Fractality–Autoencoder-Based Methodology to Detect Corrosion Damage in a Truss-Type Bridge

Abstract

Corrosion negatively impacts the functionality of civil structures. This paper introduces a new methodology that combines the fractality of vibration signals with a data processing stage utilizing autoencoders to detect corrosion damage in a truss-type bridge. Firstly, the acquired vibration signals are analyzed using six fractal dimension (FD) algorithms (Katz, Higuchi, Petrosian, Sevcik, Castiglioni, and Box dimension). The obtained FD values are then used to generate a gray-scale image. Then, autoencoders analyze these images to generate a damage indicator based on the reconstruction error between input and output images. These indicators estimate the damage probability in specific locations within the structure. The methodology was tested on a truss-type bridge model placed at the Vibrations Laboratory from the Autonomous University of Queretaro, Mexico, where three damage corrosion levels were evaluated, namely incipient, moderate, and severe, as well as healthy conditions. The results demonstrate that the proposal is a reliable tool to evaluate the condition of truss-type bridges, achieving an accuracy of 99.8% in detecting various levels of corrosion, including incipient stages, within the elements of truss-type structures regardless of their location.

1. Introduction

Civil engineering infrastructure, e.g., buildings, bridges, towers, and other facilities, is a vital component of modern infrastructure, supporting essential societal functions such as work, residence, and communication across cities and nations [1]. However, these structures inevitably undergo aging processes due to environmental factors (e.g., environmental temperature, humidity, aggressive agents, etc.) and dynamic loads (e.g., forceful winds, seismic events, human-induced perils, etc.), leading to deterioration that compromises their operational integrity [2]. Among various degradation mechanisms, corrosion is a gradual deterioration process that can diminish structural component thickness, potentially resulting in critical structural failures [3]. It is produced by a chemical or electrochemical reaction with the environment, affecting the physical and mechanical properties of the elements that integrate a civil construction, and therefore, the risk of failure increases [4,5]. Consequently, addressing corrosion requires significant financial investment for its prevention and repair [6]. Therefore, early corrosion detection is crucial in civil structures [7].

Recent studies have confirmed the critical importance of developing advanced methodologies for detecting and assessing corrosion in civil structures. For instance, Melchers contributed with empirical models and phenomenological approaches aimed at providing reliable frameworks for predicting corrosion behavior and assessing structural reliability in marine environments [8,9]. These models are particularly relevant for offshore structures, where corrosion can significantly compromise structural integrity over time [10]. On the other hand, Huras et al. [11] introduced a novel method for quantifying local stiffness loss in beams using rotation rate sensors, emphasizing the need for precise and localized monitoring techniques to ensure structural safety. These contributions highlight the ongoing efforts to enhance the understanding of corrosion and its impact on structural health, yet there remains a need for methodologies that can effectively utilize vibration signals for early detection. In this regard, the stochastic reliability-based design optimization framework proposed by Sokołowski and Kamiński [12] demonstrated the application of probabilistic methods to account for the uncertainties in corrosion effects on structural components. This approach, combined with the uncertainty modeling techniques discussed by Melchers [13], provides a comprehensive strategy for managing the risks associated with corrosion in steel structures. Despite these advancements, recent studies have concentrated on using experimental vibration signals for structural health monitoring, with a particular emphasis on applying these methodologies in real-world scenarios, where vibration-based indicators could provide an alternative for assessing civil structures exposed to or affected by corrosion-induced damage.

Furthermore, the identification of structural damage and the implementation of SHM systems have become pivotal subjects within engineering fields such as civil, structural, and construction engineering [14,15,16]. These disciplines seek effective systems or methods to assess structural health by analyzing physical variables [17,18]. A vibration-based SHM system involves three essential steps for assessing structural condition: acquisition, processing, and interpretation. The processing step is especially complex due to low signal-to-noise ratios (SNR) and non-stationary characteristics in civil structures’ vibrational responses. Structures with early- or incipient-stage damage exhibit subtle changes in their vibration responses, further complicating analyses [19]. These complexities can vary with damage levels and challenge the efficacy of signal processing techniques [20]. Therefore, the advancement and implementation of methodologies or signal processing techniques adept at discerning and extracting indicators of structural damage, particularly at its incipient stages, are imperative for evaluating structural integrity.

