Damage Classification Approach for Concrete Structure Using Support Vector Machine Learning of Decomposed Electromechanical Admittance Signature via Discrete Wavelet Transform

Abstract

The identification of structural damage types remains a key challenge in electromechanical impedance/admittance (EMI/EMA)-based structural health monitoring realm. This paper proposed a damage classification approach for concrete structures by using integrating discrete wavelet transform (DWT) decomposition of EMA signatures with supervised machine learning. In this approach, the EMA signals of arranged piezoelectric ceramic (PZT) patches were successively measured at initial undamaged and post-damaged states, and the signals were decomposed and processed using the DWT technique to derive indicators including the wavelet energy, the variance, the mean, and the entropy. Then these indicators, incorporated with traditional ones including root mean square deviation (RMSD), baseline-changeable RMSD named RMSDk, correlation coefficient (CC), and mean absolute percentage deviation (MAPD), were processed by a support vector machine (SVM) model, and finally damage type could be automatically classified and identified. To validate the approach, experiments on a full-scale reinforced concrete (RC) slab and application to a practical tunnel segment RC slab structure instrumented with multiple PZT patches were conducted to classify severe transverse cracking and minor crack/impact damages. Experimental and application results cogently demonstrated that the proposed DWT-based approach can precisely classify different types of damage on concrete structures with higher accuracy than traditional ones, highlighting the potential of the DWT-decomposed EMA signatures for damage characterization in concrete infrastructure.

1. Introduction

Health monitoring for concrete infrastructures plays a fundamental role in tracking and evaluating operational incidents, anomalies, and damages, thus ensuring structural safety, maintaining the utility functions, and extending the service lives of the target structures [1,2]. In a structural health monitoring system (SHM), structural damage characterization including damage detection, localization, classification, and assessment is a key issue in an SHM system of concrete structures. Traditional SHM approaches have been progressed using reinforced concrete (RC) structural vibrational or static characteristics such as mode shapes, mode shape curvatures, static strain, and natural frequencies [3,4]. Due to the characteristics being global in nature, these structural parameters are insensitive to being changed by localized incipient concrete damages. Instead of the global techniques, some locally non-destructive ones have been applied to detect concrete crack damages. Acoustic emission testing was reported for damage progression and crack classification in large-scale RC elements [5,6]. Ultrasonic pulse velocity measurements could generate a damage contour map that indicates the severity degree of damage in a concrete structure [7]. The use of fiber optic sensors for the SHM showed geometric adaptability to detect cracks in concrete structures [8,9]. Except these, the electromechanical impedance/admittance (EMI/EMA) technique using piezoelectric ceramic (PZT) as an actuator and sensor is emerging as a promising alternative for concrete structural damage identification and health monitoring [10,11,12]. This technique is characterized as high-frequency stimulation and fast responses to structural damage, which makes it quite sensitive to minor damages like corrosion, microcracks, mass loss, and stress-induced cracks [13,14,15,16,17,18]. Any damages or anomalies that have occurred to the structure being monitored would alter its mechanical properties including mass, stiffness, and damping, and consequently change the mechanical impedance related to the EMA signatures, thus enabling its identification in time.

However, a critical issue for concrete SHM lies in that any damages would originally change the structural mechanical impedance, which makes it difficult to identify the damage type merely relying on the EMA technique. Actually, many studies have been conducted to solve this issue over the past three decades. The impedance signatures or damage indexes of the PZT patches varied from their locations due to the decreased sensitivity of the transducer with increased damage distance; this characteristic was used to locate the damage nearby and discriminate the farer ones [19]. Using such a characteristic, the severity degree of one type of concrete structural damage was usually identified by observing the successive variations in the EMA spectrums including the alterations in the resonance peak in terms of magnitude or frequency shifts in the EMA/EMI signals and voltage responses [20,21,22,23]. With regard to the lateral and thickness modes for the PZT patches, shear and flexural-critical cracks that occurred to an RC beam could be successfully detected by using the combined root mean square deviation (RMSD) charts [24]. Other damage indicators including correlation coefficient (CC), mean absolute percentage deviation (MAPD), modified correlation coefficient (MCC), rate of change in the RMSD, and baseline-changeable RMSD named RMSDk were calculated to distinguish the stiffness or damping variation [25,26,27,28,29,30,31]. To discern the different types of concrete stress-induced crack damage, one-dimensional or two-dimensional impedance models for bonded or embedded PZT transducers were proposed to include the compression and tension effect on the EMA features [32,33]. Typical flexural monotonic/cyclic loading-induced debonding of FRP sheet and damage in RC/fiber-RC beams or frames could be discriminated from crack damage by integrating the EMA features and the RMSD index values based on the EMI method [23,34,35].

