Composite encoder–decoder network for rapid bridge damage assessment using long-term monitoring acceleration data

Abstract

In recent years, the deployment of structural health monitoring (SHM) systems has become paramount for safeguarding critical infrastructures. Notwithstanding, the development of an unsupervised deep learning framework capable of learning from long-term sensor data remains a critical challenge, particularly in accurately assessing the exact damage location. This study addresses this gap by proposing a novel approach for rapid bridge damage assessment. The proposed method employed a deep overcomplete encoder–decoder network (DOEDN) to reconstruct the acceleration data acquired from each sensor node on the bridge. The reconstruction losses generated by the DOEDN framework are then used as damage-sensitive features. Additionally, a damage indicator based on Gaussian processes is introduced to assess the damage location and evaluate its severity. The performance and sensitivity of the proposed DOEDN framework are evaluated through long-term monitoring acceleration data from a numerical highway bridge model and the well-known full-scale Z24 bridge. Furthermore, comparative assessments against the regular deep undercomplete encoder–decoder network are conducted using metrics including mean absolute error, coefficient of determination (R²), and mean intersection over union. The results show that the proposed DOEDN framework can reasonably assess the damage location and evaluate its severity across various structural scenarios in the bridge, even in the presence of temperature variations, thus providing a practical and effective solution for bridge health monitoring.

Keywords

Structural health monitoring deep learning encoder–decoder network bridge damage assessment long-term monitoring data

Introduction

Rapid damage assessment of highway bridge structural systems is carried out within hours of an incident, providing a quick evaluation of the potential hazard to critical infrastructure. This assessment aids decision-making and helps to take actions that protect the service life of bridges. Over the past few decades, researchers and engineers have paid increasing attention to the implementation of structural health monitoring (SHM) systems in critical infrastructure. As a result, SHM systems have evolved from visual inspection to smart sensor technologies and data acquisition systems.¹

Recent advancements in smart sensor technologies have enabled the acquisition of large volumes of data for assessing structural health conditions. These innovations include contact sensors,² wireless sensors,³ and vision-based remote monitoring systems.⁴ Afterward, signal-processing techniques are used to analyze the data acquired from different sensors to extract relevant features.⁵

During the past few years, the study of SHM has significantly emphasized the development of vibration-based monitoring techniques. These techniques have been used to monitor internal and invisible structural damage.⁶ Long-term structural health condition assessments rely on vibration responses acquired from sensor arrays installed on the structure under varying operational and environmental conditions. However, such operational and environmental conditions can sometimes cause changes in the structural response similar to those caused by damage, leading to false alarms.⁷

Over the last decade, machine learning, a subfield of artificial intelligence, has revealed data-driven SHM as an effective method to leverage smart sensor technologies and provide a rapid damage assessment system. This method emphasizes changes in data patterns without using structural finite element models. Data-driven SHM also utilizes either supervised or unsupervised learning algorithms to train a statistical model for evaluating the health condition of the bridge based on relevant damage-sensitive features. In the case of supervised learning, the training data comprises labeled instances of undamaged and damaged structural conditions. Consequently, supervised learning algorithms can provide details on the existence, location, type, and extent of the damage. Supervised SHM methods, such as support vector machine (SVM),^8,9 decision trees,¹⁰ neural network (NN),^11,12 recurrent neural network (RNN),¹³ and convolutional neural network (CNN),^14,15 are commonly used for data-driven SHM. These methods can also be applied to vision-based SHM, as training data are easily accessible and transfer learning is achievable.^16–19 Nevertheless, obtaining labeled data in practice can be challenging, expensive, and sometimes unfeasible. One solution is relying on a physics-based model, which may not be sufficient for complex structure damage assessment. Alternatively, labeled data can be obtained through laboratory or field experiments, which can also pose difficulties for most structures. Unsupervised learning algorithms differ from supervised learning as they do not require labeled data and instead learn from normal condition data. Various unsupervised SHM methods have been developed, such as self-organizing neural networks,²⁰ time-series models,²¹ wavelet neural networks,²² kernel-based methods⁸, Boltzmann machines,²³ synchrosqueezed wavelet transforms,²⁴ clustering methods,^25,26 time-series neural networks,²⁷ artificial neural network (ANN,)²⁸ principal component analysis (PCA),²⁹ density-based methods,³⁰ and autoencoder neural networks.^6,31,32

In recent years, there has been significant development in data-driven methods based on deep learning. The deep learning method used in the SHM field is being diversely studied based on the rapid progress in computer technology and the increased amount of monitoring data. For example, the authors developed a semi-supervised deep neural network using a one-class convolutional neural network (CNN) for damage assessment.³³ This network used bridge response data collected from a simply supported bridge subjected to a moving truck. Despite the impressive performance of the network with acceleration data, it neglected the effects of temperature on the bridge. Giglioni et al.³⁴ presented an unsupervised deep neural network for practical bridge assessment. The network utilized an encoder–decoder network to extract the damage-sensitive features from the acceleration data, along with two quantitative damage indicators (DIs). Although the network demonstrated computational efficiency and effectiveness with a limited number of sensors, it faced challenges in distinguishing the effects of single damage scenarios when the bridge experienced progressive damage. Soleimani-Babakamali et al.³⁵ proposed a method that incorporates a generative adversarial network with long short-term memory (LSTM) units, a variant of RNN, and a fast Fourier transform of raw acceleration data. This method was applied to various sensor configurations for SHM. Their findings indicated that the LSTM model outperformed the fully connected (Dense) and CNN models. Jiang et al.³⁶ introduced a decentralized end-to-end unsupervised deep neural network for detecting and locating structural damage. The network involves a deep encoder–decoder network and a normalized quantitative DI. Subsequently, they used a deep undercomplete encoder–decoder network (DUEDN) to capture hierarchical damage-sensitive features from the acceleration data and a deep overcomplete encoder–decoder network (DOEDN) to obtain the reconstruction loss of the raw data. The results indicated that the DOEDN performed better than the DUEDN regarding reconstruction loss. Although the proposed model did not rely on labeled data for training, it compromised the accuracy of damage location to some extent. Moreover, the authors presented a two-stage CNN–LSTM model that uses vibration data to assess bridge damages while considering the effects of temperatures.³⁷ The performance of the model is evaluated using simulated data generated from a highway bridge model and measured data obtained from the Z24 bridge. Although the model shows excellent prediction results for bridge damage assessment, it is trained in a supervised learning mode, limiting its practical application. Sarwar and Cantero³⁸ developed an unsupervised deep learning method for damage detection in bridges. Their method involves using a composite encoder–decoder algorithm with multiple convolutional and LSTM layers, which are trained on acceleration time series data obtained from a fleet of vehicles crossing over the bridge. The numerical results obtained from their study demonstrate the effectiveness of the proposed method in detecting damage under operational conditions. However, the proposed method has practical limitations in terms of synchronization and the collection of signals from different vehicles traversing the bridge. In a recent study, Eltouny and Liang³⁹ introduced a composite encoder–decoder network based on a CNN–LSTM architecture that operates as a DUEDN. This model is designed for the detection and localization of damages in large-scale structures within time-dependent grid settings. Nevertheless, the performance of the method significantly decreases as the number of sensors is reduced.

Although there have been significant advancements in deep learning-based SHM, there is still a lack of an unsupervised deep learning framework specifically designed to learn data acquired from each sensor node installed on a bridge structure under various operational (e.g., vehicle speed, road profile, and measurement noise) and environmental (e.g., temperature, wind, and earthquake) conditions. Furthermore, most deep unsupervised learning methods in the literature face challenges, especially in the accurate assessment of exact damage locations. To address these limitations, we propose an unsupervised method for diagnosing bridge damage utilizing long-term monitoring acceleration data. Our proposed method stands out in several key aspects compared to existing studies. First, although Jiang et al.³⁶ introduced an end-to-end unsupervised deep neural network, our method addresses the specific need for an unsupervised framework tailored to learn data from individual sensor nodes installed on a bridge. This is crucial for accommodating diverse operational and environmental conditions. Furthermore, the use of a DOEDN in our approach combines the strengths of a CNN for spatial feature extraction and a LSTM network for temporal feature extraction, which sets it apart from the composite encoder–decoder network proposed by Eltouny and Liang.³⁹ Our framework’s ability to independently preprocess short acceleration sequences from each sensor node allows it to be uniquely fine-tuned to the characteristics of each sensor, offering a highly adaptable and sensor-specific approach. These features collectively contribute to a more efficient, cost-effective, and rapid bridge damage assessment process while significantly minimizing sensor installation costs. The performance and sensitivity of our proposed method are verified using a numerical highway bridge model while considering the effects of temperatures and measurement noises and the well-known full-scale Z24 bridge. In summary, our proposed method provides a novel solution to address the shortcomings identified in existing studies, offering a tailored and unsupervised deep learning framework for bridge health monitoring that is highly sensitive to sensor-specific data and various operational and environmental factors.

The paper is organized as follows: Section “Proposed unsupervised damage diagnosis method” introduces the proposed unsupervised damage diagnosis method. Section “Numerical study” presents the numerical study and evaluates the performance of the proposed method using simulation datasets. Section “Experimental study” describes the experimental study and assesses the performance of the proposed method using the field experiment datasets. Section “Discussion” provides a discussion of the practical limitations associated with the proposed unsupervised damage diagnosis method. Finally, Section “Conclusion” summarizes the conclusions drawn from both the numerical and experimental studies.