In the last decade, a variety of parametric and non-parametric machine learning algorithms have emerged to evaluate the condition of civil structures by using vibration signals [21,22,23,24]. These algorithms typically involve two primary steps: pattern identification and categorization [25]. In the pattern identification step, vibration signals are analyzed using linear or non-linear techniques to identify patterns based on their statistical characteristics, energy, geometric complexity, and other features indicative of the structure health condition. In the categorization step, the extracted characteristics are used to train classifiers that automatically assess the health of the structure [26]. Parametric features such as modal properties (e.g., modal shapes, damping ratios, and natural frequencies) [27] and non-parametric features such as statistical methods [28], autoregressive models [29], fractal dimensions [30], entropy methods [31], wavelet transforms [32], empirical mode decomposition [33], the multiple signal classification method [34], and principal component analysis [35] have been widely explored. Various classifiers, including decision trees [36], artificial neural networks [37,38], support vector machines [39], and fuzzy logic classifiers [40], have been employed to associate these features with the structural condition. Despite promising results, these methods face limitations such as environmental sensitivity, the need for fine calibration, and the reliance on expert knowledge for feature and classifier selection [41].

Recent advances in artificial intelligence, particularly deep learning, have provided new opportunities for overcoming these limitations [42,43]. Autoencoders, a type of unsupervised learning model, have shown significant promise in this regard [44,45,46,47,48]. Autoencoders consist of an encoder, which compresses and encodes the data into a different representation space, and a decoder, which reconstructs the original data from this encoded representation. This architecture allows autoencoders to effectively extract and select features from data, minimizing reconstruction error [46,49]. When machine learning algorithms are combined with fractal analysis, which captures the complexity and self-similarity of vibration signals, they can provide a robust framework for structural health monitoring [50,51,52]. Fractal dimensions offer a detailed characterization of vibration signals, and their integration with autoencoders enhances the ability of the model for detecting subtle signs of structural damage. Despite some challenges, such as long training times and the interpretability of extracted features, ongoing research and improvements in autoencoder models hold the potential for more accurate and efficient damage detection in civil structures.

This paper proposes a novel methodology that combines fractal analysis of vibration signals with autoencoders for corrosion detection in a truss-type bridge. Initially, vibration signals were processed using six different fractal dimension algorithms, i.e., Katz [53], Higuchi [54], Petrosian [55], Sevcik [56], Box dimension [57], and Castiglioni [58], in order to generate images that capture the fractal characteristics of the vibration signals. These images were then analyzed using an autoencoder, which generates a damage indicator based on the image reconstruction error to estimate the probability of detecting corrosion damage, regardless of the structural damage location. The proposed procedure was proven using a truss-type bridge lab model exposed to different levels of corrosion damage ranging from incipient to severe. The experimental setup, located at the Autonomous University of Queretaro, Mexico, provides a comprehensive testbed for evaluating the effectiveness of the methodology under various damage conditions. The results demonstrate the potential of combining fractal analysis with autoencoders for enhancing the accuracy of corrosion detection in civil structures, thereby contributing to more effective and efficient SHM practices.

2. Theoretical Background

This section introduces the main concepts and mathematical definitions utilized in the proposed fractality–autoencoder-based methodology.

2.1. Fractal Dimension

As previously mentioned, the vibration signals generated by a civil structure contain information about its physical constitution; therefore, being able to quantitatively characterize the changes in this information will allow the criteria establishment to determine the structure’s condition, such as a healthy structure or a structure with corrosion damage. However, characterizing these changes is not trivial because of the high nonlinearities and low SNR of vibration signals.

Over the years and across different fields of science, fractal dimension, a nonlinear measurement of the complexity or fractality of a signal, has been successfully used to quantify these changes, encompassing applications from the diagnosis of electric motors [59] to the diagnosis of heart diseases [60]. This success promotes its use in this work in the topic of structural diagnosis by corrosion.