Unfortunately, despite the rough determination of the pattern of anticipated crack regions or damage severity, the usage of the raw EMA/EMI signature or its statistical indicators to classify damage was a crude manner of analysis and made it difficult to analyze the entire repertoire of data by such a cumbersome as well as computation-intensive process [36]. To this end, modern signal processing or clustering algorithms have been introduced to achieve fast identification. For example, the wavelet decomposition technique has become a powerful computational tool for damage detection attributed to the capability for automatic extraction of useful information from the monitoring data. Due to the clear existence in wavelet functions, many wavelet basis functions characterized by fast decay properties were developed [37]. Fast wavelet transform was designed for signal resolution and reconstruction [38]. Djebala et al. [39] used kurtosis as an optimization and evaluation criterion to select several parameters of wavelet multiresolution analysis for defect detection. Omar and Gaouda developed a wavelet-based technique for identifying and locating defects of gear tooth under noisy environments [40]. Wang et al. [41] applied the discrete wavelet transform (DWT) technique for tracking and evaluating the abrupt stiffness degradation of a shear structure. The DWT technique, wavelet packet decomposition, and stationary wavelet transform were also developed to extract mechanical vibration signals for mechanical fault diagnosis [42,43,44]. Once the monitoring signals have been detected, various approaches such as machine learning could be performed to classify damage types or determine the class which they belong to. Yang et al. [45] proposed a useful monitoring system for grinding wheel wear using the DWT technique and support vector machine (SVM), of which the accuracy was over 99% when using root mean square and variance of every level of decomposition as feature vectors. The SVM was employed to localize the damage and evaluate damage severity in the model test of an offshore platform or the health state of wood utility poles [46,47]. Satpal et al. [48] explored the application of the SVM to find locations of damage in a few aluminum beams using experimental and simulation data. A two-stage method for damage detection in beam structures based on SVM and swarm automatic optimization algorithms was developed to classify damage degree with high accuracy [49]. The recent application of wavelet transform techniques was also extended to the corrosion assessment of embedded rebars in RC elements using EMI [50]. Wavelet Level Decomposition (WLD) was applied to the seismic response of such historical structures, which showed the potential for the WLD-based retrospective analysis after seismic events on recorded signals [51]. Unfortunately, there were limited studies concerning the integration of the DWT/WLD technique with the SVM model for damage type classification, hence motivating the fusion of these different techniques for SHM applications, particularly dealing with the EMI signals from PZT transducers.

For damage classification in the PZT-based SHM field, numerous approaches have been developed, particularly with the aid of image processing techniques and machine learning methods, as well as intelligent algorithms, which enabled automatic inspection and analysis [52,53,54]. Min et al. [55] investigated a neural network-based approach to recognize the damage severity and type for multiple and multi-type structural damage conditions. The artificial neural network (ANN) approach was deemed a useful classification algorithm for damaged area location prediction of a composite plate using the EMI technique [56]. Selva et al. [57] presented a smart monitoring technique for damage identification and localization in aeronautical composite plates based on EMI measurements and ANNs. In the last few years, deep learning approaches, which have overcome the low efficiency of conventional ANNs in dealing with multiple input data, have emerged as a powerful tool for automatic damage detection and classification [58,59,60,61]. Once a group of monitoring signals were input and trained by the network, the corresponding damage related to a certain signal could be rapidly output [62,63]. Typically, convolutional neural networks (CNNs) were introduced to solve the rapid and accurate identification problem via deep learning of the PZT signatures. Oliveira et al. [64] split the EMI signature of an aluminum plate into a few parts by calculating the Euclidean distances to construct an RGB frame for CNN training and damage condition classification. Deep learning of 900 groups of impedance signal-based root mean square indexes of an aluminum structure demonstrated damage detection probability higher than 97% by a one-dimensional (1D) CNN model [65]. Two-dimensional (2D) and 1D CNNs were also developed for exploiting the raw EMA response to automatically detect small damages in concrete structures; comparison with a traditional back propagation neural network indicated significant superiority in terms of prediction accuracy [66,67,68,69]. Recently, applications of 1D and 2D CNNs were also extended to smart aggregate-based stress monitoring in concrete [70], the automatic evaluation of freeze–thaw-induced damage of concrete structures [71], and the autonomous damage detection of fiber-reinforced concrete higher than a mean accuracy of 95.24% [72,73,74]. Despite the rapid identification with relatively high accuracy, the major drawback of the deep learning approaches lies in the following: (1) Any network requires considerable amounts of measured signals for training and validating before they can be applied for testing. Unfortunately, the continuous and repeated measurement of the EMA signatures is always a critical issue and remains challengeable due to the time consumption, the external impact, or the varied ambient conditions in practice. (2) Using a well-trained machine learning model for identifying the unlearned signals for damage detection still remains a challenge to date, as a harsh point lies in that not all damage states can be known or learned in practice when incorporating the machine learning techniques. (3) The reported deep models attempting one-to-one identification become extensive when dealing with considerable damage conditions, particularly for those belonging to the same type. In contrast, those that possess differences in sizes and locations require attempts for classification of the damage types.

All in all, despite the similar use of the DWT/SVM technique, the differentiations from prior studies using similar pipelines in the current study lie in that the following: (1) In traditional literature, the DWT-derived EMI energy was generally used to directly detect structural damage; the other indicators including the entropy, mean, and variance have been not attempted to classify the concrete damage types. (2) The DWT-derived EMI indicators in the existing literature could be only used to detect/identify the damage level; they are not capable of classifying damage types if depending only on the traditional DWT/EMI techniques. To overcome the above limitations, this study innovatively formulated a damage classification approach for concrete structure using supervised machine learning of decomposed EMA signatures by the DWT technique. The EMA signals of the distributed PZT transducers were successively measured and decomposed using the DWT technique to derive damage indicators including the wavelet energy, the variance, the mean, and the entropy. Then these indicators incorporated with traditional RMSD, RMSDk, MAPD, and CC were taught to the SVM model, and finally damage type could be automatically classified. The major innovation and contributions of this work lie in the following:

(1) The DWT technique was incorporated with the EMA signatures to derive multiple feature-extracted damage indicators for supervised learning, which has not yet been applied for dealing with damage classification problems. Experimental tests and application results showed better performance than traditional ones.

(2) The SVM model with linear soft-margin was introduced to learn the characteristics of the damage indicators derived from the decomposed EMA signatures, which solved the hassle of damage classification merely based on the EMA or the DWT-based indicators. The results confirmed the automatic and precise damage type classification with no need for any pre-label datasets.

The rest of this article is structured in three parts. The principle of the decomposition of the EMA signals using the DWT technique for machine learning via the SVM approach is introduced in the second part. Experimental validation of the approach was performed and is compared with existing methods in Section 3. The last part summarizes the conclusion remarks.

