Damage identification of Chi River bridge based on vehicle excitation and WPEVCR

In order to identify damage of Chi River bridge’s superstructure, a damage identification indicator is implemented in the field test, which involves the wavelet packet energy analysis with the feature of dynamic response signals caused by vehicle excitation. On the basis of the field test, a series of numeric models with varied service conditions were developed. The wavelet packet analysis method was utilized to decompose the bridge’s acceleration signals at both healthy and damaged status, and the values of Wavelet Packet Energy Variance Change Rate (WPEVCR) were obtained. Then, according to the acceleration signal data measured from the field test, the damage assessment of the condition of Chi River bridge was performed by means of the obtained WPEVCR. The results demonstrate the capability of WPEVCR in localizing and quantifying the bridge damage status. Moreover, another damage indicator based on the Hilbert-Huang Transform (HHT) has been also employed to verify the assessment of WPEVCR, and both damage identification approaches indicate that the Chi River bridge is in a healthy service condition.

et al. 2020).Taha et al. suggested the use of wavelet multiresolution analysis (WMRA) as a reliable tool for digital signal processing in SHM systems (Taha et al. 2004).Bayissa et al. presented a new damage identification technique based on the statistical moments of the energy density function of the vibration responses in the time-scale (or time-frequency) domain (Bayissa et al. 2008).Ding et al. described the development of a multistage scheme for structural damage warning of the Runyang Cable-stayed Bridge using the measured dynamic responses from an online instrumentation system (Ding et al. 2010).Laboratory study shows that the feature vibration can be extracted successfully by reconstructing the wavelet coefficients (Chen and Gao 2011).Lee et al. proposed a continuous relative wavelet entropy-based reference-free damage detection algorithm for truss bridge structures (Lee et al. 2014).The wavelet analysis in one and two dimensions is applied to investigate and assess the movement behavior of a high-speed train railway bridge by applying acceleration measurements (Mohamed et al. 2017).Exploiting the dynamic response of bridge piers induced by vehicle braking excitation, Yang et al. put forward damage indicators over characteristics in acceleration signals and successfully identified the pier's scouring conditions of a specific bridge (Yang et al. 2021).Yu et al. used the ECD method of wavelet packet energy to identify damage in a replaced bridge beam of the Cangzhou Ziya River and investigated the influence of wavelet functions and decomposition levels on the identification performance (Yu et al. 2013).Wang et al. developed the approach of ECD based on wavelet packet energy to locate and identify damages in ancient wooden structures using the Xi'an Bell Tower as an engineering reference (Wang, et al. 2014).Zhu and Sun et al. introduced the WPERSS indicator and verified its accuracy with simply supported beam and cable-stayed bridge models, considering the influence of noise (Zhu and Sun 2015).She et al. adopted WPERSS combined with machine learning algorithms to assess the state of a three-span bridge (She et al. 2023a).For the damage detection of rural old arch bridges, Zhang Yu et al. introduced an optimized random forest (RF) damage identification system by means of a particle swarm (PSO) algorithm and verified the accuracy of this method by simulating a numerical damage model in a noisy environment (Xiong et al. n.d.), and She et al. utilized damage identification methods based on optimized Back Propagation Neural Network (BPNN) to realize damage evaluation of an arch bridge (She et al. 2023b).Noori et al. proposed a modified wavelet packet energy rate index to identify the location and severity of damage in steel bridges and its capability was validated by the result of numerical simulations (Noori et al. 2018).Although numerous researches have been conducted on the intersection between wavelet packet analysis and bridge damage identification, there is a long way to go to massively apply those methods to practical bridge projects due to the difficulty in obtaining the feature parameters involved and the complicated processing procedure.To this end, a simple and easy-to-implement damage identification approach is developed and applied to assess the operating condition of a specific bridge in this research.
Acceleration signals of the bridge's superstructure under vehicle excitation are collected via acceleration sensors in mobile devices, the hidden damage message contained in those signals will be extracted through wavelet packet energy analysis and finally exhibit in the way of WPEVCR.This paper takes a prestressed concrete bridge as an example to demonstrate the ability of WPEVCR in locating and quantifying structural damage.It also provides a fast, efficient, and cost-effective means of damage detection for a large number of similar projects.