Proposed unsupervised damage diagnosis method

This section presents a novel unsupervised method for diagnosing the damage locations in bridges and evaluating their structural health conditions. The method employs a deep overcomplete encoder–decoder network (DOEDN), which utilizes only the acceleration data acquired from each sensor node installed on the bridge. DOEDN is a composite encoder–decoder network consisting of CNN and LSTM architectures that can learn and extract complex spatial features (known as damage-sensitive features) from the raw acceleration data as well as effective temporal features (such as long-term dependencies) among spatial features. The proposed method focuses on the behavior of each sensor node and utilizes the effective spatiotemporal features from the reconstructed outputs for bridge damage assessment. Figure 1 shows the main overview of the proposed damage assessment method. As shown in Figure 1, only the data acquired from sensor S1 is used for the damage diagnosis. The input signals, represented in varying shades of blue, refer to the initial data provided to the composite encoder–decoder network, whereas the reconstructed outputs, depicted in shades of red, represent the data generated by the network. The composite encoder–decoder network is developed using a deep overcomplete architecture, enabling it to efficiently learn and reconstruct complex features from high-dimensional data. The architecture and hyperparameters of the network are optimized, and the mean absolute error (MAE) between the input and the reconstructed output responses is used as the reconstruction error to define the statistical baseline for damage assessment purposes. It is essential to emphasize that the reconstruction error is primarily attributed to the network’s error and does not directly relate to any physical or structural damage. Additionally, a DI based on Gaussian processes is proposed to assess the damage location and evaluate its severity. The same damage diagnosis procedure can be extended to data from sensors S_{n − 1} and S_n. The following subsections discuss the encoder–decoder network, the deep encoder–decoder network, the network architecture for the composite encoder–decoder network, and the DI based on reconstruction error.

Encoder–decoder network and deep encoder–decoder network

The encoder–decoder network is an unsupervised neural network that takes the same values from the input and output layers for dimensionality reduction and feature extraction. It is a regular feedforward neural network with two modules: an encoder and a decoder. As shown in Figure 2, the traditional encoder–decoder network, also known as the shallow model, includes an input layer, a hidden layer, and an output layer. Notably, there exist two fundamental variants of encoder–decoder networks: undercomplete and overcomplete. An undercomplete encoder–decoder network remains a fundamental architecture in the realm of neural networks. Specifically, it incorporates fewer hidden units in the bottleneck layer when compared to both the input and output layers, as illustrated in Figure 2(a). This design encourages the network to learn a more compact representation of the input data. However, due to their constrained representational capacity, this model may struggle with capturing complex, high-dimensional patterns in the data, potentially leading to information loss.⁴⁰ Moreover, it can be more sensitive to noise and variations in the input data, making it less robust in certain scenarios. Conversely, an overcomplete encoder–decoder network stands as an innovative architectural approach in the realm of neural networks. It has more hidden units in the bottleneck layer than in the input and output layers, as depicted in Figure 2(b). This allows the network to potentially learn more complex representations of the data.⁴¹ This can be particularly advantageous in scenarios where the data exhibits complex, multidimensional patterns that may be missed by a more constrained undercomplete network. While overcomplete networks may be more susceptible to overfitting in certain contexts, their ability to glean intricate details from data positions them as a powerful tool for tasks that demand a comprehensive understanding of underlying structures and features. This capacity to extract nuanced information renders the overcomplete encoder–decoder network a valuable asset in a wide array of applications, including image processing, signal analysis, and anomaly detection. Furthermore, a recent study conducted by Jiang et al.³⁶ has demonstrated that an overcomplete encoder–decoder network is capable of extracting a more effective reconstruction error from the signal when compared to an undercomplete encoder–decoder network.

The hidden layer code $(h_{i})$ and the reconstructed outputs $(y_{j})$ for any of the encoder–decoder networks are, respectively, defined as follows:

h = f (x) = Φ (Wx + b)

(1)

y = g (x) = Φ' (W' x + b')

(2)

where W, $W'$ and b, $b'$ denote the weight matrices and bias vectors for the encoder and decoder modules; x denotes the input; $Φ$ and $Φ'$ denote the activation functions of the encoder and decoder, which are usually the nonlinear activation functions sigmoid, rectified linear unit (ReLU), or hyperbolic tangent (tanh).

Compared to a traditional encoder–decoder network, a deep encoder–decoder network incorporates multiple hidden layers, as illustrated in Figure 3. This configuration allows the deep encoder–decoder network to estimate the nonlinear correlation between data and provide high feature extraction capability.⁶

In this study, a two-symmetric DOEDN shown in Figure 3(b) is used as the training unit. For the deep encoder–decoder network, the loss function between the input and its reconstructed output can be formulated as follows:

L (W, W', b, b') = \arg min_{W, W', b, b'} \frac{1}{m} \frac{1}{n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} | y_{i} - x_{i} |

(3)

where m and n denote the input layer’s size and the total number of weight matrices; $x_{i}$ and $y_{i}$ denote the ith inputs and outputs, respectively.

The deep encoder–decoder network is trained by adjusting the tuning parameters to minimize the loss function. To achieve this, a backpropagation algorithm is used along with a stochastic gradient descent (SGD) optimizer. The SGD optimizer updates the tuning parameters and can be expressed as follows:

w (t + 1) = w (t) - η \frac{\partial L}{\partial w (t)}

(4)

where w is the tuning parameters (including weights and biases), t is the index number of iterations, $η$ is the learning rate, and L is the employed loss function.

Network architecture for DOEDN

A DOEDN that combines CNNs and LSTM networks is designed for bridge damage assessment. This choice of architecture aligns with the broader trend in the literature, where CNN–LSTM hybrid networks have been widely adopted among other networks for learning and extracting spatiotemporal information characteristics from the signal. The proposed method utilizes these functions to learn the feature representation of bridge acceleration data in an unsupervised learning mode by focusing on the behavior of each sensor node, which can further be used for accurate damage assessment. Figure 4 depicts the network architecture for DOEDN used in this study. As depicted in Figure 4, the proposed DOEDN architecture comprises two CNN spatial modules and an LSTM temporal module. More details about the different modules are discussed in the following subsections.

CNN for spatial encoder–decoder

In the context of time-series data processed with 1D CNNs, spatial features refer to distinctive patterns or structures that emerge over different time intervals. These features represent meaningful information within the time-series data, providing insights into the underlying phenomena being observed. For instance, in SHM, spatial features might correspond to specific patterns of vibration that indicate particular structural behaviors or conditions. The term “spatial” here does not refer to physical space but rather to a conceptual space of patterns or structures within the time-series data. The use of 1D CNNs is particularly effective in detecting these spatial features, as they excel at identifying local patterns or motifs within the data, especially when patterns of interest occur over varying time scales.

A typical 1D CNN comprises three primary layers: a convolutional layer, an activation function, and a pooling layer. The convolutional layers use convolutional operations to learn filters from the input data. The filters are learnable weights that collect information from various spatial locations in the input. Nonlinear activation functions such as sigmoid, tanh, and ReLU are applied to each layer.⁴² The pooling layers are commonly used to reduce the feature map size, improve computational efficiency, and maintain important information. These layers perform average and maximum operations. More details regarding CNNs can be found in Abdeljaber et al.¹⁴

In the proposed framework, CNNs operate on the spatial encoder and decoder modules. Each module has a series of predefined building blocks with variable parameters. The spatial encoder module comprises a series of convolutional layers, ReLU activation functions, and max-pooling layers that enable the network to extract and reduce the dimensionality of different features from the inputs. On the other hand, the spatial decoder module is similar to the encoder module, except that it begins with an upsampling layer instead of ending with a max-pooling layer. Although the CNN network has high memory consumption, it cannot produce smooth and interpretable latent representations or bottleneck features that can be applied to reconstruct the input data. Hence, temporal features are critical for effective latent representation.

LSTM for temporal encoder

Traditional RNNs are designed for modeling sequential or time-series data by learning the temporal dependencies of features. Unlike CNNs, RNNs take the output of the previous computation as input to the next computation and are therefore distinguished by their memory. However, training RNNs can be difficult due to gradient backpropagation vanishing problems, which limit their ability to capture long-term dependencies in time-series data.⁴³ To address these limitations, LSTM has been proposed as a variant of RNNs. The LSTM network involves several hidden layers, which include LSTM layers and fully connected layers, along with the input and output layers.⁴⁴ In a standard LSTM layer, there are three different gates: forget gates, input gates, and output gates. The forget gate is responsible for removing information from the LSTM cell state. The input gate is responsible for what information should be added to the LSTM cell. The output gate is responsible for selecting important information from the current LSTM cell and showing it as output. More details on LSTMs can be found in Hochreiter and Schmidhuber.⁴⁴

In the proposed framework, LSTMs operate on the temporal encoder module. The temporal encoder module consists of multiple LSTM layers followed by ReLu activation functions. It is worth noting that the last convolutional layer of the spatial encoder module is fed into the LSTM layers to extract temporal features, including long-term dependencies among spatial features.