2. Methodology

2.1. EMA Technique Incorporated with DWT for Damage Identification

This section introduces the principle of the EMA signals decomposed by the DWT technique. The DWT technique is the most popularly applied wavelet transform among the diverse wavelet decomposition techniques [75,76] and game theory problems involve multi-decision-making under incomplete or competing information [77], which is usually carried out via a two-channel filter bank with varied levels. In this study, the DWT technique was first applied to decompose the EMA signatures and to derive damage indicators. The EMA signal is generated by a PZT patch coupled with a target structure being monitored. When a PZT patch is surface-bonded or embedded in the structure, it dynamically vibrates under voltage excitation and simultaneously transfers the mechanical deformations to the structure. Once the stimulated vibration of the structure interacts with the patch, it produces an electric signal called the EMA signature. This signature could be measured by an impedance analyzer and used for structural damage analysis. Since any damages occurring to the structure would alter the structural mass, damping, and stiffness that related to the mechanical impedance incorporated with admittance, the variations in the EMA signatures can conversely indicate the occurrence of damage or an anomaly. The EMA signature expressed as Y(ω) is a complex function with sample points in the frequency domain, which consists of the real part namely conductance and the imaginary part namely susceptance, respectively. The mathematic expression of the EMA signature could be denoted as follows [28]:

$$Y\left(\omega \right)=G\left(\omega \right)+jB\left(\omega \right)=4\omega j{\displaystyle \frac{l^2}{h}}\left({\overline{\epsilon }}_{33}^T-{\displaystyle \frac{2d_{31}^2{\overline{Y}}^E}{(1-v)}}+{\displaystyle \frac{2d_{31}^2{\overline{Y}}^E}{(1-v)}}{\displaystyle \frac{Z_{a,eff}}{Z_{a,eff}+Z_{s,eff}}}{\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}\left(\kappa l\right)}{\kappa l}}\right)$$

(1)

where G(ω) and B(ω) are the conductance and susceptance, respectively; ω is the angular frequency; v is the Poisson’s ratio of the patch; j is equal to ${\left(-1\right)}^{1/2}$; l and h denote the half-length and thickness of the PZT, respectively; ${\overline{\epsilon }}_{33}^T={\epsilon }_{33}^T\left(1-\delta j\right)$ and ${\overline{Y}}^E=Y^E\left(1+j\eta \right)$ are the electric permittivity of the PZT patch under constant stress, and the complex Young’s modulus of the PZT in constant electric field, respectively; δ and η are the dielectric loss factor and structural mechanical loss factor of the PZT, respectively; and d₃₁ denotes the piezoelectric constant. $Z_{s,eff}$ and $Z_{a,eff}$, respectively, denote the effective mechanical impedance of the structure and the PZT patch; $v$ is the Poisson’s ratio of the patch; and κ denotes the wave number. Since the parameter $Z_{s,eff}$ is directly related to structural mass, damping, and stiffness, any damages would alter the EMA values, thus indicating damage occurrence. To quantify the changes in the EMA values, statistical indexes such as RMSD, RMSDk, CC, and MAPD were usually utilized in traditional investigations. The formulas of these indexes are expressed as follows [28,29,30]:

$$RMSD=\sqrt{\left({\displaystyle \frac{{\sum }_{i=1}^N(Y_i-Y_0{)}^2}{{\sum }_{i=1}^N(Y_0{)}^2}}\right)}$$

(2)

$$RMSDk=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{\left(Y_i-Y_{i-1}\right)}^2}{{\sum }_{i=1}^N{\left(Y_{i-1}\right)}^2}}}$$

(3)

$$MAPD={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|(Y_i-Y_0)/Y_0\right|$$

(4)

$$CC={\displaystyle \frac{\frac{1}{N}{\sum }_{i=1}^N\left(Y_i-\overline{Y_i}\right)\left(Y_0-\overline{Y_0}\right)}{{\sigma }_{Y_i}{\sigma }_{Y_0}}}$$

(5)

where $Y_0$ and $Y_i$ respectively denote the baseline and the corresponding EMA spectrum of the structure under different damage conditions. $\overline{Y_0}$ and $\overline{Y_i}$ and ${\sigma }_{Y_0}$ and ${\sigma }_{Y_i}$ denote the mean values and the standard deviation values of the corresponding EMA signatures.

To extract the characteristics of the EMA signatures, this study applied the DWT technique, which is a representative of the wavelet decomposition algorithm [75]. The EMA signal denoted as x[n] is a discrete sample data array with length N; the inner product of the EMA signal can be performed by selecting the wavelet basis functions to down-sample and filter the original one. Such a process efficiently resolves the EMA signatures into multiple low-frequency sub-bands namely $x_{1,L}\left[n\right]$, as well as the high-frequency sub-bands namely $x_{1,H}\left[n\right]$ via the low-pass filter g[n] and the high-pass filter h[n], respectively. Here, subscripts L and H, respectively, denote the outcomes from the progression of the EMA signal through the low-pass filter and the high-pass filter, respectively. ↓2 signifies the down-sampling filter. After decomposing using the DWT process, low-frequency critical information indicates the high scale of the EMA signal, which is the approximation component of the EMA signal, and the high-frequency critical information signifies the high scale of the EMA signal, which is the detail of the EMA signal. In this way, the original EMA signature passes through two mutual filters to generate two responding signals. In this regard, the DWT technique resolves the EMA array x(n) with the length of N as two sub-signals with a length of N/2, i.e., the approximation parts and detail ones. The approximate part is further continuously decomposed by a continuous decomposition process, and the EMA signal could be resolved as multiple low-resolution components, which can be expressed as follows [75,76]:

$$x_{i,L}\left[n\right]={\sum }_{k=0}^{K-1}{x}_{i-1}\left[2n-k\right]g\left[k\right]$$

(6)

$$x_{i,H}\left[n\right]={\sum }_{k=0}^{K-1}{x}_{i-1}\left[2n-k\right]h\left[k\right]$$

(7)

where k denotes the discrete sample point, n denotes the discrete signal, and K is the length of the filter.

