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 is the signal of the a-th level and the b-th node, , and .
The specific implementation of the entire decomposition process is as follows:
According to the theory of multi-resolution analysis, L2(R) = ⊕Wl, l ∈ Z, where Wl 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 . The Wl 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):
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:
If X(i) has the same element—i.e., x(i + (jp − 1)f) = x(i + (jq − 1)f)—then it is sorted according to the size of i—i.e., when p ≤ q, the arrangement is x(i + (jp − 1)f) ≤ x(i + (jq − 1)f).
Therefore, any vector
X(i) can obtain a set of symbol sequences:
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, …,
jm) is one of these m! permutations. Assuming the probabilities of each occurrence of
S(
g) are
P1,
P2, …,
Pk, then according to Shannon’s entropy theorem, the permutation entropy of the sequence {
x(
k),
k = 1, 2, …,
N} can be defined as:
when
Pg = 1/
m!, PE(m) reaches its maximum value
ln(
m!). For convenience,
ln(
m!) is usually used to normalize
PE(
m).
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:
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.
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.