Damage Identification in Steel Girder Based on Vibration Responses of Different Sinusoidal Excitations and Wavelet Packet Permutation Entropy

Abstract

Damage identification, both in terms of size and location, in bridges is important for timely maintenance and to avoid any catastrophic failure. An earlier experimental study showed that damage in a steel box girder orthotropic plate can be successfully detected using the measured vibration acceleration data. In this study, the wavelet packet decomposition (WPD) method is used to analyze the measured vibration acceleration responses and then the estimation of the permutation entropy (PE) on the re-constructed signals. A damage index is then defined based on the permutation entropy difference (PED) between the damaged and the healthy conditions to detect the location and size of the damage. The method is further validated through the finite element (FE) model of a steel box girder and the computed vibration acceleration responses when subjected to the sinusoidal excitations at different frequencies. In addition, the robustness of the methodology under different white noise interference conditions is also verified. The results show that the proposed methodology can effectively identify the location of human-made damage and accurately estimate the degree of damage under different frequencies of sinusoidal excitation. The method has shown a strong anti-noise property.

1. Introduction

Damage identification for steel box beams and orthotropic plates in bridges is critical for ensuring structural integrity, safety, and longevity. Early detection of defects through Structure Health Monitoring (SHM) techniques prevents catastrophic failures [1], optimizes maintenance strategies [2], and leads to significant cost savings over the bridge’s lifespan [3]. By addressing damage promptly, engineers can enhance structural performance, minimize risks, and ensure the continued functionality of bridge infrastructure, ultimately benefiting society by maintaining safe and reliable transportation networks.

Nondestructive testing (NDT) technology offers several advantages over traditional testing methods in bridge engineering, particularly for steel box beams and orthotropic plate structures. NDT preserves structural integrity by allowing an assessment without causing harm, thus minimizing downtime and disruption to transportation networks. Unlike traditional methods, NDT can be performed efficiently and non-invasively, often without the need for bridge closures or interruptions [4]. Additionally, NDT provides real-time results and immediate feedback, facilitating timely decision making regarding maintenance and repair interventions [5]. With comprehensive inspection coverage, NDT identifies potential issues that may be overlooked by traditional methods, ensuring the thorough assessment of defects. Moreover, NDT is cost-effective and efficient, requiring fewer resources and enabling optimization of maintenance budgets, ultimately contributing to the safety, durability, and longevity of bridge infrastructure [6].

The Wavelet Packet Permutation Entropy (WPPE) technique offers several advantages for damage identification in steel box beams and orthotropic plates compared to other nondestructive testing techniques (such as using natural frequency [7], mode shape [8], and mode curvature [9] to establish the damage characteristics.). By enabling a multiscale analysis, sensitivity to complex dynamics, and nonlinear assessment, WPPE can detect subtle changes indicative of damage, even in noisy environments [10,11]. Its data-driven approach and complementary nature to other techniques enhance its versatility and effectiveness in identifying defects [10,12]. Overall, WPPE provides a powerful tool for comprehensive damage assessment, making it a valuable addition to the arsenal of nondestructive testing methods for ensuring the structural integrity and safety of bridges.

While the WPPE technique offers significant advantages for damage identification, there are potential downsides to consider [13]. These include computational complexity due to intensive signal processing, parameter sensitivity requiring careful tuning, challenges in interpreting results, and limited validation in real-world bridge inspection scenarios. Additionally, the technique’s resilience to noise may vary depending on the signal-to-noise ratio and the specific characteristics of the noise present in the data. Addressing these limitations, including enhancing noise resistance, will be essential to ensure the practical effectiveness and reliability of WPPE in nondestructive testing applications for steel box beams and orthotropic plates.