Composite CNN–LSTM for spatiotemporal encoder–decoder

The proposed composite networks are designed based on the CNN spatial encoder–decoder and LSTM temporal encoder modules. These two networks have superior representation capabilities as they can input the bridge acceleration data acquired from each sensor node and output the reconstructed acceleration data with similar spatial and channel sizes as the input. The input data are first passed through the spatial encoder during the learning process for feature extraction and dimensionality reduction. The new spatial feature is then fed into the temporal encoder to capture the temporal features. Subsequently, the last layer of the temporal encoder is flattened, connected to a fully connected layer, and subjected to a ReLu activation function to obtain a smooth bottleneck feature. The bottleneck feature is representative of outputs for the temporal encoder. Moreover, the temporal encoder output passes through the spatial decoder, resulting in the output of the spatial decoder. Note that the last convolutional layer of the spatial decoder is fed to the linear activation function to reconstruct the input data.

Damage indicator

The reconstruction error generated by the proposed DOEDN framework can serve as an efficient damage-sensitive feature for damage assessment purposes. The advantage of using MAE as the damage-sensitive feature lies in its robustness and interpretability. Unlike other metrics, MAE provides a direct and intuitive measure of the absolute differences between the reconstructed output responses and the initial input responses. This makes it particularly effective in quantifying the accuracy of damage assessment. Additionally, MAE is less sensitive to outliers compared to other metrics like mean squared error, which can be crucial in scenarios where extreme values may distort the assessment results. Overall, MAE offers a straightforward and reliable means to evaluate damage, making it a preferred choice in many applications. MAE is calculated by determining the total deviation between the reconstructed output responses and the initial input responses for each dataset using Equation (5).

MAE = \frac{1}{n} \sum_{i = 1}^{n} | y (t_{i}) - x (t_{i}) |

(5)

where $y (t_{i})$ and $x (t_{i})$ denote the reconstructed output responses and the initial input responses at a sample i for a total of n samples.

Note that the training dataset used for training the proposed DOEDN framework only includes data from the undamaged bridge condition. Therefore, the MAE distribution between the reconstructed output and initial input responses is expected to remain low for the undamaged bridge condition. In contrast, the unknown testing dataset associated with both the undamaged (baseline) and damaged bridge conditions is used to test the trained framework. As a result, the MAE distribution between the reconstructed output and initial input responses is expected to vary considerably when the bridge is damaged, leading to an effective damage-sensitive feature. This results in a stochastic distribution characterized by a normal distribution pattern, enabling the differentiation between undamaged and damaged bridge conditions. This study employs a Gaussian process-based z-score to generate a normal distribution of the MAE distribution with mean $μ$ and standard deviation $σ$ values.³²

MAE ~ N (μ, σ^{2})

(6)

A DI is introduced to quantify the inconsistency between various bridge conditions and sensor locations. The DI is evaluated by using the sum of the distances of the MAE distributions to the mean measured in standard deviations, expressed in Equation (7). This is shown in more detail through numerical simulation and field experiment results in sections “Damage assessment (Numerical study)” and “Damage assessment (Experimental study),” respectively.

D I_{i} = \frac{MA E_{i} - μ_{undamaged}}{σ_{undamaged}}

(7)

where $D I_{i}$ and $MA E_{i}$ are the DI and MAE for ith bridge condition corresponding to each sensor node. $μ_{undamaged}$ and $σ_{undamaged}$ are respectively the estimated mean and standard deviation of the MAE distributions of the undamaged bridge condition.

In this study, average DI values are found to be a superior metric for bridge damage assessment compared to individual DI values. First, they provide a more stable and reliable measure of structural health, reducing the impact of outliers or anomalies from single sensors. Additionally, leveraging collective information from each sensor distributed across the bridge offers a more comprehensive evaluation of its condition, enhancing the accuracy and reliability of damage assessments and ensuring more informed decisions for maintenance and repairs.

The process of identifying damage locations relies on the analysis of indicators that deviate from the baseline condition. A critical DI is set as a threshold, beyond which an observation is considered damaged. This threshold is often determined using statistical measures, with the 95th percentile being a common choice. The 95th percentile represents the value below which 95% of the data points fall, making it a robust measure for identifying outliers or anomalies associated with damage. When an observation exceeds this threshold, it indicates potential damage, prompting further investigation. This approach ensures accurate and reliable detection of damage locations in the bridge.

Numerical study

This section presents the vehicle-bridge coupled system utilized to generate simulation datasets for training and testing the proposed DOEDN framework. It also examines the effect of temperature on the bridge, discusses the long-term health monitoring data and outlines the configuration of the DOEDN framework. Furthermore, this section assesses the performance of the DOEDN framework for bridge health condition assessment using the simulation datasets.

Finite element model

The vehicle-bridge coupled system designed to generate the simulation datasets is shown in Figure 5. The finite element method is employed to model the bridge as a simply supported Euler–Bernoulli beam with three translational degrees of freedom. The equation of motion of the beam under a moving truck is expressed as:

M_{b} {\overset{\cdot\cdot}{X}}_{b} + C_{b} {\overset{\cdot}{X}}_{b} + K_{b} X_{b} = F_{b}

(8)

where $M_{b}$ , $C_{b}$ , and $K_{b}$ are, respectively, the global mass, damping, and stiffness matrices; ${\overset{\cdot\cdot}{X}}_{b}$ , ${\overset{\cdot}{X}}_{b}$ , and $X_{b}$ are, respectively, the acceleration, velocity, and displacement vector of the beam model; $F_{b}$ is the force vector acting on the bridge. The model accounts for damping by integrating Rayleigh damping,⁴⁵ as expressed in Equation (9):

C_{b} = α_{0} M_{b} + β_{0} K_{b}

(9)

where the two parameters $α_{0}$ and $β_{0}$ are constants and described as follows:

α_{0} = 2 ξ_{1} ω_{01} - β_{0} ω_{01} ω_{01}; β_{0} = \frac{2 (ξ_{2} ω_{02} - ξ_{1} ω_{01})}{ω_{02} ω_{02} - ω_{01} ω_{01}}

(10)

where, $ω_{01}$ and $ω_{02}$ denote the first two natural frequencies of the bridge; $ξ_{1}$ and $ξ_{2}$ denote the first two modal damping ratios of the bridge.

The vehicle system is modeled as a two-axle truck illustrated in Figure 5. This truck has four degrees of freedom analogous to the vehicle system used by Yessoufou and Zhu.³⁷ It consists of several parameters, as illustrated in Figure 5. $m_{2}$ and J represent the truck body mass and pitch moment of inertia; $m_{3}$ and $m_{1}$ represent the front and rear axle mass; $k_{2}$ , $c_{2}$ and $k_{3}$ , $c_{3}$ represent the linear suspension stiffness and viscous damping of the front and rear axles, respectively; $k_{1}$ , $c_{1}$ and $k_{4}$ , $c_{4}$ represent the linear tire stiffness and viscous damping, respectively; $l_{2}$ and $l_{1}$ represent the distances of the axles to the gravity center. The equation of motion of a four-degrees-of-freedom truck model is defined as:

M_{v} {\overset{\cdot\cdot}{Y}}_{v} + C_{v} {\overset{\cdot}{Y}}_{v} + K_{v} Y_{v} = F_{v}

(11)

where $M_{v}$ , $C_{v}$ , and $K_{v}$ are, respectively, the mass, damping, and stiffness matrices; ${\overset{\cdot\cdot}{Y}}_{v}$ , ${\overset{\cdot}{Y}}_{v}$ , and $Y_{v}$ are respectively the acceleration, velocity, and displacement vector. Further details of the variables can be found in Yessoufou and Zhu.³⁷ Note that the truck model in this study is implemented in MATLAB (version 2019a; MathWorks, Natick, MA, USA).

Based on the condition of displacement and interaction force between the truck and bridge, the dynamic equations of the truck and bridge are coupled at the contact points.⁴⁶ The coupled equation between the truck and bridge can be represented as follows:

F_{j} (t) = k_{j} Δ_{j} (t) + c_{j} {\overset{\cdot}{Δ}}_{j} (t)

(12)

where, subscript j is the jth wheel, $Δ_{j} (t)$ is the vertical displacement of the jth rear wheel relative to the bridge at the time t, as described in Equation (13):

Δ_{j} (t) = y_{j} (t) - z_{j} (t) - r_{j}

(13)

where $y_{j} (t)$ denotes the vertical displacement of the jth wheel at the time t; $z_{j} (t)$ denote the vertical displacement of the bridge in the position where the jth wheel contacts the bridge at the time t; and $r_{j}$ denote the road profile where the jth wheel contacts the bridge.

Furthermore, the Newmark-β algorithm⁴⁶ is implemented in MATLAB to solve the equation of the coupled system. More details of the vehicle-bridge interaction system can be found in Zhu and Yi⁴⁶ and Feng and Feng.⁴⁷

Bridge and truck properties

A concrete highway girder bridge characterized by a T cross-section is adopted as a numerical example. Figure 6 shows a visual representation of the bridge. The material properties of the bridge are the total span of the bridge $L = 40 m$ , Young’s modulus $E = 3.45 \times 10^{10} N m^{2}$ , linear density $ρ = 2084.37 kg / m$ , and weight $w = 83, 375 kg$ . Only the girder bridge three was modeled in ANSYS (version 15.0; ANSYS, Inc., Canonsburg, PA, USA) with a total of 100 elements, each having a length of 0.4 m. The bridge has 101 nodes, arranged from left to right. The first two modal frequencies of the bridge are 2.27 and 9.05 Hz, respectively, and a modal damping of 2% is assumed for both of these modes.^46,47 For each simulation, a two-axle truck with 20 tons³⁷ is selected to cross the bridge at a constant moving speed of v = 30 km/h. Table 1 displays the parameters of a two-axle truck modeled to cross over a bridge. A class “Good” road profile is also used in all the simulations below with a roughness coefficient $α = 0.62 \times 10^{- 6}$ .⁴⁸ As shown in Figure 5, seven sensor nodes (S1–S7) are distributed along the bridge to capture its vertical acceleration data. The locations of these sensor nodes are illustrated in black in the figure. In addition, the damage is modeled as the stiffness reduction of a single beam element. The red zones in Figure 5 show the damaged elements.

