*4.1. Reconstruction and Separation of Macro-Strain by Wavelet Transform*

After sorting the macro-strain data obtained from the vehicle–bridge coupling experiment, the overall strain diagram of the moving vehicle while passing through the bridge was obtained, as shown in Figure 7. The wavelet basis function mentioned above was used for noise reduction and reconstruction to obtain the time history curve of static strain generated by the moving vehicle-mounted action on the bridge, as shown in Figure 8a. Strain separation was then performed to obtain the macro-dynamic strain curve generated by the vehicle–bridge coupling action, as shown in Figure 8b.

**Figure 7.** Overall strain diagram.

In the case of vehicle–bridge coupling, a time history curve of static strain is generated by the weight of the vehicle, which is not easily affected by the external environment and is suitable for quantitative damage identification. However, a macro-dynamic strain is generated by vehicle–bridge coupling, which has a small value and is easily affected by the external noise environment, making it suitable for damage localization and identification. Therefore, in this study, damage localization is performed on the basis of the damage index and time history curve of the macro-dynamic strain.

Illustration: Based on the change in the wave peak, 32 strain curves generated by the force of 32 long-gauge elements in the mid-span circuit of the upper roof of the box girder as shown in Figures 9 and 10.

**Figure 8.** Strain reconstruction and separation under vehicle-bridge coupling; (**a**) Static strain time history diagram; (**b**) Dynamic strain time history diagram.

**Figure 9.** Macro-dynamic strain under damage condition for vehicle-to-bridge mass ratios; (**a**) 1.22; (**b**) 3.14.

**Figure 10.** Macro-dynamic strain cross-correlation based on damage level of 2 with different vehicleto-bridge mass ratio; (**a**) 1.22; (**b**) 1.84; (**c**) 2.45; (**d**) 3.14.

#### *4.2. Damage Localization and Identification*

Based on the separated and reconstructed macro-dynamic strain data, this section focuses on analyzing the impacts of damage degree, vehicle-to-bridge mass ratio, multiple damage locations, and other factors on the damage localization index.

#### 4.2.1. Vehicle-to-Bridge Mass Ratio

When the vehicle-to-bridge mass ratio is high, the evident vehicle–bridge coupling effect generates a macro-dynamic strain. Therefore, it is of great significance to explore the influence of the vehicle-to-bridge mass ratio on the damage localization index. Therefore, we conducted a macro-dynamic strain analysis during the actual vehicle-mounted action period; this section presents the results. Figure 9a,b show the macro-dynamic strains under the second-order damage condition with vehicle-to-bridge mass ratios of 1.22 and 3.14, respectively.

Illustration: The figures show 32 macro-dynamic strain curves indicated by different colors, which are 32 long-gauge elements in the middle span line of the upper roof.

Figure 9 shows that the macro-dynamic strain data of the 32 long-gauge elements fluctuate around 0. The vibration is most intense at 7.5 s, 10–13 s, and 14–15 s. The actual movement time of the mobile trolley is 15 s, and the trolley is controlled by a traction motor. Its speed is set to 0.2 m/s, and it takes 16 s to complete the entire process. Therefore, during the movement process of the mobile trolley, there is a condition of variable speed, that is, the 32 long-gauge elements are affected by non-stationary excitation (the process of

shifting the mobile trolley is equivalent to the impact of non-stationary excitation on the bridge, making the macro-strain data more complicated). Through the analysis of the actual damage location, the mobile trolley was found to reach the middle span of the bridge at 7.5 s, reach the damage location in approximately 10–13 s, and completely leave the bridge after 15 s. The vehicle-to-bridge mass ratios of the two types have evident and sudden oscillations at 7.5 s and 10–13 s, as shown in Figure 9a,b. However, all the 32 long-gauge elements vibrate at 7.5 s, whereas in the 10–13 s range, the oscillation is only of a few long-gauge elements, as indicated by the green line in the figure. Based on the analysis of the above characteristics and the calculation method of the damage localization index, the energy product of the macro-dynamic strain cross-correlation element is obtained, as shown in Figure 10.