The measurement of the fractality of a signal has been proposed with different approaches, the most common being Petrosian (PFD), Higuchi (HFD), Sevcik (SFD), Box counting (BFD), Katz (KFD), and Castiglioni (CFD) fractal dimensions [53,54,55,56,57,58]. As each of these indicators quantifies the fractality of the signal in a different manner, it is essential that all of them are analyzed and taken into account to characterize the changes in the fractality of the vibration signals. In particular, the ability of Petrosian’s method to identify abrupt changes makes it ideal for signals experiencing peaks or sudden discontinuities. The Sevcik method is simple to implement and provides good discrimination for signals with varying levels of complexity. Higuchi’s method is highly effective for analyzing non-stationary signals and is sensitive to changes in signal roughness. Box counting offers a clear visualization of the spatial distribution of the signal and is suitable for signals with complex geometric structures. Katz’s method is particularly useful for capturing the complexity of signals that exhibit repetitive patterns or cycles. Finally, Castiglioni’s method offers a robust measure of signal complexity, particularly useful for signals with noise. Therefore, the simultaneous use of these methods allows a more comprehensive characterization of the geometric structure of vibration signals. Since each method emphasizes different aspects of signal complexity, their combined application may maximize the ability to detect structural damage, providing a robust and multifaceted characterization of changes in vibration signals. In the next subsections, the six fractals used in this work are detailly described.

2.1.1. Katz Fractal Dimension

The KFD of a signal sampled from the time domain (x_i, y_i) with N samples is computed by the following [53]:

$$\mathrm{KFD}={\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}\left(N\right)}{\log \left(N\right)+\log \left(\frac{d}{L}\right)}}$$

(1)

where d represents the maximum distance from the first sample to the others, and L is the Euclidian distance separating adjacent samples. These values are computed as follows:

$$L={\sum }_{i=2}^N\sqrt{{\left(x_i-x_{i-1}\right)}^2+{\left(y_i-y_{i-1}\right)}^2}$$

(2)

$$d=max\left({\left(x_i-x_1\right)}^2+{\left(y_i-y_1\right)}^2\right)$$

(3)

2.1.2. Higuchi Fractal Dimension

For calculating the fractality according to HFD, the input signal x(t) must first be decomposed into new signal sequences according to the following [54]:

$$S_k^m=\left\{S_m, S_{m+k}, S_{m+2k},\dots , S_{m+ak}\right\}$$

(4)

where k = 1, 2, …, k_max, and a = (n − m)/k for m = 1, 2, …, k. Once the new sequences are obtained, their length must be calculated as follows:

$$L_k^m={\displaystyle \frac{\sum _{i=1}^a\left|S_{m+ik}-S_{m+(i-1)k}\right|(N-1)}{ak}}$$

(5)

where N represents the number of samples for $S_k^m$. With these values, the average value is obtained through the following:

$$L_k={\displaystyle \frac{1}{k}}{\sum }_{m=1}^k{L}_k^m$$

(6)

To conclude, create a log-log plot of L(k) against (1/k) and determine the best-fit line for the slope of the curve. Thus, the slope value of this line is the HFD.

2.1.3. Box Counting Fractal Dimension

To calculate BFD of the 1-dimensional signal x(t) with N samples in the time domain, it is necessary to consider a grid of boxes that enclose the signal. In this way, the fractality will correspond to the boxes that are occupied. The BFD of a signal can be approximately calculated [57]:

$$BFD\cong -{\displaystyle \frac{ln\left(\sum _{i=1}^{N-1}\raisebox{1ex}{$\left|x_{i+1}-x_i\right|$}\!\left/ \!\raisebox{-1ex}{$∆t$}\right.\right)}{ln\left(∆t\right)}}$$

(7)

where $∆t$ is the sampling time.