Table 1.

Parameters of a two-axle truck.

Parameter	Value	Parameter	Value
m₁/kg	1500	m₃/kg	1000
m₂/kg	17,500	J/(kg·m⁻¹)	1.47 × 10⁵
k₁/(N·m⁻¹)	4.60 × 10⁶	c₁/(N·m⁻¹/s)	4.30 × 10³
k₂/(N·m⁻¹)	4.23 × 10⁶	c₂/(N·m⁻¹/s)	4.00 × 10⁴
k₃/(N·m⁻¹)	2.47 × 10⁶	c₃/(N·m⁻¹/s)	3.00 × 10⁴
k₄/(N·m⁻¹)	3.47 × 10⁶	c₄/(N·m⁻¹/s)	3.90 × 10³
l₁/m	1.5	l₂/m	2.5

Effect of temperature on the bridge

Bridges are susceptible to temperature variations, impacting their material properties, vibration response, and modal frequencies. These effects can sometimes be mistaken for damage, leading to false alarms.^49,50 There exists a linear correlation between temperature and Young’s modulus,⁵¹ consistent across various reinforced concrete decks.^50,52 Meanwhile, a bi-linear correlation has been observed for reinforced concrete bridges, where temperature changes below 0°C can cause varying degrees of softening and stiffening due to asphalt-wearing surfaces.⁵³ This study employs the bi-linear model developed by Behmanesh and Moaveni⁵⁴ to adjust the temperature-shifted Young’s modulus of the bridge $(E_{m})$ . This model is deemed appropriate to depict real-life scenarios where temperature changes affect the bridge behavior.

E_{m} = Θ E_{0}

(14)

where $Θ$ and $E_{0}$ are the modification factor and the reference Young’s modulus utilized to adjust the temperature-shifted Young’s modulus $E_{m}$ .

Θ = Q + S T_{t} + R (1 - \erf (\frac{T_{t} - k}{τ}))

(15)

where the parameters Q and S represent the linear correlation for temperatures above freezing ( $Q = 1.0129$ and $S = - 0.0048$ ); parameters k and $τ$ represent the transition around the freezing point ( $k = - 1.1012$ and $τ = 3.1466$ ); $T_{t}$ is the temperature in degrees Celsius; erf represents the error function as specified in Equation (16).

\erf (x) = \frac{2}{\sqrt{π}} d t \int_{0}^{x} e^{t^{2}} d t

(16)

In this study, 25-month temperature datasets from the 1st July 2019 to the 31st July 2021, were obtained from the Meteoblue meteorological service⁵⁵ for the city of Basel, Switzerland. For each simulation, a random daily temperature value was chosen from this dataset and employed to adjust the temperature-shifted Young’s modulus for the bridge based on Equation (14). The average daily temperature variation for the city of Basel from July 2019 to July 2021, along with the training and assessment periods, is depicted in Figure 7.

Long-term simulation monitoring data

This section presents a comprehensive dataset obtained from rigorous daily simulations spanning an extensive 25-month period, from July 2019 to July 2021. The data encompass a range of critical factors, including temperature-shifted Young’s modulus and diverse structural scenarios. Notably, each simulation precisely mimicked the real-world scenario of a single two-axle truck crossing the bridge, employing parameters detailed in Table 1. A key distinction was made between an 18-month training phase (July 2019–December 2020) where the bridge remained assumedly undamaged and a subsequent 7-month assessment period. During the assessment phase, damage severity gradually increased by 5% each month, from February to July 2021. This damage was modeled as a stiffness reduction in a single beam element. The study explored four distinct damage scenarios (damage case 1 to damage case 4) at different bridge locations, including one-eighth span, quarter-span, three-eighths span, and midspan. At each location, damage severity was further investigated, ranging from 5% to 30%. This rigorous analysis includes a total of 25 structural scenarios, outlined in Table 2, along with summarized simulation results for each sensor.

Table 2.

Simulated bridge structural scenarios.

Structural scenarios	Duration (months)	Description	Number of datasets
Undamaged case	18	Baseline (D0)	548
Damage case 1 (DC1)	7	Damage at one-eighth span (L/8) with six different damage severities (D1_5%, D1_10%, D1_15%, D1_20%, D1_25%, and D1_30%)	848
Damage case 2 (DC2)	7	Damage at quarter-span (L/4) with six different damage severities (D2_5%, D2_10%, D2_15%, D2_20%, D2_25%, and D2_30%)	848
Damage case 3 (DC3)	7	Damage at three-eighths span (3L/4) with six different damage severities (D3_5%, D3_10%, D3_15%, D3_20%, D3_25%, and D4_30%)	848
Damage case 4 (DC4)	7	Damage at midspan (L/2) with six different damage severities (D4_5%, D4_10%, D4_15%, D4_20%, D4_25%, and D4_30%)	848

The vehicle parameters are assumed to be independent of temperature, and the bi-linear correlation between temperature and Young’s modulus is considered to remain unchanged even under damage conditions. While the vehicle crosses the bridge, acceleration data are recorded from each sensor node. This process yielded 548 data points for the initial 18-month period and 3392 data points for the subsequent 7-month period. This study considered only the acceleration data from sensors S1 to S4 due to the symmetry of the bridge. Specifically, 5.28-second-long acceleration signals are analyzed, resulting in 111 data points. Figure 8 illustrates the acceleration data collected from sensor S4 in both time and frequency domains when a two-axle truck crosses an undamaged bridge.

The raw acceleration data for each sensor is preprocessed using min–max normalization,³⁷ which scales the values between 0 and 1. This preprocessing step improves the generalization of the DOEDN framework and accelerates the training process. Additionally, three distinct levels of white Gaussian noise (1%, 2%, and 3%) were intentionally introduced to the normalized acceleration data.³⁷ This strategic introduction of noise allows the DOEDN framework to effectively learn features from undamaged condition data while also taking into account uncertainties in the data. Following this noise interference operation, the simulation datasets from the training period consisted of 2192 datasets for each sensor. Out of these, 1754 (80%) are randomly chosen for training, and the remaining 438 (20%) are employed for validation. This split operation is performed separately for each sensor data. Conversely, all simulation datasets from the assessment period (totaling 3392 datasets) were designated for testing purposes. The detailed characteristics of the simulation dataset are presented in Table 3. This thorough and strategic dataset serves as the basis for the subsequent analysis and development of the proposed DOEDN framework.

Table 3.

Simulation dataset characteristics.

Datasets	Duration (month)	Frequency (Hz)	Sliding window (s)
Training	18	111	5.28
Testing	7	111	5.28

The training samples for the framework have an input and output shape of [1754, 111, 1], with 1754, 111, and 1 representing the training samples, data points, and features, respectively. The spatial encoder module of the framework has three convolutional layers with 32, 64, and 128 filters, two max-pooling layers, and ReLu activation functions. The kernel size is 10 for all convolution layers. Simultaneously, the temporal encoder module integrates two LSTM layers with 128 LSTM units and fully connected layers, followed by ReLu activation functions. This combination imparts the framework with a heightened capacity to capture complex temporal relationships within the data. The spatial decoder module has an equivalent number of convolutional layers as the spatial encoder, but they are arranged in the opposite order. Instead of a max-pooling layer, an upsampling layer is used, and a linear activation function is applied to the output layer. Table 4 summarizes the architecture of the proposed framework for each sensor.

Table 4.

Architecture of the proposed DOEDN framework for each sensor.

Layer type	Output shape	Kernel size	Activation function
Spatial encoder
Input	(111 × 1)	—	—
Conv1D	(111 × 32)	(1 × 10)	ReLU
MaxPooling1D	(111 × 32)	(1 × 10)	—
Conv1D	(111 × 64)	(1 × 10)	ReLU
MaxPooling1D	(111 × 64)	(1 × 10)	—
Conv1D	(111 × 128)	(1 × 10)	ReLU
Temporal encoder
LSTM	(111 × 128)	—	ReLU
LSTM	(111 × 128)	—	ReLU
Flatten	(14,208)	—
Fully connected	(32)	—	ReLU
Spatial decoder
Fully connected	(14,208)	—	ReLU
Reshape	(111 × 128)	—
Conv1D	(111 × 64)	(1 × 10)	ReLU
Up-Sampling1D	(111 × 64)	(1 × 10)	-
Conv1D	(111 × 32)	(1 × 10)	ReLU
Up-Sampling1D	(111 × 32)	(1 × 10)	-
Output	(111 × 1)	—	Linear

Conv1D: 1-dimensional convolutional layer; ReLU: rectified linear unit; DOEDN: deep overcomplete encoder–decoder network; LSTM: long short-term memory.