After the decomposition of the EMA signal, its features could be further extracted out for damage characterization. Generally, the wavelet feature indicators encompassing the mean $\overline{{\omega }_j}$, the variance $\sigma$, the entropy $H_k$, and the energy $E_{j,k}$ are usually utilized, respectively, and expressed as follows:

$$\overline{{\omega }_j}={\displaystyle \frac{{\sum }_{L_j}{\omega }_{j,k}}{L_j}}$$

(8)

$$\sigma ={\displaystyle \frac{{\sum }_{L_j}{\left({\omega }_{j,k}-\overline{{\omega }_j}\right)}^2}{L_j}}$$

(9)

$$H_k=-{\sum }_{L_k}{E}_{n,k}\mathit{ln}{E}_{n,k}$$

(10)

$$E_{j,k}={\displaystyle \frac{E_{j,k}}{{\sum }_k{E}_{j,k}}}\times 100\%$$

(11)

where ${\omega }_{j,k}$ and L_j, respectively, denote the wavelet coefficient and the length of the jth wavelet coefficient; k denotes the position of the wavelet coefficient; and j is the jth wavelet coefficient. Regarding these indicators, wavelet energy signifies the energy of the EMA signal in various sub-bands and is usually applied to analyze the frequency domain characteristics of a signal. For the 1D DWT technique, the expression for computing wavelet energy is denoted as $E_j={\sum }_{Lj}{\left|{\omega }_{j,k}\left(n\right)\right|}^2$, where E_j denotes the energy part of the jth wavelet coefficient. Here, the entropy denotes the signal analysis approach depending on the wavelet transform utilized to evaluate the irregularity and complexity of a signal, which quantifies the complexity of the signal by calculating its entropy. The wavelet entropy is a description of the statistical properties of the EMA signal, which is applied in different fields such as data mining, signal processing, and biomedical engineering. The calculation algorithm for wavelet entropy involves decomposing the signal into wavelet coefficients at different frequencies using wavelet transform. A much higher value for the entropy means greater irregularity or complexity in the signal and vice versa. Commonly used wavelet entropy measures include Shannon entropy-based wavelet entropy, Tsallis entropy-based wavelet entropy, and Renyi entropy-based wavelet entropy [75,76]. Here, Shannon entropy-based wavelet entropy is applied for EMA decomposition. Since these indicators contain crucial information about the EMA signals, they are further applied for damage classification, as illustrated in the following subsection.

2.2. SVM-Based Damage Classification Approach

This subsection deals with the damage classification approach using the damage-sensitive indicators derived from the DWT technique for machine learning by the SVM. The SVM is a supervised machine learning model commonly used for classification or regression analysis. The principle of the SVM algorithm is to classify data by constructing hyperplanes in high-dimensional/infinite-dimensional space and maximizing the margin between different classes, thereby making the classification boundary more robust. Suppose there is a training dataset containing N samples, each composed of a feature vector $x_i$ and the corresponding class label $y_i$. Here, $x_i$ belongs to the feature space $R^d$, and $y_i$ belongs to the set {−1,1}. In a normal SVM model, if the data are $p$-dimensional vectors, a ($\mathit{p}-\mathbf{1}$)-dimensional hyperplane can be found to separate these data points. Whereby, the hyperplane can be represented as follows:

$${\mathit{w}}^{T}x+b=0$$

(12)

where $\mathit{w}$ represents the normal vector of the hyperplane and $\mathit{b}$ denotes the bias term. For a 2D plane representation of the SVM model, support vectors refer to the training sample points that influence the decision boundary during the training process. They could be deemed as essential elements in constructing the decision boundary and are typically located between the two parallel decision boundaries defined by ${\mathit{w}}^{T}x+b=1$ and ${\mathit{w}}^{T}x+b=-1$. The margin is the distance between those two decision boundaries. According to the principles of the SVM, for data points with positive class labels $y_i=+1$, they should lie above or on the upper boundary, while for data points with negative class labels $y_i=-1$, they should lie below or on the lower boundary. Therefore, the SVM should satisfy the following constraints:

$$\left\{\begin{array}{c}{\mathit{w}}^{T}x+b>+1,y_i=+1\\ {\mathit{w}}^{T}x+b<-1,y_i=-1\end{array}\right.$$

(13)

According to the principles of linear algebra, the margin is calculated using the following formula:

$$\mathit{d}={\displaystyle \frac{2}{‖\mathit{w}‖}}$$

(14)

where $\mathit{d}$ represents the Euclidean norm (magnitude) of the weight vector $\mathit{w}$. The objective of the SVM algorithm is to maximize the margin between classes, which is equivalent to the following optimization problem:

$$\begin{array}{c}\underset{w,b}{\mathit{min}}{\displaystyle \frac{1}{2}}{‖\mathit{w}‖}^2\\ s.t.y_i\left({\mathit{w}}^T{x}_i+b\right)\ge 1,i=1,2,\dots ,n\end{array}$$

(15)