2.1.4. Petrosian Fractal Dimension

PFD must be computed over a thresholder signal; i.e., the input signal x(n) = x₁, x₂, …, x_N with N samples has to be converted through the following [55]:

$$z_i=\left\{\begin{array}{c}1 x_i>mean\left(x\right)\\ -1 x_i\le mean\left(x\right)\end{array}\right. for i=1, 2, \dots , N$$

(8)

Thus, the PFD value is computed by the following:

$$PFD={\displaystyle \frac{log\left(N\right)}{log\left(N\right)+log(N/(N+0.4N_{∆}\left)\right)}}$$

(9)

where $N_{∆}$ represents the total changes of z_i and is computed as follows:

$$N_{∆}={\sum }_{i=1}^{N-1}\left|{\displaystyle \frac{z_{i+1}-{ z}_i}{2}}\right|$$

(10)

2.1.5. Sevcik Fractal Dimension

In general, the SFD index is considered a variation of KFD since it also depends on the Euclidian distance; however, for SFD, the input signal (x_i, y_i) with N samples must be initially normalized [56]:

$$x_i^{*}={\displaystyle \frac{x_i-min\left(x\right)}{max\left(x\right)-min \left(x\right)}}$$

(11)

$$y_i^{*}={\displaystyle \frac{y_i-min\left(y\right)}{max\left(y\right)-min \left(y\right)}}$$

(12)

where the functions min(·) and max(·) compute the minimum and maximum values of x and y, respectively, for the sample i.

Once the input signal is normalized, the SFD is determined as follows:

$$SFD=1+{\displaystyle \frac{log\left(L\right)+log\left(2\right)}{log(2·(N+1\left)\right)}}$$

(13)

where L is computed by the following:

$$L={\sum }_{i=2}^N\sqrt{{\left(x_i^{*}-x_{i-1}^{*}\right)}^2+{\left(y_i^{*}-y_{i-1}^{*}\right)}^2}$$

(14)

2.1.6. Castiglioni Fractal Dimension

The CFD is also a modified version of KFD, where L, the total distance between adjacent points, and d, the non-stable range of the analyzed time-series signal, are computed as follows [58]:

$$L={\sum }_{i=1}^{N-1}\sqrt{{{\left(y_{i+1}-y_i\right)}^2+\left(x_{i+1}-x_i\right)}^2}$$

(15)

$$d=max\left(y_i\right)-min\left(y_i\right)$$

(16)

Once these values are obtained, the CFD value is computed:

$$\mathrm{CFD}={\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}(N-1)}{\log \left(N-1\right)+\log \left(\frac{d}{L}\right)}}$$

(17)

2.2. Autencoder

An autoencoder, a tool used in machine learning, is a neural network designed to learn in an unsupervised way to reconstruct the input data at the output as shown in Figure 1 [61]; i.e., the output image of an autoencoder is a reconstructed version of the input image. This neural network architecture comprises two main components: an encoder and a decoder. The encoder is tasked with converting the input data a(n) into the compressed/encoded representation b(m) through the function b(m) = enc(a(n)). In this work, the input data are the fractal dimension values of the analyzed signal. Conversely, the decoder reconstructs the original input from this encoded representation by applying the function g(n) = dec(b(m)). Given that b(m) serves as a compact representation of a(n), autoencoders are commonly employed for dimensionality minimization tasks. Based on these points, an autoencoder is defined as follows [62]:

$$dec\left(enc\right(a\left(n\right)\left)\right)=c\left(n\right)$$

(18)

where c(n) is as similar as possible to a(n).

To be more specific regarding the functions enc(·) and dec(·), the following vector operations have to be carried out:

$$\mathrm{b}=h\left({\mathrm{W}}^{\mathrm{E}}\mathrm{a}+{\mathrm{B}}^{\mathrm{E}}\right)$$

(19)

where h, W, and B are the transfer function, the weight matrix, and the bias vector, respectively. As transfer function, the logsig function is utilized in this work [63]. The superscript E denotes the numbers associated with the encoder stage. After encoding the input data, the decoder generates an approximation of the input data through the following:

$$\mathrm{c}=h\left({\mathrm{W}}^{\mathrm{D}}\mathrm{b}+{\mathrm{B}}^{\mathrm{D}}\right)$$

(20)

where the superscript D refers to the numbers related with the decoder stage. To adjust the bias and weight values during the training, the loss function is evaluated utilizing the cross-entropy function and the scaled conjugate gradient backpropagation method [63].