The training process of the proposed DOEDN framework is carried out independently for each sensor node (S1, S2, S3, and S4) using the training samples, which included short undamaged acceleration sequences. The training process for each sensor node involved the use of the Adam optimizer,⁵⁶ with a learning rate of 0.001 and a batch size of 32 samples. The total number of epochs is 500, and these parameters are found to be suitable for this study. During training, a backpropagation algorithm is used to adjust the tuning parameters to minimize the loss function $L (W, W', b, b')$ described in Equation (3). The trained DOEDN framework is further used for feature extraction of the unknown testing samples (e.g., undamaged and damaged acceleration data). All training and testing operations are conducted in the Google Colaboratory environment using Keras in conjunction with Python 3.9 (Open Source, Python Software Foundation (PSF), Beaverton, OR, USA). It should be noted that the proposed DOEDN framework for each sensor node required approximately 8 min of training.

Network training performance (Numerical study)

As mentioned in section “Long-term simulation monitoring data,” the performance of the training DOEDN framework is conducted with four different sensor node information scenarios. Each DOEDN framework is trained using the training samples (e.g., undamaged acceleration data) for a single sensor node. The effectiveness of the proposed DOEDN framework (e.g., deep overcomplete encoder–decoder network) is compared to the regular DUEDN framework (e.g., DUEDN). As discussed in section “Encoder–decoder network and deep encoder–decoder network,” the main difference between the two frameworks is that the DOEDN adopts a higher dimensionality of the hidden layer, making it feasible to capture more effective reconstruction losses. It should be noted that the spatial encoder and decoder modules similar to the DOEDN are used for the DUEDN, but the number of filters for the convolutional layers is in the opposite direction. This meant the convolution used 128, 64, and 32 filters instead of 32, 64, and 128. Also, the two LSTM layers in the temporal encoder consist of 32 LSTM units. The training process of the DUEDN is the same as that of the DOEDN. Figures 9 and 10 show the distribution of the reconstruction losses (MAE) of the DOEDN and DUEDN with epochs for sensors S1, S2, S3, and S4. As shown in Figures 9 and 10, the distributions of the training losses match well with the distributions of the validation losses in both the DOEDN and DUEDN. Also, the temperature variations and measurement noise did not significantly affect the distribution of the reconstruction losses. The performance results of the two frameworks on the training samples for sensors S1, S2, S3, and S4 are shown in Table 5.

Table 5.

Performance results of the two frameworks on the training samples.

Sensors	DUEDN		DOEDN
Sensors	Training	Validation	Training	Validation
S1	4.36 × 10⁻³	4.42 × 10⁻³	1.21 × 10⁻³	1.22 × 10⁻³
S2	3.07 × 10⁻³	3.05 × 10⁻³	1.41 × 10⁻³	1.39 × 10⁻³
S3	3.05 × 10⁻³	3.06 × 10⁻³	1.06 × 10⁻³	1.03 × 10⁻³
S4	2.65 × 10⁻³	2.66 × 10⁻³	1.26 × 10⁻³	1.29 × 10⁻³

DUEDN: deep undercomplete encoder–decoder network; DOEDN: deep overcomplete encoder–decoder network.

Network prediction performance (Numerical study)

The prediction performance of the two trained networks on the unknown testing samples (e.g., undamaged and damaged acceleration data) provides an indication for damage assessment. The prediction performances of the two frameworks for each sensor node are evaluated based on three different metrics: MAE, coefficient of determination (R²), and mean intersection over union (IoU). The smaller the MAE, the greater the accuracy of the trained networks. The higher the R², the more accurate the predictions. The higher the mean IoU, the lower the levels of false positives and false negatives, signifying that the network’s predictions (reconstructed output responses) align well with the initial input responses. Table 6 illustrates the performance results of the two frameworks for sensors S1, S2, S3, and S4 in the case of an undamaged bridge. As shown in Table 6, the trained DOEDN performs the best result with the smallest MAE (1.82 × 10⁻³, 1.78 × 10⁻³, 1.62 × 10⁻³, and 1.88 × 10⁻³) and the highest R² (0.9998, 0.9999, 0.9999, and 0.9999) corresponding to the sensors S1, S2, S3, and S4, respectively. Notably, both the trained DOEDN and DUEDN demonstrate an identical mean IoU score of 0.9909 for all sensors. These results demonstrated that the DOEDN might be more accurate for rapid bridge damage assessment. Hence, the following paragraphs discuss the use of the DOEDN for rapid bridge damage assessment.

Table 6.

Performance results of the two frameworks to the undamaged bridge case.

Sensors	DUEDN			DOEDN
Sensors	MAE	R ²	mean IoU	MAE	R ²	mean IoU
S1	4.30 × 10⁻³	0.9996	0.9909	1.82 × 10⁻³	0.9998	0.9909
S2	3.10 × 10⁻³	0.9997	0.9909	1.78 × 10⁻³	0.9999	0.9909
S3	3.00 × 10⁻³	0.9997	0.9909	1.62 × 10⁻³	0.9999	0.9909
S4	3.10 × 10⁻³	0.9997	0.9909	1.88 × 10⁻³	0.9999	0.9909

DUEDN: deep undercomplete encoder–decoder network; DOEDN: deep overcomplete encoder–decoder network; MAE: mean absolute error

Taking advantage of the proposed DOEDN, it is feasible to assess the bridge health conditions by calculating the MAE and R² for different sensors S1, S2, S3, and S4 using unknown testing samples, including undamaged and damaged acceleration data. Table 7 shows the performance results obtained by applying the proposed DOEDN framework to undamaged and damaged bridge cases corresponding to sensors S1, S2, S3, and S4. As shown in Table 7, the trained DOEDN can produce distinctive MAE and R² for sensors S1, S2, S3, and S4 with different bridge health conditions. For sensor S1, the MAE is significantly increased as the bridge health conditions change from the undamaged case (1.82 × 10⁻³) to different damage cases (8.42 × 10⁻³, 9.57 × 10⁻³, 17.9 × 10⁻³, and 24.8 × 10⁻³). On the other hand, the R² degrades as the bridge health conditions change from the undamaged case (0.9998) to different damage cases (0.9943, 0.9772, 0.9455, and 0.9031). Similar results are obtained for most sensors S2, S3, and S4. These results allow for the use of the undamaged acceleration data from the testing samples as a baseline for damage assessment. It is also worth noting that the MAE for each sensor is significantly increased for midspan damage, whereas R² decreases significantly. The MAE is positively correlated with the type of damage condition. Nevertheless, the R² is negatively correlated with the type of damage condition. Thus, the reconstruction losses (MAE) are further used as an evaluation criterion for DIs.

Table 7.

Performance results of the DOEDN framework to undamaged and damaged bridge cases.

Sensors	Undamaged case		Damage case 1		Damage case 2		Damage case 3		Damage case 4
Sensors	MAE	R ²	MAE	R ²	MAE	R ²	MAE	R ²	MAE	R ²
S1	1.82 × 10⁻³	0.9998	8.42 × 10⁻³	0.9943	9.57 × 10⁻³	0.9772	17.9 × 10⁻³	0.9455	24.8 × 10⁻³	0.9031
S2	1.78 × 10⁻³	0.9999	5.06 × 10⁻³	0.9981	8.31 × 10⁻³	0.9791	14.6 × 10⁻³	0.9632	20.9 × 10⁻³	0.9295
S3	1.62 × 10⁻³	0.9999	5.88 × 10⁻³	0.9952	6.99 × 10⁻³	0.9799	17.1 × 10⁻³	0.9433	23.2 × 10⁻³	0.9003
S4	1.88 × 10⁻³	0.9999	1.92 × 10⁻³	0.9998	6.17 × 10⁻³	0.9805	16.2 × 10⁻³	0.9444	22.0 × 10⁻³	0.8999

MAE: mean absolute error; DOEDN: deep overcomplete encoder–decoder network.

Based on the correlation described above, a normal distribution of the MAE is established to evaluate the differences between various bridge conditions. Figure 11 shows the normal distributions of the MAE for all structural scenarios corresponding to sensor S1. As shown in Figure 11, the centers of the normal distributions move to the right of the baseline in a very slight way as the damage severity increases. Also, the normal distributions of the baselines are narrow and high, whereas the normal distributions of various damaged severities (D1_5%–D6_30%) become wide and short (Figure 11). Similar results are obtained for sensors S2, S3, and S4. The statistical parameters, such as the mean and standard deviation of the normal distribution, are then used to compute the DI discussed in section “Damage indicator.”

Damage assessment (Numerical study)

To further evaluate the damage location performance, we calculate the DI for each sensor node using Equation (7). Figure 12 depicts the distributions of the average DI values for sensors S1, S2, S3, and S4 when considering various structural scenarios. D0 represents the baseline condition, indicating that the bridge is in a healthy state with no structural damage detected. D1_5%–D1_30% represent the six different levels of damage severity, ranging from 5% to 30% in damage case 1, with D1_30% indicating a critical level of damage. Similarly, D2_5%–D2_30% refer to the six different levels of damage severity corresponding to damage case 2, with D2_30% indicating a critical level of deterioration. D3_5%–D3_30% represent the six different levels of damage severity corresponding to damage case 3, with D3_30% indicating a critical level of damage. D4_5%–D4_30% indicate the six different levels of damage severity corresponding to the scenario presented in damage case 4, with D4_30% indicating a critical level of deterioration.