Solving Equation (15) is a convex optimization problem, and a common approach is to apply the Lagrange multiplier approach to construct the Lagrangian function and then solve its dual problem to acquire the optimal solution to the original problem. Through introducing the Lagrange multiplier α and the Lagrangian function, Equation (15) could be transformed into

$$\begin{array}{c}\underset{\alpha }{\mathit{max}}L\left(w,b,\alpha \right)⟺\underset{\alpha }{\mathit{min}}{\displaystyle \frac{1}{2}}\sum _{i=1}^n\sum _{j=1}^n{\alpha }_i{{\alpha }_{j}y}_i{y_{j}x}_i^T{x}_j-\sum _{i=1}^n{\alpha }_i\\ s.t.\sum _{i=1}^n{\alpha }_i{y}_i=0,i=1,2,\dots ,n\end{array}$$

(16)

Using Equation (16), one can obtain the optimal $\hat{\mathit{w}}$ and $\hat{\mathit{b}}$ as follows:

$$\begin{array}{c}\hat{\mathit{w}}=\sum _{i=1}^n{\hat{\alpha }}_i{y}_i{x}_i\\ \hat{\mathit{b}}=y_i-\sum _{i=1}^n{\hat{\alpha }}_i{y}_i\left({x_j^{T}x}_i\right)\end{array}$$

(17)

According to the KKT complementary condition, at least one of the conditions ${\alpha }_i=0$ and $y_i·f\left(x_i\right)-1=0$ must hold. If condition ${\alpha }_i=0$ holds, the sample point $x_i$ is not a support vector and has no effect on the model. If condition ${\alpha }_i>0$ holds, the sample point $x_i$ lies on the maximum margin boundary and is a support vector. However, in practical applications, this assumption often does not hold true. One method to address this issue is to allow the SVM algorithm to induce errors on a few samples, i.e., not satisfying the constraints $y_i\left({\mathit{w}}^T{x}_i+b\right)\ge 1$. Therefore, the soft-margin is usually introduced into the SVM algorithm, which adopts a slack variable ${\xi }_i$ and a penalty parameter $C$, where the optimization problem in Equation (15) becomes

$$\begin{array}{c}\underset{w,b}{\mathit{min}}{\displaystyle \frac{1}{2}}{‖\mathit{w}‖}^2+C\sum _{i=1}^n{\xi }_i\\ s.t.\left\{\begin{array}{c}{y}_i\left(w^T{x}_i+b\right)\ge 1\\ {\xi }_i\ge 0\end{array},i=1,2,\dots ,n\right.\end{array}$$

(18)

Using the Lagrangian function and taking partial derivatives with respect to $\xi$,$\alpha$, and $\beta$, the dual optimization problem can be further simplified as follows:

$$\begin{array}{c}\underset{w,b}{\mathit{min}}{\displaystyle \frac{1}{2}}\sum _{i=1}^n\sum _{j=1}^n{\alpha }_i{{\alpha }_{j}y}_i{y}_j\left(x_i^T{x}_j\right)-\sum _{i=1}^n{\alpha }_i\\ s.t.\left\{\begin{array}{c}\sum _{i=1}^n{\alpha }_i{y}_i=0\\ 0\le {\alpha }_i\le C\end{array},i=1,2,\dots ,n\right.\end{array}$$

(19)

The difference in this equation from Equation (16) lies in the additional term $0\le {\alpha }_i\le C$ in the constraints of Equation (19). Solving the optimization problem in Equation (19) yields the optimal solution $\hat{\mathit{\alpha }}$, and the solution for the soft-margin SVM is expressed as follows:

$$\begin{array}{c}\hat{\mathit{w}}=\sum _{i=1}^n{\hat{\alpha }}_i{y}_i{x}_i\\ \hat{\mathit{b}}=y_i-\sum _{i=1}^n{\hat{\alpha }}_i{y}_i\left({x_j^{T}x}_i\right)\end{array}$$

(20)

Following the above procedure, once the EMA signatures are obtained under different damage conditions, they could be decomposed to obtain the DWT-derived indictors and input the SVM model, and then damage classification is automatically achieved. It is worth mentioning that the SVM model adopted here is formulated by a tuning process, where critical parameters are determined through the trial-and-error method. The most important hyperparameters such as penalty parameter or the kernel selection influence the classification results. The penalty parameter is adjusted in a range from 0.01 to 10 based on the default value of 1.0 after a trail-and-error process; the value of this parameter actually affects the accuracy of the results, the larger the value, the more rigorous the damage classification and the less the value, the more wrong the values; however, it could improve generalization ability. As for the kernel selection, we adopted the Gaussian kernel because it is the most widely used one among a few kernels. The KernelScale is used to control the influence of data points, which varies from 0.1 to 10 because when it reaches 0.1, there may be overfitting and when it reaches 10, there are too many wrong points. After the establishment of the model, the workflow of the formulated framework for damage classification of the RC slab structures employed in this study is shown in Figure 1, which contains five steps: Step 1: a sensor array consisting of a few PZT patches is distributed on the RC slab being monitored; Step 2: the EMA signals are measured for the PZT patches under varied damage cases, which are designed as different types with approximate severity; Step 3: the measured EMA signatures are decomposed by the DWT technique; Step 4: the EMA signatures are applied to compute the damage indicators including the RMSD, CC, RMSDk, and MAPD indexes, and for the calculation of the variance, the entropy, the mean, and the energy indicators; Step 5: damage identification and classification using the damage indicators taught to the SVM model. Following such a process, any damages with different severities that occurred on the slab structure can be automatically classified into their corresponding categories. The next section deals with the experimental test of the approach.