The WPPE method was found to be useful in an experimental study on a steel box girder to identify the damage location when sinusoidally excited at 10 Hz [10]. This study addresses the challenges associated with the WPPE technique by considering the finite element model of a steel box girder and analyzing the vibration acceleration responses under sinusoidal excitation at frequencies such as any frequencies or at the natural frequencies. The study further verifies the validity of WPPE for damage identification in steel box beams and orthotropic plates. Additionally, the robustness of the methodology is tested under various white noise interference conditions, providing insights into its effectiveness in real-world scenarios. The results show that the proposed methodology can effectively identify the location of human-made damage and accurately estimate the degree of damage under different frequencies of sinusoidal excitation. The method has also shown a strong anti-noise property.

2. Methodology

The framework of the orthotropic plate damage identification method proposed in this study is shown in Figure 1.

By applying a sinusoidal excitation load, the original vibration response can be obtained from sensors already arranged on a steel box girder or an orthotropic plate. However, these raw vibration signals should not be used directly, but should be pruned, as shown in Figure 1a. This is due to the fact that in signal processing and analysis, pruning and removing the initial portion of the data when sinusoidal excitation loads are applied to the structure helps to more centrally and accurately evaluate the dynamic behavior and integrity of the structure under sinusoidal excitation. Firstly, it helps remove transient effects, such as initial settling or response onset, ensuring that subsequent analysis focuses on the steady-state response of the structure under excitation. Secondly, by isolating the steady-state response, it enables a clearer analysis of structural behavior and extraction of relevant information, such as excitation frequency and resonant frequencies. Additionally, this process aids in noise reduction by eliminating noise or artifacts present in the initial portion of the data, thereby improving the signal-to-noise ratio and facilitating more accurate detection and analysis of structural responses.

Then, the trimmed data are decomposed by Wavelet Packet Decomposition (WPD). In this process, WPD first uses a set of wavelet basis functions to represent the signal. These wavelets are localized in both time and frequency and are used to capture different features of the signal across various scales. WPD then decomposes the signal into a tree-like structure, allowing decomposition at scale and frequency bands, as shown in Figure 1b. At each layer of the decomposition tree, a filter bank is used to segment the signal into sub-bands or nodes. This segmentation process divides the signal into different frequency bands, with each node representing a specific frequency range. The decomposition process is recursively applied to each node of the tree, generating further sub-bands or nodes at deeper levels in the tree. This recursive decomposition continues until the desired level of detail or frequency resolution is reached. At the end of the decomposition process, the signal is represented as a set of wavelet packet coefficients that capture the amplitude and phase information of the signal at different frequency bands and scales. Analyzing these coefficients can extract meaningful features or features of the signal, such as dominant frequencies, transient components, or structural anomalies.

If $U_b^a$ is the signal of the a-th level and the b-th node, $U_{2b}^{a+1}=\mathrm{L}\mathrm{o}\mathrm{w}\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}\left(U_b^a\right)$, and $U_{2b+1}^{a+1}=\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}\left(U_b^a\right)$.

The specific implementation of the entire decomposition process is as follows:

$$W_l=U_{l-k}^{2k}\oplus U_{l-k}^{2^k+1}\oplus \cdot \cdot \cdot \oplus U_{l-k}^{2^{k+1}-1}, l, k\in Z$$

(1)

According to the theory of multi-resolution analysis, L²(R) = ⊕W_l, l ∈ Z, where W_l is the l-th wavelet subspace, l is the current level of analysis, and k is the amount of step or level change in the current decomposition process. And L²(R) is a real valued function space that can be squared integrally and can be decomposed into a direct sum of a series of wavelet subspaces $W_l$. The W_l decomposition is further decomposed by wavelet packet decomposition, which gives a better division of the whole analysis frequency band, thus improving the frequency resolution. Wavelet packet decomposition can be expressed as follows:

Complexity is an important nonlinear feature to measure the degree of disorder of a system. The measurement of the complexity of vibration signals of orthotropic steel bridge panels is beneficial to excavate the damage state of orthotropic steel bridge panels from the complex vibration signals and judge the damage location effectively. This study focuses on the energy of the signal after decomposition. By observing the proportion of energy in each frequency band, the frequency band with the highest proportion of energy is selected, as shown in Figure 1c. The inverse wavelet transform is used to reconstruct the signal, as shown in Figure 1d. The specific calculation process of permutation entropy algorithm adopted in this study is as follows:

For a time series of length N: {x(i), i = 1, 2, …, N}, reconstruct any element x(i) according to the phase-space delay coordinate method, take m consecutive sample points of each sampling point, and obtain the m-dimension reconstruction vector of point x(i):

$$\mathit{X}\left(\mathit{i}\right)=\left\{x\right(i),x(i+f),\cdot \cdot \cdot ,x(i+(m-1)f\left)\right\}$$

(2)

where m ≥ 2 is the embedding dimension, f is the time delay, i = 1, 2, …, N, and the m reconstructed components in X(i) are arranged in the order from smallest to largest:

$$\left\{x\right(i+(j_1-1)f)\le x(i+(j_2-1)f)\le \cdot \cdot \cdot \le x(i+(j_m-1)f\left)\right\}$$

(3)

If X(i) has the same element—i.e., x(i + (j_p − 1)f) = x(i + (j_q − 1)f)—then it is sorted according to the size of i—i.e., when p ≤ q, the arrangement is x(i + (j_p − 1)f) ≤ x(i + (j_q − 1)f).

Therefore, any vector X(i) can obtain a set of symbol sequences:

$$S\left(g\right)=(j_1,j_2,\cdots j_m)$$

(4)

Among them, j = 1, 2, …, m (m ∈ N). According to the theorem of permutation and combination, there are a total of m! different permutations of symbol sequences that are different from the m-dimensional phase space mapping, and S(g) = (j₁, j₂, …, j_m) is one of these m! permutations. Assuming the probabilities of each occurrence of S(g) are P₁, P₂, …, P_k, then according to Shannon’s entropy theorem, the permutation entropy of the sequence {x(k), k = 1, 2, …, N} can be defined as:

$$PE(m)=-\sum _{g=1}^k{P}_{g}In{P}_g$$

(5)

when P_g = 1/m!, PE(m) reaches its maximum value ln(m!). For convenience, ln(m!) is usually used to normalize PE(m).

$$0\le H_P=H_P/In(m!)\le 1$$

(6)

The randomness of the acceleration vibration signal is measured by the magnitude of the PE value. The smaller the PE value, the more regular the time series is; on the contrary, the larger the PE value, the higher the degree of disorder and signal complexity of the time series.

The permutation entropy difference (PED) of under different states can be expressed as:

$$PED=\left|P{E}^u-P{E}^d\right|$$

(7)

where PE^u and PE^d represent the sample entropy under nondestructive and damaged conditions, respectively.

For the selection of embedding dimension, when the embedding dimension is too small (m < 4), the range of change in permutation entropy is too small to effectively distinguish damaged vibration signals from normal vibration signals. However, when the embedding dimension is too large (m > 8), the operation time of permutation entropy becomes longer, and it will lead to a decrease in the range of change in permutation entropy, making it difficult to accurately measure signal amplitude [14]. This article selects an embedding dimension value of m = 6 to determine the damage location of the orthotropic steel bridge deck by calculating the PED value.

3. Experimental Results

3.1. Experimental Conditions

In order to prove the feasibility of WPD to identify potential damage in an orthotropic steel bridge girder, the single U rib model of an orthotropic steel bridge panel is used [10]. The test setup, experiments, and observation from the earlier study [10] are presented here to aid in better understanding. This specimen size is 1500 mm × 900 mm × 780 mm. The specimen has a U-shaped longitudinal stiffener. This model adopts the constraint mode of two-end anchorage, and each anchorage point area is arranged with a diaphragm. The material of the whole specimen is Q345D, and the thickness of the top plate and U rib are 4 mm and 2 mm, respectively. The test model is shown in Figure 2a,b.