As depicted in Figure 12(a), sensor S1, located at one-eighth span, demonstrates remarkable proficiency in detecting damage case 1 at the same location. It can also reliably identify damage cases 2, 3, and 4, located at considerable distances from its position, as shown in Figure 12(b) to (d). This demonstrates a proportional increase in the distribution of DI with higher damage severity levels. For instance, average DI values of 6.85, 7.53, 27.04, and 33.59 are observed for damage severities D1_30%, D2_30%, D3_30%, and D4_30%, respectively. Notably, a critical DI threshold of 0.05 is established, beyond which an observation is classified as damaged, providing a clear criterion for accurate assessment. As illustrated in Figure 12(f), sensor S2, positioned at quarter-span, demonstrates a high level of confidence in detecting damage case 2 at its precise location. It can also accurately identify damage cases 1, 3, and 4, even when they occur at significant distances from its placement, as demonstrated in Figure 12(e) to (h). The DI distribution consistently reveals a positive correlation with increasing damage severity levels. This is evidenced by the average DI values of 6.02, 9.11, 28.46, and 36.53, corresponding to damage severities D1_30%, D2_30%, D3_30%, and D4_30%, respectively. However, that sensor S2 does not detect the lowest damage severity (D1_5%) in damage case 1 (Figure 12(e)). These results emphasize that sensor S2 can accurately detect the location of the damage cases 2, 3, and 4 in the bridge, particularly when the cutoff is set at 0.07. As shown in Figure 12(k) and (l), sensor S3, located at three-eighths span, exhibits a high level of confidence in detecting damage case 3 at its precise location, as well as damage case 4 at midspan. This is supported by the average DI values of 30.31 and 41.44, corresponding to damage severities D3_30% and D4_30%, respectively. Notably, for damage cases 1 and 2, the average DI value becomes evident after D1_10% and D2_10% damage severities, respectively (Figure 12(i) and (j)). Nevertheless, these results emphasize that sensor S3 can reasonably detect the location of damage cases 3 and 4 in the bridge when the cutoff is set at 0.07. As depicted in Figure 12(p), sensor S4 positioned at midspan demonstrates a high level of confidence in detecting damage case 4 at its precise location, with an average DI value of about 43.23 when D4_30% damage severity exists. Note that the sensor S4 failed to detect the damage severity D1_5%, D1_10%, D1_15%, and D1_20% in damage case 1 (Figure 12(m)). Additionally, it encounters some difficulties in capturing the lowest damage severities in damage cases 2 (D2_5%, D2_10%) and 3 (D3_5%), as shown in Figure 12(n) and (o). Results indicate that sensor S4 can effectively detect the location of damage case 4 in the bridge when the cutoff is set at 0.04.

Therefore, these results demonstrate the performance of sensors S1, S2, S3, and S4 in accurately assessing the damage location and evaluating specific structural scenarios in the numerical bridge model. Notably, the proposed DI varies significantly for each sensor node, depending on both the type and location of the specific damage case within the bridge structure.

Experimental study

This section provides specific details of the Z24-bridge used to generate the field experiment datasets for both training and testing purposes of the proposed DOEDN framework. This section also discusses the long-term continuous monitoring data and the configuration of the proposed DOEDN framework. Additionally, the effectiveness of the proposed DOEDN framework to assess progressive bridge damage is discussed.

Details of the Z24 bridge

The Z24 bridge, built between 1961 and 1963, is a classical post-tensioned concrete two-cell box girder with three spans, connecting the villages of Koppigen and Utzenstorf in Switzerland. Located between Bern and Zurich, the bridge traverses the A1 highway, featuring a classical design with a main span of 30 m and two side spans of 14 m (as depicted in Figure 13(a)). The bridge had a skew configuration consisting of concrete columns, girder connections, and piers. More-detailed information about the Z24 bridge can be found in Reynders and Roeck.⁵⁷

Despite being in relatively good condition, the bridge was demolished at the end of 1998 to make way for a new one that required a larger side span. Before a complete demolition, a long-term continuous monitoring test was conducted from November 1997 to September 1998, capturing environmental factors and acceleration data through sensors. The bridge was primarily monitored based on environmental factors. However, starting from August 5, 1998, progressive damage scenarios were inflicted on the bridge over a month to study the impact of damage on its dynamic behavior. These structural scenarios included various types of damage, such as pier settlement, foundation tilt, concrete spalling, abutment landslide, concrete hinge failure, anchor head failure, and tendon rupture. Moreover, the bridge was subjected to excitation by shakers positioned at both the mid and side spans. Sixteen accelerometers were placed at different locations on the bridge to measure acceleration time-series data in different directions, whereas other sensors recorded environmental factors such as air temperature, humidity, rain, wind speed, and wind direction. The accelerometers recorded vibration responses at a frequency of 100 Hz for approximately 10 min every hour, resulting in a total of 65,536 acceleration values per sample. Figure 13(c) and (d) illustrates the specific positions of the 16 accelerometers along the bridge deck and pier. However, during the monitoring period, eight accelerometers malfunctioned, and two additional accelerometers were excluded from the analysis due to data corruption. The sensors utilized in this study are indicated by the red-filled boxes in Figure 13(c), specifically sensors S5, S6, S7, S12, S14, and S16.

Long-term continuous monitoring data

The test program includes a total of 11 different damage scenarios and one undamaged scenario. Table 8 summarizes the structural scenarios on the Z24 bridge and the number of datasets associated with each scenario. To capture essential information and limit the computational efforts of the network, 10-second-long acceleration signals are considered, resulting in 1024 data points. As shown in Table 8, 4800 acceleration data points were collected, with 2688 undamaged, and 2112 damaged samples. Figure 14 presents a representative example of the acceleration data collected from sensor S6, showing both the time and frequency domains. Note that, the data were obtained from the undamaged state of the bridge.

Table 8.

Z24 bridge structural scenarios.

Structural scenarios	Date (1998)	Description	Number of datasets^a
Undamaged case	4 August	Baseline (D0)	2688
Damage case 1(DC1)	10 August12 August17 August18 August	Settlement of pier Koppigen, 20 mm (D1_20)Settlement of pier Koppigen, 40 mm (D1_40)Settlement of pier Koppigen, 80 mm (D3_80)Settlement of pier Koppigen, 95 mm (D4_95)	192192192192
Damage case 2(DC2)	25 August26 August	Spalling of concrete at soffit, 12 m² (D2_12)Spalling of concrete at soffit, 24 m² (D2_24)	192192
Damage case 3(DC3)	2 September3 September	Failure of 2 anchor heads, (D3_2)Failure of 4 anchor heads (D3_4)	192192
Damage case 4(DC4)	7 September8 September9 September	Rupture of 2 tendons (D4_2)Rupture of 4 tendons (D4_4)Rupture of 6 tendons (D4_6)	192192192

MAE: mean absolute error; DOEDN: deep overcomplete encoder–decoder network.

10-second-long acceleration signals are used to evaluate the performance of the DOEDN framework.

The measurement data from each sensor undergoes preprocessing using min-max normalization,³⁷ scaling the values between 0 and 1. Only the measurement samples from the undamaged bridge are used to train the framework. Out of the 2688 measurement samples, the initial 2152 samples are designated as training sets, whereas the remaining 250 and 192 samples are assigned as validation and baseline sets, respectively. On the other hand, all measurement samples from the damaged bridge (2112 samples) are used as testing sets. Field experiment dataset characteristics are summarized in Table 9. The input and output shapes of the framework are arranged in [2152, 1024, 1] for the training sets, where 2152, 1024, and 1 represent the training samples, the data points, and the features, respectively.

Table 9.

Field experiment dataset characteristics.

Datasets	Duration (week)	Frequency (Hz)	Sliding window (s)
Training	1	100	10
Testing	4	100	10

A network architecture similar to that used in the numerical study was adopted for each sensor. The training process was conducted independently for each sensor, including sensors S5, S6, S7, S12, S14, and S16, using the training samples. The training process for each sensor involved the Adam optimizer,⁵⁶ with a learning rate of 0.001 and a batch size of 64 samples. A total number of 100 epochs, along with these optimizer settings, were deemed suitable for this study. Throughout the training, the backpropagation algorithm was utilized to adjust the tuning parameters and minimize the loss function defined in Equation (3). The trained DOEDN framework was then utilized for feature extraction from unknown testing samples. All training and testing operations for each sensor were performed in the Google Colaboratory environment. Furthermore, the proposed DOEDN framework for each sensor took approximately 10 min of computational time.

Network training performance (Experimental study)

The performance evaluation of the training DOEDN framework is carried out under six different sensor node information scenarios. Each DOEDN framework is independently trained for each sensor using the training samples, which consist of undamaged acceleration data. The effectiveness of the proposed DOEDN framework is compared to the regular DUEDN framework. Figures 15 and 16 display the distribution of the reconstruction losses (MAE) of the DOEDN and DUEDN with epochs, focusing on sensors S5, S6, S12, and S16. The results show that the distributions of the training losses and validation losses are nearly overlapped in both the DOEDN and DUEDN, even in the presence of temperature variations. Similar results are observed for sensors S7 and S14. The performance results of both frameworks on the training samples for sensors S5, S6, S7, S12, S14, and S16 are summarized in Table 10.

Table 10.

Performance results of the two frameworks on the training samples (Experimental study).