3. Experimental Validation and Discussions

3.1. Experimental Procedure

This section covers the experimental investigation for the validation of the proposed damage classification approach. To sufficiently verify the formulated approach, two experimental tests including prior test and practical application were conducted: Test#1 was performed on a lab-scale smooth RC slab and Test#2 was conducted on a practical shield-tunnel segment for damage classification. The RC slab in Test#1, with dimensions of 1000 mm × 500 mm × 60 mm (length × width × height), was casted for severe and minor crack classification. The material characteristics of the slab were designed as a strength of 30.0 MPa, water ratio of 0.38, and sand ratio of 0.42. The aggregate type for concrete was primarily rubble. Longitudinal and transverse reinforcement of the slab are shown in Figure 2. Longitudinal reinforcement comprised six steel bars with diameters of 6 mm within an interval of 190 mm, and transverse reinforcement comprised four steel bars with the same diameter and interval of 100 mm, with a strength of 300 MPa. After a normal 28-day curing process of the specimen, it was prepared to obtain the EMA signals. Two patches of PZT-5 with sizes of 10 mm × 10 mm × 0.5 mm were prepared to attach onto the slab for data measurements. The main properties of the PZT patches are listed in

Table 1. Physical properties of PZT patch.

Item	Density ($\mathbf{g}/{\mathbf{c}\mathbf{m}}^3$)	Dielectric Constant	Electromechanical Coupling Coefficient	Piezoelectric Coefficients ($\mathbf{C}/\mathbf{N}$)	Insulation Resistance ($\mathbf{m}\mathbf{\Omega }$)	Curie Temperature ($\mathbf{℃}$)
PZT-5	7.86	1400 ± 10%	0.8	≥400 × 10−12	≥1000	≥330

. Then two PZT patches were surface bonded on the top surface of the slab using an epoxy adhesive layer and were set for a couple of days before conducting measuring tests. The thickness of the adhesive layer for each patch was cautiously controlled within a thickness of approximately 0.1 mm through continuously pressing the patches tightly to make the redundant adhesive spill out. The material properties of the epoxy adhesive used in this test are listed in

Table 2. Physical properties of epoxy adhesive.

Item	Density ($\mathbf{g}/{\mathbf{c}\mathbf{m}}^3$)	Thickness (mm)	Poisson’s Ratio	Mechanical Loss Factor	Static Shear Modulus ($\mathbf{G}\mathbf{P}\mathbf{a}$)
Epoxy adhesive	1.70	0.3	0.40	0.34	1.24

.

Before applying for damage detection and classification, the two PZT patches were tested for signal collection. The EMA measuring system consisted of an Agilent 4294A impedance analyzer connected to a laptop for EMA data installation. Through a local inter-network generated by the analyzer, it was shared by the Agilent Connection Expert to the notebook computer. The scanning frequency for each condition was selected from 40 Hz to 400 kHz, with 801 sample points. The measuring duration of each EMA spectrum took approximately 60 s for one scan. Such a frequency band was determined mainly according to the previous literature [10,11,12,13,14], as a frequency band less than 400 kHz was deemed appropriate for concrete damage identification, but higher than 500 kHz was deemed more effective to determine the condition of the adhesive condition or the material properties of the PZT instead of structural damages. A pre-test on the PZT transducers was first performed before conducting crack damage. Figure 3 shows the conductance signatures of the two patches in a frequency range of 40 Hz–400 kHz. It is shown that the primary resonant peaks of the patches are primarily near 200 kHz, while significant differences existed in the resonance magnitudes with deviation of approximately 1.4 mS. Such differences may be mainly ascribed to individual variations in the PZT materials and local property differences in the RC slab. Hence the patches could be further applied for damage detection. To minimize the adverse influence of the bonding irregularities of the PZT patches on the signals and the latter decomposed signals, the bonding layer is carefully controlled as 0.01 mm thickness by compacting the patches tightly after their attachment. Since the impedance measurement is obtained in the high-frequency range, the ambient noise is avoided by controlling the environment by shutting off the possible electromagnetic interference machines. Actually, to minimize the noise influence, the mother wavelet for the DWT technique in this study was selected as Biorthogonal wavelets named bior2.4 because it has better performance for noise reduction compared with Daubechies, Haar, Symlets, and Coiflet. In addition, it is worth mentioning that all the EMA measurements for damage classification are under a lab temperature near 20 °C to mitigate ambient interference; the ambient temperature fluctuations during the tests were less than 2 °C, which will not affect the original or the decomposition signals [78,79]. As mentioned above, two aspects of experimental application were conducted to sufficiently verify the approach, which included damage classification on a lab-scale RC slab with totally different damage types, and on a practical tunnel segment structure with similar damage and different damage severity degrees, as explained in the following sections.

3.2. Damage Classification on Lab-Scale RC Slab

3.2.1. Qualitative Detection of Crack and Shock Damage

In the first aspect of validation, damage classification using the proposed approach was conducted on a lab-scale RC slab. Two types of damages namely crack and shock damages were conducted on the concrete surface by using a cutting machine, as shown in Figure 3. These two types of damages were introduced randomly in cross proceeding for the purpose of sufficient verification of the approach. For the severe crack group, four cracks as damage cases #1, #2, #5, and #6 were cut with sizes of 500 mm in length, 3 mm in width, and 10 mm in depth, and for the shocking group, the four corners of the slab were knocked off, as shown in Figure 3. The damage sequence is also displayed in Figure 3. Figure 4 displays the conductance spectrums in the whole and the selected critical frequency bands for the two PZT patches, as these peaks ranging from 150 to 250 kHz represent the dominant structural responses sensitive to damage. It can be seen that the resonance peaks of the conductance spectrums for each PZT transducer behave predominantly with leftward and upward (LU) shifts (i.e., decrease in resonant frequencies accompanied with increase in resonance peaks) along with the progression of the cracking and shocking damages. Despite the general moving trend, it is obvious that damage cases #1, #2, #3, and #7 have induced significant shifts in the conductance peaks of PZT 1 due to the smaller distances, while damage cases #2, #3, #6, and #8 have a more significant impact on that of PZT 2. Generally, shocking damage is more severe and could be detected by the PZT transducers, particularly for damage cases #3 and #7. However, it is difficult to divide the damage type depending on these qualitative analyses; this issue is expected to be resolved through applying the proposed framework in the following section.