Static displacement and dynamic response signals of orthotropic steel bridge panels were collected by TZT3828E (dynamic and static signal acquisition instrument). The displacement sensor layout is shown in Figure 3a. The layout of acceleration sensors is shown in Figure 3b. It should be noted that in order to effectively collect acceleration response signals, the sampling frequency of acceleration sensors is set at 1000 Hz.

In the test, a harmonic force of 120 N, 10 Hz is applied to the center of the top plate of the orthotropic steel bridge panel by the shaker, and the acceleration response signal of the orthotropic steel bridge panel is collected under nondestructive conditions. Then, an area of rectangular damage of size 1 cm (width) × <5 mm (depth) × 1 mm (height along thickness) is manufactured between sensor 2-4 and sensor 2-5, which is denoted as working Condition 1, as shown in Figure 4. And the same excitation is applied to the center of the panel with the shaker to collect the acceleration response signal. After the signal collection is completed, the human-made damage width is extended to 3 cm, which is denoted as working Condition 3. The same excitation is then applied to the center of the roof, and the acceleration response signal is also collected. The test conditions are shown in

Table 1. Experimental conditions.

Condition	Width of Damage (cm)	Depth of Damage (mm)	Note
Condition 1	0	0	Healthy
Condition 2	1	<5	Minor damage
Condition 3	3	<5	Serious damage

.

3.2. Experimental Data

To enable the quantitative visualization of the state of the structure, the PE value is used to characterize the state of the structure. The reconstructed acceleration response signal is obtained using Equations (2)–(6) to obtain the PE value of the signal. The PE values obtained from each acceleration sensor under three working conditions. PED value for each measured location is estimated by substituting the PE values of each accelerometer under two different damage conditions and the corresponding PE values of the accelerometer under healthy conditions into Equation (7). These PED values are plotted in Figure 5. It can be seen that the PED value close to the damage location showing a sudden change, and the degree of change between two damage cases. This suggests that the indicator can identify damage at the damaged location. As the degree of damage increases, the PE value at the same location on the steel bridge deck is also observed to be higher. This indicates that the damage indicator can also estimate the degree of damage.

4. Mathematical Validation

While previous experiments have sufficiently demonstrated the feasibility of the method for detecting damage in steel box beams and orthotropic plates, its practical utility and robustness still face significant challenges. Therefore, this study established numerical models for steel box beams and orthotropic plates, and analyzed the vibration acceleration responses of steel box beams under sinusoidal excitations at different frequencies. Initially, the sinusoidal excitation at 10 Hz is used to validate the experimental observations and then further tested at the first two natural frequencies, f₁ and f₂, to demonstrate the effectiveness of the WPPE method in damage identification. Additionally, the robustness of the method is also tested under various white noise interference conditions, thereby verifying its feasibility in real-world applications.

4.1. Finite Element Model

A finite element (FE) model of the U-shaped ribbed plate for the steel box beam is developed using Abaqus 2020 software. The FE model is entirely constructed using solid elements, with hexahedral elements employed for the main plate and tetrahedral elements for the transverse diaphragms to simplify computations. The dimensions, material properties, and boundary conditions of the FE model are consistent with those of the experimental model. Rectangular notches (1 mm × 10 mm × 1 mm and 1 mm × 30 mm × 1 mm) are introduced onto the bridge deck panel using cutting and filing methods to simulate damage induced by human actions, as depicted in Figure 6. This approach enables the straightforward simulation of the three conditions outlined in Table 1. Under Condition 1, the FE model of the orthotropic steel bridge deck is illustrated in Figure 7. The first four computed natural frequencies and mode shapes of this structure are also shown in Figure 7.

4.2. Validation (10 Hz)

The sinusoidal excitation load is illustrated in Figure 8. The amplitude of this sinusoidal excitation is ±60 N, and the input frequency is 10 Hz, corresponding to a cycle period T of 0.10 s. The loading position for this sinusoidal excitation load is the center point of the bridge deck panel surface.