Theoretical background
The bridge dynamic response signal is related to the structure's intrinsic properties such as stiffness, damping, and mass distribution itself, which can also be regarded as determined by those critical characteristics.Acceleration signals, as a form of the dynamic response of structures, contain structural damage information as well.However, the structure-related damage information is not directly observable from the signal curve, so frequency domain transformation methods are applied to extract the defect features from it.Wavelet packet analysis is a powerful analytical tool that is capable of decomposing the acceleration signals into different frequency bands to obtain the energy values of each band signal and form the characteristic parameter set (Lee and Yamamoto 1994).Damage indicators reflecting structural damage conditions can be obtained by analyzing and processing the energy values in each frequency band.In this paper, the characteristic energy values of healthy structures are compared with those of structures in different damage scenarios, and the WPEVCR metric is derived to serve as a damage assessment criterion.
For the acceleration response signal x(t), the total energy E x under the j-layer wavelet packet decomposition is given by the following equation.where x i j (t) is the signal in the i-th frequency band under the layer of j.Using the orthogonality condition, the total energy of the signal in each frequency band is obtained as follows.
where E x i j is the signal energy in the i-th frequency band under the layer of j.
When a structure is damaged, the dynamic response signal of its superstructure will fluctuate and the energy of the corresponding frequency band will either increase or decrease after wavelet packet decomposition (Kim and Melhem 2004).According to this principle, the superstructure dynamic response signals corresponding to various structural damage conditions are processed relying on the test bridge.Then, the signal wavelet packet node energy and the characteristic parameters under each frequency band are analyzed and compared.Finally, the energy's variance change rate of the free decay section in the acceleration response signal is proposed as the identification indicator to determine the location of the structural damage.
The signal energy variance of the bridge acceleration response is obtained from Eq. ( 3). (1) where E a j is the mean value of wavelet packet energy of the decomposed spectrum of layer j of the measured signal.The formula for the damage identification indicator is as follows.
where WPEVCR is the change rate of the wavelet packet energy variance of the acceleration response signal x(t) at scale i. σ 2 h , σ 2 d are the signal energy variances of the acceleration response signal when the designated structure is in healthy condition and damage condition, respectively.

Validation of WPEVCR
To validate the damage identification ability of WPEVCR, the Finite Element (FE) model of a simply supported beam is established and its structural acceleration signals are collected by conducting explicit dynamic analysis.The validation beam in Fig. 1 is discretized into 16 beam elements at 1-m intervals.The beam's length is 16 m with a rectangular 2 × 1.5m cross section.The density, modulus of elasticity and the damping ratio for each order of modalities are 2500 kg/m 3 , 31.5 GPa and 0.02, respectively.It is assumed that there are 15 accelerometers equally spaced on the model to capture as many structural vibration modes as possible.To yield the history data at each measurement point, the rand command in MATLAB is employed to generate random Gaussian white noise with mean 0 and variance 1, as shown in Fig. 2, which acts as an excitation force load on the 15 nodes distributed across the span.The example adopts the reduction of element stiffness to simulate the damaged state of the structure.Additionally, the damage conditions cover single-damage cases and multi-damage cases and the specific damage settings of varied cases are listed in Table 1.
In this paper, the db20 wavelet basis function with a decomposition level of 3 is selected to decompose the free decaying segments of the corresponding element acceleration signals.Therefore, the signal frequency band is uniformly divided into 8 frequency bands.Taking condition D3 as an example, the acceleration signal of sensor 8 for the healthy case and 10% damaged case is displayed in Fig. 3. Obviously, the variation of the beam before and after the damage cannot be seen directly from the acceleration signals.With Eq. ( 1), the corresponding subspace signal energy can be calculated for each (3) Fig. 1 The layout of the simply supported beam frequency band composed of the structural signal energy spectrum vector.Finally, the WPEVCR value of each measurement point is calculated using Eq. ( 4).The wavelet analysis module of MATLAB software realizes the energy feature extraction of the collected acceleration signals.The corresponding WPEVCR values of each sensor for three damage cases are calculated by algebraic coding, as shown in Fig. 4. It can be observed that the peak WPEVCR appears at the measurement point of the beam  Fig. 3 The acceleration of sensor 8 element where the stiffness is compromised in both single-damage and multi-damage conditions.This result suggests that the damage indicator adopted in the paper is capable of accurately locating structural damage and its engineering applicability is as illustrated.