To obtain the damage indicators, the conductance spectrums in Figure 4 were decomposed via a three-level DWT technique, as shown in Figure 5. It can be seen that the low-frequency information of the raw conductance signatures of PZT 1 and 2 that are represented by “A3” in Figure 5 are approximately identical to the original signals, and the extracted high-frequency responses represented by “D3” behave significantly, as indicated in the conductance signatures. Such a three-level DWT technique chosen for the decomposition process depended on the trial-and-error method by observing the most representative information extracted from the original signals, despite that the decomposition of the conductance signals in this experiment could be continuously performed by nine times given that it generally comprises 801 discrete data points. Here, the three-level DWT technique is adopted mainly due to the fact that ignorable variations are observed between the original conductance and reconstructed EMA signatures namely A3 after three-level decomposition, where the error is very small, as shown in Figure 5a,b. Additionally, approximate coefficient is applied here as it performs more regular characteristics when compared with the detail coefficient. Depending on the approximate coefficient, the parameters, including the mean, variance, and energy, are accordingly calculated. In addition, for dataset size per class and the test/training split, three groups of the EMA dataset were collected per class with two of them trained and the remaining one tested for damage classification. Data normalization is performed by using a Z-scored algorithm before they are input into the SVM model. As for the feature selection strategy, we used the Chi-square method for automatic feature selection in MATLAB 2017 software, which is suitable for discrete features and categorical labels of the EMA signatures. Nevertheless, other multiple wavelet types and orders should be conducted to evaluate the systematic performance of the indicators [80]. To quantitatively assess the variations in the original and the extracted information in Figure 5, the traditional damage indicators including the RMSD, the MAPD, the RMSDk, and the CC, as well as the wavelet-related indicators including the mean, the variance, the energy, and the entropy, were computed in normalization and are displayed in Figure 6. It can be seen that there are uniform increases for the RMSD, the MAPD, and conversely the CC values, which indicate the damage growth well. The random changes in the RMSDk values indicate the different impact caused by the different types of damage. However, despite the changes in the values, it is unable to determine the damage types. As for the wavelet-related indicators displayed in Figure 6, the entropy, the variance, and the mean show approximate values for the two types of damage, which demonstrate that these indicators have captured the inherent difference in the slab structure caused by the shock damage and the crack damage, as they involve damage-sensitive information of the decomposing EMA signals. In this respect, these indicators are superior to the traditional ones and could be applied for damage classification, as discussed in the next subsection.

3.2.2. Damage Classification Using the Proposed Approach

This subsection covers damage classification of the slab structure by using the proposed approach. Before inputting the SVM model for damage classification, all the indicators were normalized to optimize the fast convergence and stability for the SVM model and enhance its generalization ability and interpretability using the Z-scored algorithm, as expressed by

$$Z={\displaystyle \frac{X-\mu }{\sigma }}$$

(21)

where $\mu$ and $\sigma$ denote the mean and variance of damage index X, respectively. By substituting the values in Figure 6 into Equation (21), the values of the eight indictors are tabulated in

Table 3. The normalized RMSD, MAPD, RMSDk, and CC and the entropy, mean, variance, and energy indicators from the conductance signals of PZT 1 under different damage cases.

PZT1	RMSD	MAPD	RMSDk	CC	Entropy	Mean	Variance	Energy
Case 1	−1.17	−1.11	0.12	0.91	−1.15	−1.18	−1.11	1.31
Case 2	−1.12	−1.15	0.19	0.90	−0.82	−0.92	−0.39	−1.43
Case 3	−0.39	−0.43	0.52	0.46	−0.92	−0.85	−1.14	0.01
Case 4	−0.24	−0.35	−1.04	0.35	−0.71	−0.66	−0.90	0.36
Case 5	−0.01	0.05	−0.31	0.23	0.39	0.42	0.45	0.75
Case 6	0.04	0.12	−0.98	0.25	0.96	0.94	1.31	0.88
Case 7	1.35	1.34	2.06	−1.40	1.13	1.10	0.99	−0.91
Case 8	1.54	1.52	−0.56	−1.71	1.14	1.15	0.79	−0.96

and

Table 4. The normalized RMSD, MAPD, RMSDk, and CC and the entropy, mean, variance, and energy indicators from the conductance signals of PZT 2 under different damage cases.

PZT1	RMSD	MAPD	RMSDk	CC	Entropy	Mean	Variance	Energy
Case 1	−1.88	−1.80	−0.50	1.51	−1.83	−1.87	−1.85	0.56
Case 2	−0.91	−0.67	0.99	1.08	−0.57	−0.49	−0.58	1.58
Case 3	−0.09	−0.34	1.20	0.27	−0.41	−0.43	−0.36	−0.82
Case 4	−0.02	−0.29	−1.48	0.18	−0.37	−0.37	−0.36	−1.07
Case 5	0.22	0.14	−1.20	−0.06	0.19	0.21	0.22	−0.67
Case 6	0.73	0.79	−0.10	−0.69	1.01	1.00	1.06	−0.91
Case 7	0.73	0.87	0.74	−0.73	0.89	0.94	0.77	0.98
Case 8	1.22	1.31	0.36	−1.56	1.09	1.02	1.10	0.34

.

