup>2</sup> and *ct*<sup>2</sup> are the damping of vehicle rear axle suspension system and tires, respectively. *ys* is the vertical displacement of the car body. *θ* represents the rotation angle of the car body. *yt*<sup>1</sup> is the vertical displacement of vehicle front axle suspension system; *yt*<sup>2</sup> is the vertical displacement of vehicle rear axle suspension system; *yc*<sup>1</sup> and *yc*<sup>2</sup> are the vertical displacements of the front and rear axle tires, respectively.

**Figure 2.** Coupling effect of the vehicle and bridge.

The bridge vibration equations of the vehicle–bridge coupled vibration model shown in Figure 2 can be expressed as:

$$\left[ \left[ M\_{\upsilon} \right] \left\{ \ddot{y}\_{\upsilon} \right\} + \left[ \mathbb{C}\_{\upsilon} \right] \left\{ \dot{y}\_{\upsilon} \right\} + \left[ K\_{\upsilon} \right] \left\{ y\_{\upsilon} \right\} = \left\{ F\_{\upsilon - b} \right\} + \left\{ F\_{G} \right\} \tag{1}$$

$$[M\_b]\{\ddot{y}\_b\} + [\mathbb{C}\_b]\{\dot{y}\_b\} + [\mathbb{K}\_b]\{y\_b\} = \{F\_{b-v}\} \tag{2}$$

where [*Mv*], [*Cv*], and [*Kv*] are the mass, damping, and stiffness matrices of the vehicle, respectively; [*Mb*], [*Cb*], and [*Kb*] are, respectively, the mass, damping, and stiffness matrices of the bridge; {*yb*} and {*yv*} are the displacement vectors of the bridge and vehicle, respectively; {*Fv*−*b*} and {*Fb*−*v*} are, respectively, the combined force components of the vehicle–bridge coupling acting on the vehicle and bridge; {*FG*} is the gravity vector acting on the vehicle.

Based on the dynamic balance and displacement coordination relationship, the following matrix can be established:

$$
\begin{bmatrix} M\_{\upsilon} & \\ & M\_{b} \end{bmatrix} \begin{Bmatrix} \ddot{y}\_{\upsilon} \\ \ddot{y}\_{b} \end{Bmatrix} + \begin{bmatrix} \mathbb{C}\_{\upsilon} & \mathbb{C}\_{\upsilon-b} \\ \mathbb{C}\_{b-\upsilon} & \mathbb{C}\_{b} + \mathbb{C}\_{b-b} \end{bmatrix} \begin{Bmatrix} \dot{y}\_{\upsilon} \\ \dot{y}\_{b} \end{Bmatrix} + \begin{bmatrix} K\_{\upsilon} & K\_{\upsilon-b} \\ K\_{b-\upsilon} & K\_{b} + K\_{b-b} \end{bmatrix} \begin{Bmatrix} y\_{\upsilon} \\ y\_{b} \end{Bmatrix} = \begin{Bmatrix} F\_{\upsilon-b} + F\_{G} \\ F\_{b-\upsilon} \end{Bmatrix} \tag{3}
$$

In this formula, *Cv*−*b*, *Cb*−*v*, *Cb*−*b*, *Kv*−*b*, *Kb*−*v*, *Kb*−*b*, *Fv*−*b*, and *Fb*−*<sup>v</sup>* are the coupling terms generated by vehicle–bridge coupling. Generally, *Newmark* − *β* is used to solve the above coupling equation.

After solving for the displacement *yb* at each point in the bridge using this method, the angular displacement *ϕ<sup>b</sup>* of the point can be obtained by differentiating the displacement [30].

$$\varphi\_b(\mathbf{x}, t) = -\frac{\mathbf{d}y\_b(\mathbf{x}, t)}{\mathbf{d}x} \tag{4}$$

$$\overline{\varepsilon}(L\_m, t) = \frac{\varrho\_b(\mathbf{x}, t) - \varrho\_a(\mathbf{x}, t)}{L\_m} h\_m \tag{5}$$

where *ε* is the theoretical macro-strain, *Lm* is the gauge length, *hm* is the height of the neutral axis. *ϕa*(*x*, *t*) is the angular displacement of the point *a*, *ϕb*(*x*, *t*) is the angular displacement of the point *b*.

#### *2.2. Reconstruction of Macro-Strain by Wavelet Transform*

The measured macro-dynamic response *ε*- (*Lm*, *t*) of the bridge under environmental excitation and moving load excitation includes complex response signals, such as the vehicle load and environmental excitation [31,32], as expressed in Equation (6). Therefore, it is necessary to eliminate the influence of environmental excitation, such as noise, and extract useful strain signals of vehicle load from the complex macro-dynamic strain signals, so that the time-varying bridge damage can be effectively identified.

$$
\overline{\varepsilon}'(L\_{m\prime}t) = \overline{\varepsilon}(L\_{m\prime}t) + \overline{\zeta}(t) \tag{6}
$$

where, *ε*- (*Lm*, *t*) is the actual measured macro-strain, *ξ*(*t*) is the macro-strain response signal generated by environmental excitation.

In the actual engineering signal acquisition, the signal mostly contains many mutations and spikes, and the noise signal is not a stationary white noise signal. Therefore, when de-noising this non-stationary signal, the traditional Fourier transform cannot give the mutation of the signal at a certain time point, and it is difficult to effectively distinguish the mutation of the signal in the time domain, which makes it difficult to realize the accurate de-noising of non-stationary signals by Fourier transform. However, the wavelet transform is different. It can conduct signal analysis in both time and frequency domains at the same time, which can effectively distinguish the noise part of the signal and the mutation of the signal on the time axis, so as to complete the reasonable noise reduction of nonstationary signals. In the following, the measured macro-strain *ε*- (*Lm*, *t*) is decomposed

and reconstructed by wavelet transform to remove the influence of environmental noise on macro-strain.

$$\mathbb{E}'(L\_{\mathfrak{m}}, t) = \mathfrak{c}\_1 A\_1 + \mathfrak{c}\_1 D\_1 + \mathfrak{c}\_2 A\_2 + \mathfrak{c}\_2 D\_2 \cdots \mathfrak{c}\_i D\_i \tag{7}$$

where *i* is the number of decomposition layers, *ciAi* is the approximate part of the decomposition, *ciDi* is the detailed part of the decomposition, and the noise part is typically included in *ciDi*. The de-noise separation can be applied to obtain the reconstructed macro-strain *εcg*(*Lm*, *t*), as expressed in Equation (8).

$$\mathbb{E}\_{\mathbb{Q}}(L\_m, t) = c\_1 A\_1 + c\_2 A\_2 + \dots \cdot c\_i A\_i \tag{8}$$