3. Methodology

Civil infrastructure, e.g., bridges, buildings, and towers, among others, are constantly experiencing dynamic or changing forces or excitations, i.e., earthquakes, wind, and traffic, among others. These dynamic excitations provide an opportunity to assess the structural behavior through vibrational responses recorded or acquired during these events [64]. Vibration signals play a crucial role in structural evaluation, as they can capture changes or patterns indicative of damage within the structure. The extent of these changes varies with the damage severity level; for instance, an initial damage known as incipient or light damage may cause subtle alterations in the signals [65]. Hence, the capability of a method or technique to identify relevant patterns in the measured signals is essential for accurate structural condition assessment.

Figure 2 illustrates the steps of the proposed method for determining the condition of a truss structure in bridge configuration. It consists of four main steps. As a first step, the vibration signals are measured in three axes, represented by Vx, Vy, and Vz, under two conditions: healthy and damaged (with elements affected by corrosion). These signals are acquired from a 3D nine-bay truss structure configurated as a bridge, which is subjected to low-amplitude dynamic excitations (comparable to ambient vibrations) imposed by an electromechanical system known as electrodynamic shaker [66]. Then, the acquired vibration responses are evaluated by mean of six FD methods: SFD, KFD, CFD, BFD, HFD, and PFD, with the aim of calculating features or patterns that can effectively determine the bridge’s condition. It is worth mentioning that the monitored vibration signals of the truss-type bridge, which are collected for 20 s during each test (this time window was chosen to ensure that the electromechanical system can reliably generate an excitation range from 0 to 100 Hz, allowing capturing the relevant natural frequencies of the structure), are segmented into time windows of 1 s. Each of these segmented signals is analyzed by the proposed method, leading to a total of 360 values (60 values per FD method, considering the three axes).

The third step involves the creation of images through the FD values obtained in the previous stage (see Figure 3a), which can be processed by an unsupervised algorithm, specifically an autoencoder. The obtained 360 FD values for each test are visualized using a surface plot, where the X and Y axes represent the number of FD values estimated and the FD methods employed, respectively, while the Z-axis indicates the amplitude or magnitude of these FD values (see Figure 3b). The “turbo” color map was chosen for its smooth color transition. To create the image, a top-down view is utilized, where variations in color depict the estimated FD values, and the dimensions of the image indicate the number of FD values estimated and the FD methods employed, respectively, ensuring clarity and minimal overlap between data elements (as illustrated in Figure 3c). The generated image is converted to grayscale to reduce computational load compared to an RGB format (Figure 3d) [67]. The generated images have dimensions of 666 by 531 pixels.

Finally, in the fourth step, an autoencoder model is trained for reconstructing input images as outputs, making its training an unsupervised learning task. It is crucial to note that the autoencoder is trained exclusively on non-damaged FD values to establish a baseline model. Therefore, a significant reconstruction error indicates the model’s difficulty in reconstructing images generated by FD values obtained from a damaged structure. This error serves as a damage indicator, DI, to identify or determine the bridge’s condition.

4. Experimental Setup

The experimental test rig used to evaluate the integrity condition of the bridge studied in this article is shown in Figure 4 and is situated at the Autonomous University of Queretaro, Campus San Juan del Rio, Mexico. This experimental prototype contains 162 bar elements, creating 9 bays (see Figure 4a), similar to the model studied in [68]. The material utilized for this truss-type bridge is aluminum grade 6061-T6 (Mg~0.8–1.2%, Al~95–98%, Cu~0.15–0.40%, and Si~0.4–0.8%) [69], and its general dimensions are length of 6.40 m, height of 0.71 m, and width of 0.71 m. On the other hand, the diameter of the bars is 19 mm, the horizontal and vertical elements have 0.70 m length, and the diagonal bars have corresponding lengths of 0.70√2 m. With the aim of assessing the structural state of this experimental platform (see Figure 4b), an electrodynamic shaker (Labworks model ET-126B, Labworks, Costa Mesa, CA, USA) along with a linear amplifier (Labworks model PA-138, Labworks, Costa Mesa, CA, USA) were used to generate continuous dynamic excitations of low-intensity/frequency into the structure, simulating, in this way, the vibratory behavior of real bridges [70]. So as to produce the corresponding dynamic signals, 200 Hz sampling frequency white Gaussian noise was used; hence, a 0–100 Hz bandwidth with a frequency content in the whole range was generated. Taking into account that the central region of this experimental bridge is the one that registers the highest vibration amplitudes [71,72], the fourth bay, which is located in this zone, was selected for acquiring the respective dynamic signals; in this way, valuable data were monitored and analyzed. A Kistler tri-axial accelerometer (Vx, Vy, and Vz) model 8395A with ±10 g of measuring range, 400 mV/g of resolution, and 0–1000 Hz bandwidth, was utilized for measuring the structure dynamic response. This sensor was placed on the fourth bay, as can be seen Figure 5, indicated by a blue dot. Lastly, a National Instruments data acquisition system (DAS) model NIUSB-6002 with a 16-bit resolution analog-to-digital converter was employed to store and transfer the measured vibrational signals into a computer.