A total of 32 long-gauge elements were pasted on the top plate of the beam. Through the damage localization index calculation, we find 1024 cross-correlation long-gauge elements, including 32 cross-correlation long-gauge groups as shown in matrix (12). The meaning of this element group is that the mechanical vibration of each long-gauge element and the first long-gauge element are correlated. If the first long-gauge element is damaged, the macro-strain response generated by the force will be different from the macro-strain generated by the other long-gauge elements, and the calculation based on the damage localization index will make it different from the other elements. Figure 10 shows that the energy product of the cross-correlation element has an evident mutation in the (26, 26) cross-correlation long-gauge element group. Therefore, it is possible to locate the damage in the 26th long-gauge element from the sudden change in the figure.

As the vehicle-to-bridge mass ratio increases, the abrupt value of the energy product of the cross-correlation element also increases. Under the different vehicle-to-bridge mass ratios, the macro-dynamic strain generated by the vehicle–bridge coupling can realize damage location identification based on the damage localization index and the higher the vehicle-to-bridge mass ratio, the more evident the damage localization effect.

#### 4.2.2. Analysis of the Influence of Multiple Damage Locations

Based on the reconstructed and separated macro-dynamic strain data, an identification analysis was conducted at multiple damage locations. Through the calculation of the damage localization index, the energy product values of the cross-correlation elements with different damage degrees at two damage locations were obtained, as shown in Figure 11.

Figure 11 shows mutations in the 6th and 26th cross-correlation long-gauge element groups, and the mutation rules are the same as those described in Section 4.2.1. More specifically, (6, 6) has a sudden change in the energy product at both (26, 26) and the two cross-correlation long-gauge element groups. The mutation becomes more and more evident as the damage level increases at both damage locations. However, when both damage levels reach grade four damage, they do not reach the maximum of (6, 6) and (26, 26) at the same time. Therefore, when there are multiple damages, the cross-correlation element energy product between the two damages will be weakened, which will lead to the damage localization index. The damage localization effect is reduced, particularly for minor damages. However, it is still possible to locate the damage at the 6th and 26th long-gauge elements based on the above rules. The damage localization index can realize the localization and identification of multiple damages based on the macro-dynamic strain. The greater the damage, the better the localization effect.

**Figure 11.** Macro-dynamic strain cross-correlation based on two damage locations with different damage levels; (**a**) Level 1 + Level 4; (**b**) Level 2 + Level 4; (**c**) Level 3 + Level 4; (**d**) Level 4 + Level 4.

#### 4.2.3. Analysis of the Influence of the Damage Degree

To further analyze whether the damage degree would affect the positioning identification of the localization index, damage positioning research was performed with the localization index based on the macro-dynamic strain under the condition of D0–D4 for a 32 kg mobile vehicle. Through the calculation, the energy product of the macro-dynamic strain cross-correlation element was obtained, as shown in Figure 12.

Figure 12 shows the damage identification effects of the damage location index under different damage degrees. As it can be seen from Figure 12a, when the bridge is in the condition of non-destructive, the maximum value is at (16, 16), where the mid-span element of the beam is located. By comparison with Figure 12b–e, it can be seen that with the increase in damage degree, the value at (26, 26) gradually increases, and the value at (16, 16) gradually decreases. Therefore, it can be seen that damage occurred at the 26th long-gauge element, and the greater the damage degree, the more obvious the damage localization effect.

**Figure 12.** Macro-dynamic strain cross-correlation based on a single damage location with different damage levels; (**a**) Non-destructive; (**b**) Level 1; (**c**) Level 2; (**d**) Level 3; (**e**) Level 4.

In conclusion, for the macro-dynamic strain after wavelet reconstruction and separation, the damage localization index can still be used for damage localization and identification. In terms of the vehicle–bridge mass ratio, multiple damage location, and damage degree, the higher the vehicle-to-bridge mass ratio, the greater the damage degree, the more evident the positioning effect of the damage localization index. After the reconstruction and separation, the macro-dynamic strain was not further de-noised by wavelet transform; therefore, the damage localization index based on the maximum cross-correlation number has certain anti-noise performance. Macro-dynamic strain is generated by vehicle–bridge coupling action; this confirms the feasibility of the damage localization index and identification method employed for the time-varying bridge based on the damage localization index under vehicle–bridge coupling action.