Sensors	DUEDN		DOEDN
Sensors	Training	Validation	Training	Validation
S5	2.38 × 10⁻²	2.22 × 10⁻²	1.35 × 10⁻²	1.50 × 10⁻²
S6	1.68 × 10⁻²	1.96 × 10⁻²	1.40 × 10⁻²	1.58 × 10⁻²
S7	1.99 × 10⁻²	2.14 × 10⁻²	1.51 × 10⁻²	1.88 × 10⁻²
S12	1.52 × 10⁻²	1.56 × 10⁻²	1.42 × 10⁻²	1.56 × 10⁻²
S14	1.27 × 10⁻²	1.31 × 10⁻²	1.26 × 10⁻²	1.22 × 10⁻²
S16	9.50 × 10⁻³	1.13 × 10⁻²	6.74 × 10⁻³	8.67 × 10⁻³

DOEDN: deep overcomplete encoder–decoder network; DUEDN: deep undercomplete encoder–decoder network.

Network prediction performance (Experimental study)

The prediction performance of the two trained frameworks is evaluated using unknown testing samples, which consist of both undamaged and damaged acceleration data. The assessment is based on three different metrics, namely MAE, R², and mean IoU for each sensor node. Table 11 presents the performance results of the two frameworks for sensors S5, S6, S7, S12, S14, and S16 in the case of an undamaged bridge. As indicated in the Table 11, the trained DOEDN achieves better performance with smallest MAE (1.02 × 10⁻¹, 5.38 × 10⁻², 1.11 × 10⁻¹, 7.45 × 10⁻², 1.06 × 10⁻¹, and 1.49 × 10⁻¹) and highest R² (0.6781, 0.9296, 0.7495, 0.7972, 0.6374, and 0.7385) values for sensors S5, S6, S7, S12, S14, and S16, respectively. Remarkably, both the trained DOEDN and DUEDN exhibit an identical mean IoU score of 0.9947 for sensors S5, S7, S12, S14, and S16, and 0.9919 for sensor S6. These results confirm that the DOEDN might be more accurate for rapid bridge damage assessment.

Table 11.

Performance results of the two frameworks to the undamaged Z24 bridge case.

Sensors	DUEDN			DOEDN
Sensors	MAE	R ²	Mean IoU	MAE	R ²	Mean IoU
S5	1.29 × 10⁻¹	0.6606	0.9947	1.02 × 10⁻¹	0.6781	0.9947
S6	6.79 × 10⁻²	0.9181	0.9919	5.38 × 10⁻²	0.9296	0.9919
S7	2.77 × 10⁻¹	0.5291	0.9947	1.11 × 10⁻¹	0.7495	0.9947
S12	7.89 × 10⁻²	0.7794	0.9947	7.45 × 10⁻²	0.7972	0.9947
S14	1.08 × 10⁻¹	0.6239	0.9947	1.06 × 10⁻¹	0.6374	0.9947
S16	3.27 × 10⁻¹	0.7259	0.9947	1.49 × 10⁻¹	0.7385	0.9947

DOEDN: deep overcomplete encoder–decoder network; DUEDN: deep undercomplete encoder–decoder network; MAE: mean absolute error; IoU: intersection over union.

In order to assess the bridge health conditions, the MAE and R² for different sensors S5, S6, S7, S12, S14, and S16 are computed using unknown testing samples, including both the undamaged and damaged acceleration data. Table 12 presents the performance results obtained by applying the proposed DOEDN framework to undamaged and damaged bridge cases corresponding to these sensors. It is observed that the trained DOEDN network generates distinctive MAE and R² values for sensors S5, S6, S7, S12, S14, and S16 under different structural scenarios. For sensor S6, the MAE is significantly increased as the structural scenarios change from the undamaged case (5.38 × 10⁻²) to different damage cases (1.12 × 10⁻¹, 8.47 × 10⁻², 8.93 × 10⁻², and 7.74 × 10⁻²). In contrast, the R² degrades as the structural scenarios change from the undamaged case (0.9296) to different damage cases (0.6313, 0.7897, 0.7320, and 0.8320). Similar results are obtained for sensors S5, S7, S12, and S16, indicating their ability to capture different damage cases by analyzing 10-second-long acceleration signals.

Table 12.

Performance results of the DOEDN framework to undamaged and damaged Z24 bridge cases.

Sensors	Undamaged case		Damage case 1		Damage case 2		Damage case 3		Damage case 4
Sensors	MAE	R ²	MAE	R ²	MAE	R ²	MAE	R ²	MAE	R ²
S5	1.02 × 10⁻¹	0.6781	1.71 × 10⁻¹	0.6184	1.73 × 10⁻¹	0.5241	1.70 × 10⁻¹	0.6203	1.75 × 10⁻¹	0.5390
S6	5.38 × 10⁻²	0.9296	1.12 × 10⁻¹	0.6313	8.47 × 10⁻²	0.7897	8.93 × 10⁻²	0.7320	7.74 × 10⁻²	0.8320
S7	1.11 × 10⁻¹	0.7495	1.53 × 10⁻¹	0.7170	1.59 × 10⁻¹	0.7166	1.67 × 10⁻¹	0.6240	1.44 × 10⁻¹	0.7125
S12	7.45 × 10⁻²	0.7972	1.29 × 10⁻¹	0.7390	1.37 × 10⁻¹	0.7234	1.54 × 10⁻¹	0.6772	1.43 × 10⁻¹	0.7119
S14	1.06 × 10⁻¹	0.6374	9.07 × 10⁻²	0.7927	9.22 × 10⁻²	0.6956	1.13 × 10⁻¹	0.6334	9.75 × 10⁻²	0.6550
S16	1.49 × 10⁻¹	0.7385	1.81 × 10⁻¹	0.7101	1.95 × 10⁻¹	0.7040	2.01 × 10⁻¹	0.6127	2.13 × 10⁻¹	0.6092
S14^*	1.38 × 10⁻²	0.9523	8.79 × 10⁻²	0.8400	9.04 × 10⁻²	0.7496	1.08 × 10⁻¹	0.7016	1.06 × 10⁻¹	0.7047

MAE: mean absolute error; DOEDN: deep overcomplete encoder–decoder network.

41-second-long acceleration signals are used to improve the anomaly detection capability of sensor S14.

Meanwhile, for sensor S14, the MAE is significantly decreased as the structural scenarios change from the undamaged case (1.06 × 10⁻¹) to different damage cases (9.07 × 10⁻², 9.22 × 10⁻², 1.13 × 10⁻¹, and 9.75 × 10⁻²). Conversely, the R² significantly increased as the structural scenarios changed from the undamaged case (0.6374) to different damage cases (0.7927, 0.6956, 0.6334, and 0.6550). These results indicate that sensor S14 is not able to capture different damage cases by analyzing 10-second-long acceleration signals. However, when considering 41-second-long acceleration signals, its MAE significantly increased as the structural scenarios change from the undamaged case (1.38 × 10⁻²) to different damage cases (8.79 × 10⁻², 9.04 × 10⁻², 1.08 × 10⁻¹, and 1.06 × 10⁻¹). In contrast, its R² degrades as the structural scenarios change from the undamaged case (0.9523) to different damage cases (0.8400, 0.7496, 0.7016, and 0.7047), as illustrated in Table 12. These results highlight the significance of signal duration in evaluating the anomaly detection capabilities of sensor S14.

Since the MAE for sensors S5, S6, S7, S12, S14, and S16 shows a positive correlation with the type of damage condition, their undamaged acceleration data in the testing samples can be utilized as a baseline for damage assessment. Hence, the reconstruction losses (MAE) are adopted as an evaluation criterion for the DI.

According to the above correlation, a normal distribution of the MAE is established to evaluate the differences between various structural scenarios. Figure 17 illustrates the normal distribution of the MAE for all structural scenarios corresponding to sensor S6. As shown in Figure 17, the normal distributions of the baselines are narrow and high, whereas the normal distributions of various damaged severities become wide and short. Similar results are obtained for sensors S5, S7, S12, S14, and S16. Therefore, it is evident that the mean and standard deviation of the normal distribution are effective in computing the DI outlined in section “Damage indicator.”

Damage assessment (Experimental study)

To assess the damage location related to the Z24 bridge, we calculate the DI for sensors S5, S6, S7, S12, and S16 using Equation (7). These sensors have demonstrated proficiency in detecting various damage cases by analyzing 10-second-long acceleration signals. Figure 18 shows the distributions of the average DI values for sensors S5, S6, S7, S12, and S16 when considering various structural scenarios. D0 refers to the baseline condition, indicating the absence of damage to the bridge. D1_20, D1_40, D1_85, and D1_95 denote varying degrees of damage severity in damage case 1 associated with settlement of 20, 40, 80, and 95 mm of pier Koppigen, respectively. D2_12 and D2_24 indicate the two independent damage severities corresponding to damage case 2 (with 12 and 24 m² of concrete spalling at the soffit). D3_2 and D3_4 represent the two independent levels of damage severity corresponding to damage case 3 (with failure of 1 and 4 anchor heads). D4_2, D4_4, and D4_6 refer to the three independent damage severities corresponding to damage case 4 (with rupture of 2, 4, and 6 tendons).