4.2.1. Without Noise

Firstly, the feasibility of WPPE without ambient noise is tested by simulation. The excitation load in Figure 8 was input into the finite element model shown in Figure 6, and the trimmed vibration acceleration response was obtained, as shown in Figure 9a–c. Figure 9a, Figure 9b, and Figure 9c, respectively, show the acceleration time history curve of the position of sensor #2-4 under three damage conditions.

Figure 10 shows the sub-band energy ratio of the vibration response signals at the measured location (2-4) under three working conditions after wavelet packet decomposition and normalization. The component with the highest normalized energy is selected for reconstruction.

In order to achieve quantitative visualization of the structural state, PE values are used to characterize the state of the structure. The PE value of the signal can be obtained by substituting the reconstructed acceleration response signal into Equations (2)–(6).

Table 2. PE value under the three conditions.

Case	#2-1	#2-2	#2-3	#2-4	#2-5	#2-6	#2-7	#2-8
Case 1	1.1552	1.1552	1.1552	1.1552	1.1552	1.1552	1.1552	1.1552
Case 2	1.1551	1.1551	1.1551	1.1507	1.1509	1.1551	1.1551	1.1551
PEDcase 1,case 2	0.0001	0.0001	0.0001	0.0045	0.0043	0.0001	0.0001	0.0001
Case 3	1.1550	1.1550	1.1551	1.1492	1.1493	1.1551	1.1551	1.1550
PEDcase 1,case 3	0.0002	0.0002	0.0001	0.0060	0.0059	0.0001	0.0001	0.0002

shows the PE values of each acceleration sensor under three damage conditions. It can be seen that the PE value of the signal can reflect the state of the structure. Then, the PE value in the damaged state and the PE value in the nondestructive state of the structure are substituted into Equation (7), and the PED value can be obtained (Figure 11). Consistently with the results of the real experiment, in the simulation experiment, when the steel bridge deck is damaged, the PED value near the damaged location will mutate, and the degree of change is relatively large. This means that the degree of damage at the damaged location can be identified by the indicators obtained after processing the vibration data at the location of the sensor in Figure 3b using WPPE technology.

4.2.2. With 10–30 dB Noise

On the basis of the above research, environmental background noise is further considered. Ambient noise with a signal-to-noise ratio (SNR) of 10, 20, and 30 dB was added to the original signal respectively, as shown in Figure 12.

The signals in Figure 12 are transformed to the frequency domain by FFT analysis. The spectra plots are shown in Figure 13a,c,e,g. The averaged spectra plots show a distinct frequency peak at 10 Hz even for the most noisy data at a SNR of 10 dB. This is because the averaging process during the FFT computation has minimized the noise content [15]. The proportion of energy plots in each corresponding frequency band are also shown in Figure 13b,d,f,h. Similarly, the sub-band recombination signal with the highest energy proportion is selected for each case and substituted into Equations (2)–(6) to calculate its PE and PED values, as shown in Figure 14.

As can be seen in Figure 14, no matter how the environmental noise changes within the SNR range of 10–30 dB, the PED values of the locations of sensors 2-4 and 2-5 will increase significantly, and PED values of damage Condition 3 are significantly higher than those of Condition 2. Therefore, even under the interference of environmental noise, WPPE theory can still be as good as that without noise interference. The precise location of the damage is clearly identified (between sensors 2-4 and 2-5), with high anti-noise robustness.

5. Further Validation

Testing the damage recognition theory at different frequencies can verify its robustness. A good theory should show similar performance at multiple frequencies, not just be effective at a particular frequency. The structure will be stimulated by various frequencies in the actual operation, so the damage identification theory must be able to cope with the vibration at different frequencies to ensure the effectiveness in practical applications. By testing the performance at different frequencies, the robustness of the damage recognition theory can be evaluated more comprehensively. The first- and second-order natural frequencies are used as the excitation frequencies to test the proposed method. By testing these frequencies, it is possible to better understand the effects of damage on different modes of vibration, and to ensure its robustness and overall performance.