Chi River bridge
Chi River bridge crosses the Chi River in Shiquan County, and its total length is 86.94 m with a deck of 7m width as presented in Fig. 5.The bridge consists of a 2 × 10m castin-place reinforced concrete slab approach bridge and a 4 × 16m precast prestressed concrete slab.The cast-in-place reinforced concrete slabs, cover beams, abutment caps, piers and other bridge components are mainly made of C30 concrete.The aerial view is shown in Fig. 6.In the meantime, the concrete of the prestressed concrete slab, bridge deck and cast-in-place layer is C40.Additionally, the prestressing steel bars are made of

Numeric analysis
To accurately simulate the field test process of the main span of this structure, taking the layout of measurement points in field test and bridge type characteristics into consideration, so the finite element software Abaqus was adopted to build a span of 16m precast bridge.This model is discretized into 18,468 C3D8R elements with a mesh size of 20cm as shown in Fig. 7, which is in accordance with the practical situation of the bridge.The density and modulus of elasticity of the main girder are 2460 kg/m 3 and 32.5 GPa, respectively.Sixteen measurement points, 1m between each point, were set up on the FE model in total.The VDLOAD subroutine module is employed to define the position and weight of the front and rear axle, as well as the speed of selected vehicle.Through changing the properties of the damage elements set, the damage reduction of the local structure is achieved, i.e., different damage degrees correspond to various stiffness reduction.Finally, the acceleration response signal at the measurement points is output by performing explicit dynamic analysis on the model.
In the previous study, it is found that even with the same damage level setting, the damage indicators obtained by processing the acceleration response signals at different damage locations can be significantly varied, which indicates that the damage indicators have unique correspondence in particular operating cases.Therefore, the damage situations shown in Table 2 are set up for the FE model of the Chi River bridge to figure out the correlation between the structural damage and the damage identification indicator of this bridge.
With a vehicle speed of 20 km/h, the damage conditions C2, C4, C7, and C8 were executed in the simulation software, and the damage information of 16 points was collected to calculate the WPEVCR values, which are plotted in Fig. 8(a, b).It can be observed that the locations of the maximum value of the damage identification indicator in these damage cases coincide with the location of the stiffness discount setting, which means that WPEVCR can accurately locate the structural damage.
With the aim of discussing the sensitivity of the damage identification indicator to different damage degrees of Chi River bridge, the single damage of 4 parts and the multi-damage of 4 parts and 13 parts were chosen as examples to set 3 different levels of damage discounting.The WPEVCR values were obtained by processing the collected data as displayed in Fig. 8(c, d).From the graphs, it is evident that the peak of the damage identification index still appears at the location of structural damage, while the peak value also tends to become larger with the increase of the damage degree.

Effects of vehicle axle weight on damage identification index
To demonstrate the efficacy of the damage identification indicator applied in this paper in identifying the location and extent of structural damage regardless of the vehicle axle weight, as well as investigate the effect of vehicle axle weight on the damage identification indicator, the vehicle's weight was changed according to the "Road vehicle outer dimensions, axle load, and mass limits", and the single damage cases were set up in Sect.4, 8, and 9 as shown in Table 3.After that, finite element analysis is performed keeping the speed of vehicle other variables stable, and WPEVCR values are calculated for each measurement point at various damage cases.
Figure 9(a) is the damage identification indicator obtained by setting a vehicle with a single axle weight of 15,500N moves along the bridge when the damage occurred in Sect.8, while Fig. 9(b) is the damage identification indicator obtained by setting a vehicle with a single axle weight of 20,500N moves along the bridge when the damage occurred in Sect.9. From these figures, it is obvious that the peak value of the damage identification indicators for different axle weight conditions still appears at the same location of the structural damage and the peak value tends to become more significant with the increase of the degree of damage.Following the data listed in Table 4, altering only the axle weight of the vehicle does not significantly increase or decrease the values of the damage identification indicators obtained with the same settings of damage location and damage degree.However, in the field test, the variation of the damage identification indicator may be more significant due to the self-oscillation noise of different axle-weight vehicles and the interference of environmental noise.