According to the basic principle of soft interval SVM expressed in Equation (20), this study first randomly selected two damage indicators as research objects and input them into the model. Taking the data of PZT 1 as an example, the damage classification results of the combination of some damage indicators are shown in Figure 7 and Figure 8. Figure 7 shows the classification results using the traditional damage indicators taught to the proposed SVM model; it is seen that there are two cases of shock damages wrongly classified to the crack damage group no matter which two indicators are utilized. However, if only one wavelet-related indicator is used for the SVM model, the classification accuracy could reach 100%, as shown in Figure 8. This result indicates the superiority of the proposed wavelet-related indicators compared to the traditional ones for damage classification. It can be noted that the contribution of all the indicators in pairs for classification result analysis shows that the variance of the wavelet coefficient index has the best recognition ability, with the highest frequency of occurrence in the combination of damage indicators for successfully identifying the damage type, indicating its significant role in the recognition of damage types. To comprehensively assess the validity of all the eight damage indicators, two are randomly combined for computation. The results are shown in Figure 9, which indicates that with the increased use of the number of damage indicators, the ability of the SVM model to accurately identify damage types is also improved. When the number of damage indicators reaches 7, all the damage conditions could be precisely classified to their type with an accuracy of 100%. This result confirms the efficacy of the proposed approach to identify the damage type when the traditional EMI technique fails, which could be potentially extended to practical infrastructural damage classifications.

3.3. Application Process of Practical Tunnel Segment Slab Structure

To further validate the effectiveness of the approach and to enhance its generalizability, this section discusses applying the approach to classify the more complex damage types including severe cracking damage and minor crack/shock damages for a practical tunnel segment RC slab structure. Figure 10 shows the date measuring system for the practical tunnel segment RC slab with two bonded PZT patches. The segment structure with dimensions of external/internal diameter of 4000 mm/500 mm, width of 500 mm, and length of 1000 mm was successively cut by eight transverse surface cracks as severe damage, eight short cracks with approximate dimensions of 10 mm × 2 mm × 5 mm (length × width × depth) as minor damage, and knocking off the surface concrete as ten cases of shock damage, as shown in Figure 10. Considering the minor loss of concrete caused by the shock damage, the minor cracks and shock damages are classified as the minor damage group. The conductance spectrums of all the PZT patches are shown in Figure 11. It is seen that for all the PZT transducers, the most variations in the conductance spectrums are observed for the initial severe damage conditions, which show drastically leftward shifts in the resonance peaks with magnitude fluctuations. And for the shock and minor crack damages, the conductance signatures have relatively small variations due to the fewer mechanical variations caused by the small cracks and relatively farer distances of the shock damage from the PZT transducers. According to these results, the shock and minor crack damages are deemed to be the minor damage type, validating the proposed approach.

Then the conductance signatures in Figure 11 are decomposed by using the DWT technique, and the wavelet-related damage indicators including the entropy, mean, variance, and energy could be calculated to establish a classification model. Figure 12 displays these indicators under different damage levels. It is found that there are obvious changes in the mean, variance, and entropy indicators with damage progression; these indicators exhibit changes for different PZT transducers under varied damage levels. However, it is difficult to determine the damage type and damage severity based on these indicators. Then these indicators are input in the proposed model for damage classification. Figure 13 shows the classification results by the proposed model via the hybrid RMSD/MAPD with entropy; it can be seen that all of the severe damage conditions and the minor damage conditions could be accurately classified into the right group by using the proposed model for PZT1 and only one sample is wrongly classified near the boundary line for PZT 2. These results again demonstrate the superiority of the proposed model for damage classification, thus presenting excellent generalizability and promising potential for damage classification on practical civil engineering infrastructures.

4. Conclusions

This study presented an innovative damage classification approach for concrete structures by integrating supervised machine learning with wavelet-decomposed EMA signatures. The proposed method employed the DWT technique to decompose the EMA signals acquired from distributed PZT transducers, extracting both conventional damage indicators (RMSD, RMSDk, MAPD, and CC) and novel wavelet-based features (wavelet energy, variance, mean, and entropy). These indicators were subsequently processed using an SVM model to achieve automated damage classification. The methodology was rigorously validated through experimental testing on a laboratory-scale reinforced concrete slab and by applying to a practical tunnel segment RC slab structure instrumented with multiple PZT patches, designed to distinguish between two distinct concrete damage types including severe cracking damage and minor crack/shock damage damages. Initial results using traditional damage indicators with the SVM model revealed two misclassifications where shock damages were incorrectly identified as crack damages, regardless of the indicator combinations. In contrast, incorporating the hybrid wavelet-derived indicators achieved perfect classification accuracy as high as 100%, demonstrating the superior discriminative capability of wavelet-based features over conventional indicators. Performance evaluation conducted by systematically assessing random combinations of damage indicators showed that the classification accuracy of the SVM model improved progressively with the number of incorporated indicators. Remarkably, when seven indicators were utilized, all damage conditions were correctly classified with 100% accuracy. Concerning the application of damage to the practical tunnel slab structure, the classification results cogently validated the high accuracy as well. These results cogently confirmed the effectiveness of the proposed approach for identifying the damage type when using the traditional EMI technique failed. The promising results in this study established a machine learning-based framework for damage classification that eliminated the need for pre-labeled datasets. This approach has significant potential for extension to broader infrastructural health monitoring applications in future research. Despite of the promising results, there still remain several limitations that should be investigated in future works: (1) Synthetic noise effect on the original EMA and decomposition signals should be investigated despite the ambient one has been reasonably avoided. (2) The use of the SVM should be better supported in future investigations, as for instance, a Relevance Vector Machine is a compelling alternative to the SVM especially for structural health monitoring tasks involving classification or regression based on EMI signal data. (3) Despite the involvement of the wavelet-based indicators, techniques such as recursive feature elimination or permutation feature importance could be used to quantify which indicators are most critical to the classifier’s success. These issues will be systematically investigated in future work.