As a means to simulate corrosion damage in the bar elements at different levels, a diameter reduction of 1 mm was induced for incipient damage (Figure 5b), 4 mm for moderate damage (Figure 5c), and 8 mm for severe damage (Figure 5d). The selected sampling frequency for each experiment was 200 Hz, resulting in a total of 4000 samples considering the 20 s duration per test. Likewise, the experiments were performed 100 times for each condition; that is, 100 times for the healthy case and 100 times for each damage scenario (corrosion level) at each of the nine bays, generating a total of 2800 tests. It should be noted that each corrosion condition (incipient, moderate, and severe corrosion level) was applied one at a time into the structure and one after the other, replacing a healthy element with a corroded bar. Thus, the experimental procedure consisted of substituting a healthy bar at a certain location by another damaged bar until completing the three damaged scenarios, beginning in the first bay and, subsequently, repeating the process for the other eight bays of the structure. For each bay, the selection of the corroded element’s location was random (see Figure 5) so that the efficacy of the proposed methodology to detect damage in the experimental bridge could be verified regardless of the damage position.

Damaged Elements

The natural corrosion process can take a long time [73]. Hence, for the purpose of this research, the corrosion process was accelerated in order to assess the effect of this kind of damage at different levels in the behavior of the experimental truss-type structure; thus, in order to produce external corrosion [74,75,76], the bars’ ends were submerged in hydrochloric acid (HCl). In this way, the three damaged levels obtained in the bars included diameter reductions of 1 mm, 4 mm, and 8 mm for incipient, moderate, and severe damage, respectively, with the corresponding mass reductions of 19.7 g, 49.7 g, and 84.4 g with respect to the healthy element. The controlled diameter reductions were chosen to ensure that the integrity of the elements was not compromised [77]. In Figure 6, pictures of a healthy bar (19 mm of diameter and 445 g of mass) and the three damaged bars with the different corrosion levels (incipient, moderate, and severe) can be observed.

5. Results

Figure 7 illustrates the vibrational responses of the Vx, Vy, and Vz axes monitored from the fourth bay location for both the healthy state and the incipient damage condition. As depicted, both vibrational responses exposed similar visual patterns, underscoring the necessity of employing robust algorithms like FD methods to discover reliable patterns or differences into the monitored signals for accurate structural condition assessment.

Once the different FD methods were applied to diverse conditions, the values obtained from these FD methods for each condition were used to create images that were employed to determine the bridge’s condition. For illustrative purposes only, Figure 8 presents the images obtained for the conditions of a healthy structure and a structure exhibiting an incipient corrosion damage located in the fourth bay, respectively. As depicted, both images exhibit similar visual patterns, emphasizing the importance of using robust algorithms such as autoencoders to detect subtle differences or patterns in the signals to accurately assess the bridge’s reliability.

With the purpose of establishing a baseline model, the autoencoder was exclusively trained with images obtained by a healthy or initial condition. It is essential to investigate the adequate number of hidden neurons in the autoencoder to minimize computational costs while maintaining high performance. To perform this, we conducted research, spanning from 10 to 100 neurons, incremented by 10. The obtained results indicated that using 30 neurons yields high performance and a low error rate, suggesting that the reconstructed images closely match the input images generated from FD values obtained by a healthy structure.