Field test
The field test is conducted to study the self-vibration characteristics of the bridge structure and the joint vibration characteristics of the vehicle and bridge.The dynamic signals collected in the field test is an essential factor to assess the operation condition and load-bearing characteristics of Chi River bridge.Vehicle excitation has been implemented as show in Fig. 10, in which the detail of vehicle includes weighted 36.06 kN with a wheelbase of 3.3 m, a front axle weight of 17.50 kN and a rear axle weight of 18.56 kN.The acceleration signals at each measurement point are recorded using a portable acceleration sensor with a sampling frequency of 100Hz when the vehicle moves at the speed of 20 km/h.Moreover, MATLAB is employed to plot the acceleration-time curve of the partial acceleration signal.The layout of measurement points in the field test is given in Table 5.

WPEVCR of the test
In the former subsection, it is learned that the measurement points of the field test are distributed in the middle or at the end of the bridge span.In order to diagnose the real condition of the Chi River bridge, the FE model was used to simulate the different damage states of the corresponding measurement points and calculate their WPEVCR values.Meanwhile, the acceleration signals of different measurement points collected from the field test were converted to WPEVCR values are listed in Table 6.A conclusion can be drawn from the above Table that the damage level of each measurement section on the bridge is less than 5%, which means the assessed structure is in a healthy condition.

HHT damage identification of Chi River bridge
For analyzing nonlinear and unstable data, Huang et al. proposed the Hilbert-Huang Transform (HHT) as a time-frequency signal analysis method combining Empirical Modal Decomposition (EMD) part and Hilbert Transformation part (Feldman 2011).As   the core of this algorithm, EMD is able to decompose complex data into a finite number of internal modal functions (IMFs) and then performs a Hilbert transform to obtain the instantaneous frequencies.The original signal is then transformed into a Hilbert spectrum to diagnose structural damage (Huang and Wu 2008).
If HHT is performed on the original signal and its marginal spectrum is calculated.Setting the resulting frequency interval [ω min , ω max ] and dividing the interval into fre- quency bands �ω i (i = 1, • • • , n) .Then the spectral value of energy in each band is (ω i ), i = 1, • • • , n .The maximum value of n depends on the signal length and the sam- pling frequency of the signal, the larger the value, the more subdivided the frequency band is.
The definition of the modal component vector is as follows.
where ξ is the relative proportion of each frequency band component of an IMF sig- nal.The degree of nonlinear deformation of the structural response signal can be represented by numerically processing the change of the modal component vector ξ when the structure is healthy or damaged.ξ h , ξ d are defined as the modal component vectors of the healthy bridge structure and the bridge structure in the real state, respectively, then the damage identification index, the modal vector Euclidean distance, is calculated as below.
The larger the value of the damage indicator corresponds to the greater variation in the frequency band distribution of each mode in the structural response of the bridge, i.e., the more severe the damage is.Conversely, a damage indicator value approaching zero signifies a healthy and undamaged bridge structure.
To ensure the reliability of the damage identification results as well as to validate the damage identification results of WEPVCR, MATLAB was implemented to code the process of calculating the damage indicatorD, which is based on the Hilbert-Huang transform.The values of D calculated from the signal outputs of different damage conditions in the FE model and the field tests are listed in Table 7.The results show that the damage (5) ξ = h(ω i ) level of each measurement point of this bridge is less than 5%.In other words, the measured bridge section is in healthy condition and this outcome is similar to the damage identification results based on wavelet packet analysis.

Conclusions
• Chi River Bridge is tested and the acceleration signals excited by vehicle load are obtained as the primary information for structural damage analysis.The structural health information of the bridge superstructure is extracted in the form of damage identification indicator WPEVCR, which is able to localize the damage accurately by means of maximum value.• Comparing the damage identification results based on WPEVCR and HHT, it's obvious that the damage level of the measured superstructure is less than 5% and it's in a healthy service condition.At the same time, the adopted damage identification method based on wavelet packet analysis is effective.• By altering the axle weight of the vehicle, the values of WPEVCR are obtained, which indicates the independent relationship between the damage identification indicator and the axle weight.

Fig. 6
Fig. 6 Aerial view of Chi River bridge

Fig. 9
Fig.9The WPEVCR value of damage cases with various axis weight

Fig. 10
Fig.10 The picture of the field test

Table 1
List of damage cases in the simply supported beam

Table 2
The damage cases in Chi River bridge

Table 3
The damage cases of various axis weight

Table 4
The maximal WPEVCR value of various axis weight

Table 5
The layout of measurement points

Table 6
The WPEVCR values of the field test

Table 7
The D values of the field test