As depicted in Figure 18, it is evident that the average DI values exhibit distinct variations across different damage cases, with each sensor demonstrating a high level of proficiency in accurately identifying all damage cases when the cutoff is set at the 95th percentile threshold. To determine the sensor that predominantly recognizes a particular damage case, a thorough analysis of the distribution of average DI values for each sensor is conducted. Sensor S5, as illustrated in Figure 18(b) and (c), exhibits good performance in accurately detecting damage case 2, indicating different spalling of concrete (with average DI values of D2_12 = 0.69 and D2_24 = 1.02) and damage case 3, involving various anchor head failures (with average DI values of D3_2 = 0.79 and D3_4 = 0.83). Notably, a selected critical DI threshold of 0.03 serves as the indicator beyond, which an observation is labeled as damaged. However, sensor S5 struggled to accurately evaluate damage severities corresponding to damage cases 1 and 4, as shown in Figure 18(a) and (d). Sensor S6, depicted in Figure 18(g) and (h), demonstrates exceptional performance in accurately detecting damage case 3 (with average DI values of D3_2 = 0.34 and D3_4 = 1.03) and damage case 4, characterized by various tendon ruptures (with average DI values of D4_2 = 0.3, D4_4 = 0.44, and D4_6 = 0.58). Notably, since D0 = 0.04 is already at or above the 95th percentile, it can be considered as the cut-off DI value. However, sensor S6 faced challenges in accurately evaluating damage severities for damage cases 1 and 2, as shown in Figure 18(e) and (f). Sensor S7, shown in Figure 18(j) to (l), exhibits good performance in detecting damage case 2 (with average DI values of D2_12 = 0.35 and D2_24 = 0.73), damage case 3 (with average DI values of D3_2 = 0.6 and D3_4 = 0.65), and damage case 4 (with average DI values of D4_2 = 0.29, D4_4 = 0.37, and D4_6 = 0.42). The designated cut-off DI value for this scenario is 0.02. However, it is worth noting that sensor S7 encounters difficulties in accurately evaluating damage severities for damage case 1, as depicted in Figure 18(i). Sensor S12, as illustrated in Figure 18(m), demonstrates good performance in accurately detecting damage case 1, which involves various settlements of pier Koppigen. The distribution of DI values proportionally increases with the settlement distance, as evidenced by average DI values of D1_20 = 0.51, D1_40 = 0.70, D1_80 = 0.97, and D1_95 = 1.31. Additionally, sensor S12 effectively identifies damage case 3 (with average DI values of D3_2 = 1.14 and D3_4 = 1.43) as shown in Figure 18(o). In this context, the selected critical DI value is set at 0.03. However, sensor S12 faced challenges in accurately evaluating damage severities for damage cases 2 and 4, as demonstrated in Figure 18(n) and (p). Finally, sensor S16, depicted in Figure 18(r) and (s), displays exceptional performance in accurately detecting damage case 2 (with average DI values of D2_12 = 0.33 and D2_24 = 0.50) and damage case 3 (with average DI values of D3_2 = 0.26 and D3_4 = 0.67). The chosen critical DI value is set at 0.01. However, sensor S16 addresses significant challenges in accurately evaluating damage severities for damage cases 1 and 4, as shown in Figure 18(q) and (t).

Since sensor S5 (Figure 18(b)) consistently displays higher average DI values in damage case 2 compared to sensor S7 (Figure 18(j)) and sensor S16 (Figure 18(r)), it demonstrates remarkable effectiveness in both assessing the location and evaluating the severity of concrete spalling at the soffit. The same rigorous procedures are employed to determine the results for the remaining damage cases. Sensor S6 (Figure 18(g) and (h)) exhibits proficiency in assessing the location and evaluating the severity of both the failure of anchor heads and the rupture of tendons compared to other sensors (Figure 18(c), (k), and (l), (o), and (s)). Sensor S12, located close to the damaged pier, excels in its ability to assess the damage location and precisely evaluate the settlement of pier Koppigen in the bridge, as depicted in Figure 18(m). These findings demonstrate the efficiency of sensors S5, S6, and S12 in accurately assessing the damage location and evaluating specific structural scenarios in the Z24 bridge.

Discussion

The proposed unsupervised novelty damage diagnosis method presents potential advantages for rapid bridge damage assessment. This study highlights the importance of utilizing the dynamic responses from each sensor node installed on the bridge, considering various operational and environmental conditions. The contribution of this study can be summarized in four key points. (1) The proposed damage diagnosis method is an unsupervised learning method that utilizes a DOEDN and long-term monitoring acceleration data, which can reduce sensor installation costs and computational efforts. (2) The proposed method can directly leverage the behavior of each sensor node installed on the bridge and extract more effective reconstruction losses for rapid bridge damage assessment. (3) Both the numerical and experimental studies provide a comprehensive analysis of the training and prediction performances of the proposed DOEDN framework for each sensor node. The results show that the proposed DOEDN framework outperforms the regular DUEDN framework with the smallest MAE and the highest R². Reconstruction losses (MAE) can be used as an evaluation criterion for DIs. (4) The study also introduces the DI for each sensor node to access the damage location and evaluate its severity performances under different structural scenarios. The proposed DI can effectively assess the damage location and evaluate specific structural scenarios on the numerical highway bridge model and the full-scale Z24 bridge, even in the presence of temperature variations.

Despite the promising results of the proposed unsupervised damage diagnosis method, some limitations to its practical application should be considered. These limitations can be summarized as follows: (1) Sensor requirements, as the proposed method is only tested with a limited number of sensors. For more complicated structures with torsional, bending, lateral, and longitudinal modes, it is essential to design an appropriate sensor arrangement. (2) The influence of traffic, as vibrations induced by traffic events significantly affect the service life of bridges. Hence, including realistic traffic data during the training and testing process can be valuable in enhancing the effectiveness of the proposed method. Weigh-in-motion systems can provide this type of data. (3) Good training performance, as the accuracy of the proposed method relies on the training process. Bridge owners should collect as much data as possible from structures equipped with long-term health monitoring systems to further improve the training process of the proposed framework. (4) Experimental validation, as further experimental studies are needed to evaluate the proposed damage diagnosis method for its accurate implementation on different in-service bridges, considering various operational and environmental conditions.

Conclusion

This study proposes an unsupervised novelty bridge damage assessment method by utilizing the DOEDN framework and long-term monitoring acceleration data. Specifically, the proposed method focuses on the behavior of each sensor node installed on the bridge. The DOEDN framework is a composite encoder–decoder network combining CNN and LSTM architectures that can learn and extract complex spatial and temporal features from acceleration data. The DOEDN framework is independently trained and tested for each sensor using training and testing samples. The reconstruction losses (MAE) made by the framework are then used as damage-sensitive features for bridge damage assessment. The Gaussian processes of these MAE distributions are used to calculate the DI, enabling the assessment of bridge damage location and the evaluation of its severity. The performance and sensitivity of the proposed framework are verified through a numerical highway bridge model and the well-known full-scale Z24 bridge. Moreover, the performance result of the proposed DOEDN framework (e.g., deep overcomplete encoder–decoder network) is compared to the regular DUEDN framework (e.g., DUEDN) using three distinct metrics: MAE, R², and mean IoU. The main conclusions of this study are drawn as follows:

(1) Each DOEDN framework is trained with short acceleration sequences from a single sensor installed on the bridge under varying operational and environmental conditions. This approach reduces sensor installation costs, limits computational efforts, and provides rapid bridge damage assessment.

(2) The distributions of the training losses match well with the distributions of the validation losses in both the DOEDN and DUEDN. Additionally, temperature variations and measurement noise did not have a substantial impact on the distribution of the reconstruction losses.

(3) The trained DOEDN performs the best result with the smallest MAE and the highest R² in both the numerical and experimental studies. Notably, both the trained DOEDN and DUEDN demonstrate an identical mean IoU score. These results show that the DOEDN framework might be more accurate for rapid bridge damage assessment.

(4) The MAE has a positive correlation with the type of damage condition, whereas the R² has a negative correlation. Thus, the MAE can be used as an evaluation criterion for DIs.

(5) A normal distribution showed an excellent representation of the MAE distributions, which provided benefits for quantifying the differences between various bridge conditions.

(6) The performance of the proposed DI for each sensor node can reasonably assess the damage location and evaluate its severity across various structural scenarios in both the numerical highway bridge model and the full-scale Z24 bridge.

Overall, the findings demonstrate that the proposed unsupervised novelty bridge damage assessment method is a practical and effective solution for bridge health monitoring. Despite its findings, this study acknowledges a potential issue with the connection between 1D CNN and LSTM units, which may impact the effective utilization of temporal dependencies of the features. Introducing a channel attention module as a skipped connection shows promise in mitigating this limitation.^58–60 This module acts as a dynamic filter, allowing the network to selectively emphasize or de-emphasize certain features, enhancing its ability to capture critical patterns in the data. In future studies, we will explore the integration of this module to further enhance the proposed method, potentially leading to even more accurate and reliable bridge damage assessment results.

Footnotes

Acknowledgements

The authors would like to thank researchers at the Katholieke Universiteit Leuven for providing the Z24 bridge datasets to conduct this experimental study.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is financially supported by the National Key R&D Program of China (2018YFB1600300 and 2018YFB1600301), the National Natural Science Foundation of China (51578370, 52078333), and the Tianjin Transportation Science and Technology Development Plan Project (G2018-29). Any opinions, findings, conclusions, or recommendations expressed in this article are those of the authors and do not necessarily reflect those of the sponsor.

ORCID iD

Jinsong Zhu

Data availability

The simulation and field experiment datasets generated during and/or analyzed during the current work are available from the corresponding author on reasonable request.

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