Table 1. Estimated mean values obtained for loss function and reconstruction error for different number of neurons.

No. Neurons	Loss Value	Estimated Error
100	1.8456 × 10−4	1.7902 × 10−4
90	2.3020 × 10−4	1.9357 × 10−4
80	3.9727 × 10−4	3.2216 × 10−4
70	4.2435 × 10−4	4.0953 × 10−4
60	2.2616 × 10−4	2.1101 × 10−4
50	2.6931 × 10−4	2.5435 × 10−4
40	2.6117 × 10−4	2.3685 × 10−4
30	1.6913 × 10−4	1.5919 × 10−4
20	2.2099 × 10−4	2.0457 × 10−4
10	2.4064 × 10−4	2.0065 × 10−4

summarizes the mean values obtained for the loss function (formulated as the mean squared error combined with a regularization term) during the training phase and the reconstruction error (calculated as the mean squared error between the input data and the reconstructed output) during the testing phase of the autoencoder. It highlights that using 30 neurons results in the lowest loss and error rates (see Figure 9). These results are derived from training with 70 images and testing with 30 images. Additionally, five-fold cross-validation was employed for mitigating a bias and assessing the methodology’s efficacy [78].

Once the most appropriate number of neurons for the autoencoder method used in evaluating the bridge condition was identified, it helped to establish a baseline model for determining whether the truss bridge exhibits a damage. In this context, a DI is defined based on the extent of deviations between predicted and actual structural responses, where higher DI values suggest potential damage or deterioration. Mathematically, it is defined as follows:

$$DI={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2$$

(21)

where n is the total number of values or observations, and ŷ_i and y_i denote the predicted values and the actual observed values, respectively. A lower DI value implies a better fit of the predictive model to the observed data, suggesting minimal discrepancies and, consequently, less structural damage.

In structural applications, the DI can be employed to identify changes in the vibrational responses of civil structures over time, providing insights into potential damage progression. By establishing baseline DI values during initial assessments, engineers can monitor deviations that may signal structural deterioration or failure. Therefore, DI serves not only as a measure of model performance but also as a vital tool for assessing the overall health of structures. A baseline or threshold is established by comparing DI values obtained from a healthy bridge with those from a slightly damaged one (i.e., incipient damage condition).

Figure 10 displays a box plot of the calculated DI values for both the healthy bridge condition, denoted by the letter H, and for incipient, moderate, and severe levels of corrosion in each bay, represented as Bi (where I = 1, …, 9 corresponds to the bay number). The obtained results in Figure 10 demonstrate that the DI effectively identifies the bridge’s condition across all three levels of corrosion, particularly the last two levels of damage, regardless of the damage location. However, it is important to note that the DI values for bays B1 to B6 are higher than those for bays B7 to B9 due to their proximity to the sensor. The farther bays show a decrease in DI values, particularly for incipient damage, while the DI values remain higher for moderate and severe damage, indicating that these more severe conditions are detected with greater accuracy. Despite this variation, the DI, estimated through fractality and the autoencoder, can successfully identify the presence of initial or incipient damage. Based on the results obtained, a threshold value, represented by a dotted blue line in Figure 10, was established according to the maximum DI value obtained for the undamaged structure in order to determine the bridge condition automatically. A DI value greater than 1.788 × 10⁻⁴ is established as an indicative of damage, while a DI value less than or equal to 1.788 × 10⁻⁴ is established as a healthy condition.

Figure 11 shows a box plot diagram with the data distribution of the calculated DI values for the three levels of corrosion and healthy bridge investigated in this study. It should be noted that the estimated DI values from all nine bays for each damage level were combined to demonstrate the method’s capability for determining the truss structure condition regardless of the corrosion location. Along with this figure, the computed DI values for a healthy bridge and the moderate and severe corrosion levels do not overlap, implying that the proposal can recognize these two levels of corrosion irrespective of its location. However, the incipient corrosion condition shows a slight overlap. This overlap does not impede the identification of this condition, as it is shown in the confusion matrix (

Table 2. Obtained results for determining the truss condition based on the calculated DI values and threshold selected.