5.1. First-Order Natural Frequency

When the sinusoidal excitation frequency of the input structure changes from 10 Hz to the first-order natural frequency of the structure, the vibration response of the structure after damage is shown in Figure 15. The same processing method as in Section 4 is adopted, and the spectra plots obtained by the FFT are shown in Figure 16a–d. Then, WPD technology is used to obtain the energy ratio of each frequency band, as shown in Figure 16e–h.

As can be seen from Figure 16, in Condition 2 and Condition 3, although the SNR of background noise changes from 20 dB to 10 dB (the actual noise amplitude increases by about 3.16 times), the frequency band with the highest energy ratio can still be clearly and accurately found after WPD and used for signal reconstruction. This shows that the new method has strong robustness under the condition of low SNR (<20 dB).

Corresponding to the data in Figure 16, the calculated PED result has reached the expected value, as shown in Figure 17. The results show that, consistently with the results in the condition without noise interference, the PED values between the location of sensor 2-4 and sensor 2-5 show a significant surge when the ambient noise of 10–30 dB is applied in Condition 2 and Condition 3. The PED value of Condition 3 is slightly higher than that of Condition 2, but the gap is small. Therefore, when the input sinusoidal load excitation is consistent with the first-order natural frequency of the structure, WPPE technology still shows its good noise resistance in diagnosing structural damage.

5.2. Second-Order Natural Frequency

When the sinusoidal excitation frequency of the input structure changes to the second natural frequency of the structure, the vibration responses (with different noise contents) of the structure after damage are shown in Figure 18. Similarly, the spectra plots obtained by FFT analysis are shown in Figure 19a–d. Then, WPD technology is used to obtain the energy ratio of each frequency band, as shown in Figure 19e–h.

As can be seen from Figure 19, regardless of damage Condition 2 or Condition 3, although the SNR of background noise changes from 20 dB to 10 dB, the frequency band with the highest energy ratio can be clearly and accurately found after WPD and used for signal reconstruction.

The results in Figure 20 show that the PED value trend when ambient noise of 10–30 dB is applied is basically consistent with the results without noise interference. When the input frequency of the excitation load matches the second-order natural frequency of the structure, significant spikes in the PED values between the location of sensors 2-4 and 2-5 are observed for both Condition 2 and Condition 3. And the PED value of Condition 3 is slightly higher than that of Condition 2. This is consistent with when the input frequency of the excitation load is the first-order natural frequency of the structure. Therefore, when the input sinusoidal excitation load coincides with the structure’s second-order natural frequency, the robustness of the WPPE technique is still evident despite interference from background noise.

6. Conclusions

In order to address the current issues with the WPPE technique and further validate its effectiveness in identifying damage in box beams and orthotropic plates, this study examines a finite element model of a steel box beam. Expanding upon previous experiments, the vibration acceleration responses of the steel box beam under sinusoidal excitations at frequencies of 10 Hz, first- and second natural frequencies are analyzed in greater detail. Moreover, the robustness of this theory is assessed under various white noise interference conditions, providing further evidence of its practical applicability. The main conclusions of this study are outlined below:

The newly proposed theory proves its effectiveness in identifying the location of damage and estimating its severity under different sinusoidal excitation frequencies. This shows that the theory performs consistently well across various frequencies, not just at specific ones. Consequently, the WPPE damage identification theory shows great potential for practical applications in handling vibrations at different frequencies.

The WPPE technique demonstrates impressive noise resistance capabilities. The test results revealed that with background noise levels reaching 10 dB, or even almost masking the vibration response, WPPE can still accurately assess the location and severity of damage in steel box beams or orthotropic plates, highlighting its remarkable robustness.