Truss Structure Condition		Predicted		Effectiveness (%)
		Healthy	Corrosion
Actual	Healthy	30	0	100
	Corrosion	10	2690	99.6
	Total accuracy (%)			99.8

). Nevertheless, is noteworthy, as it may result from minimal changes in the structure’s dynamics since only one element is damaged, and it can be far away from sensor location when the vibration responses are measured.

Table 2 exhibits a confusion matrix with the estimated classification results. It can be observed that the proposed DI accurately evaluates the bridge condition, achieving a 99.8% accuracy for distinguishing between a healthy truss-type bridge and one with corrosion damage. It is important to note that 30 tests were considered for the healthy condition (as 70 tests were used to train the autoencoder) and 900 tests for each damage condition, resulting in 2700 (incipient, moderate, and severe corrosion).

6. Discussion

Although that the results of this research were obtained from a lab model under controlled conditions (without load or environmental variations), it is significant to point out that the method proposed in this work is promissory and could be a useful SHM tool for evaluating the structural condition of civil constructions such as bridges for three main reasons: (a) Only one sensor is required to determine the structure global condition regardless of the defect level and location, which is an important advantage and even more considering that other methodologies need multiple sensors to detect damage, and in some cases, the instrumentation is installed close to the damage region, which would not be helpful for real cases [79,80]. (b) The complexity as well as the computational load are low, due to the fact that neither pre-processing nor domain transformation are required for the measured signals, whereas other methods need to carry out diverse signals’ pre-processing and obtain non-linear features to define the integrity condition of a structure [70,77,81,82,83]. (c) Considering the previous advantages, the method proposed in this article could be implemented on site to assess the structural state of bridges in real time. Therefore, in order to avoid tragic collapses of bridges, this methodology could evaluate the health condition of civil structures permanently, and once a damage is detected, more detailed analyses will have to be performed to define the damage location and establish/execute appropriate maintenance plans.

In future works, the operational variations as well as acquisition time window, which can generate uncertainties that affect the methodology performance, will be modeled and considered for the analyses; in this way, the proposed methodology can help avoid tragic collapses of bridges by determining the integrity condition of real-life civil structures permanently, and once a damage is detected, more detailed analyses will have to be performed to define the damage location and establish/execute appropriate maintenance plans. In addition, with the aim of optimizing the computational resources of the proposed methodology and gaining a clear understanding of the contribution of each fractal method and each vibration axis to the proposal, statistical tests such as ANOVA or Kruskal–Wallis, will be applied to rank these indicators according to their discrimination ability between classes (e.g., healthy condition and damage condition) as well as some metaheuristic optimization algorithms for the optimal selection of the number and type of indicators.

7. Conclusions

Corrosion negatively impacts the functionality of civil structures. This paper investigates the integration of fractal dimension with autoencoders for determining the health condition of a tridimensional truss-type structure of nine bays configurated as a bridge exposed to forced excitations. The proposed DI revealed the model’s difficulty in reconstructing images generated from FD values obtained from a damaged structure, allowing an effective assessment of the bridge’s condition.

The estimated results prove that integrating FD and autoencoders produces a DI capable of accurately detecting a structural damage, regardless of its location within the bridge. This indicates the model’s proficiency in identifying reliable features from vibration signals, achieving a 99.8% accuracy in determining the bridge condition. Consequently, the effectiveness of the proposed method in corrosion detection, compared to the healthy condition, is reliable and outperforms existing methodologies. Notably, unlike other approaches, the proposal utilizes a single sensor for damage detection, eliminating the need for pre-processing vibration signals to extract features. These attributes indicate that it can be a tool for real-time SHM systems, offering simplicity in classifier usage and feature extraction. Furthermore, future evaluations will explore the application of these FD algorithms integrated with autoencoders or other deep learning methods for detecting other types of damage (e.g., loosened bolts, cracks, etc.) as well as varying the quantity of sensors employed to avoid a sensitivity loss of extracted features across various civil infrastructures (e.g., towers, houses, buildings, etc.) in order to determine the most reliable FD-based features for different damage scenarios.