A Deep Learning-Based Structural Damage Identification Method Integrating CNN-BiLSTM-Attention for Multi-Order Frequency Data Analysis

Abstract

Structural health monitoring commonly uses natural frequency analysis to assess structural conditions, but direct frequency shifts are often insensitive to minor damage and susceptible to environmental influences like temperature variations. Traditional methods, whether based on absolute frequency changes or theoretical models like PCA and GMM, face challenges in robustness and reliance on model selection. These limitations highlight the need for a more adaptive and data-driven approach to capturing the intrinsic nonlinear correlations among multi-order modal frequencies. This study proposes a novel approach that leverages the nonlinear correlations among multi-order natural frequencies, which are more sensitive to structural state changes. A deep learning framework integrating CNN-BiLSTM-Attention is designed to capture the spatiotemporal dependencies of multi-order frequency data, enabling the precise modeling of intrinsic correlations. The model was trained exclusively on healthy-state frequency data and validated on both healthy and damaged conditions. A probabilistic modeling approach, incorporating Gaussian distribution and cumulative probability functions, was used to evaluate the estimation accuracy and detect correlation shifts indicative of structural damage. To enhance the robustness, a moving average smoothing technique was applied to reduce random noise interference, and damage identification rates over extended time segments were calculated to mitigate transient false alarms. Validation experiments on a mass-spring system and the Z24 bridge dataset demonstrated that the proposed method achieved over 95% damage detection accuracy while maintaining a false alarm rate below 5%. The results validate the ability of the CNN-BiLSTM-Attention framework to effectively capture both structural and environmental nonlinearities, reducing the dependency on explicit theoretical models. By leveraging multi-order frequency correlations, the proposed method provides a robust and highly sensitive approach to structural damage identification. These findings confirm the practical applicability of deep learning in damage identification during the operational phase of structures.

1. Introduction

The long-term performance of civil infrastructure is inevitably affected by environmental degradation, material aging, and extreme loads. To ensure structural safety and mitigate potential casualties and economic losses, structural health monitoring (SHM) systems have been widely implemented to collect real-time structural response data and assess the structural conditions [1,2]. From a structural dynamics perspective, damage is typically characterized by changes in physical parameters such as stiffness, damping, and mass, which in turn lead to variations in modal properties, including frequency, damping ratio, and mode shape [3,4,5]. Current structural damage identification methods primarily rely on analyzing variations in these dynamic parameters based on SHM data [6,7,8]. Due to their non-destructive nature, real-time applicability, and high sensitivity, modal parameter-based damage indicators have gained significant attention in SHM applications [9,10,11]. However, during the operational phase of structures, external load data are generally unavailable, and environmental factors such as temperature fluctuations introduce uncertainties in structural response, making it challenging to distinguish environmental effects from actual damage.

Among these modal indicators, frequency-based damage identification methods have been widely used due to their simplicity [12,13]. Early studies often directly employed changes in natural frequency as damage indicators. Additionally, as natural frequencies are global properties, they lack spatial resolution, making it challenging to accurately localize damage [14]. To enhance the effectiveness of frequency-based damage identification, recent studies have explored the integration of multi-source long-term monitoring data and the correlation among different modal frequencies [15,16]. Compared with frequency alone, mode shapes exhibit higher sensitivity to localized stiffness changes due to their spatial distribution characteristics [17]. For instance, Yoon et al. proposed a global fitting approach based on displacement mode shape changes, achieving both damage identification and localization [18]. Several alternative techniques, such as the modal confidence criterion, modal curvature, and modal flexibility-based methods, have also demonstrated strong capabilities in detecting local stiffness variations [11,19]. Furthermore, strain mode shapes extracted from dynamic strain data can sometimes offer even higher sensitivity to damage. Xu et al. introduced a strain mode-based damage detection method for large-span lattice structures, verifying its effectiveness in stiffness reduction identification [20]. However, despite their advantages, mode shape-based methods are highly dependent on sensor placement, as only mode shapes at sensor locations can be accurately measured.

A key challenge in frequency-based damage identification is the influence of environmental factors such as temperature fluctuations and wind speed, which cause variations in modal frequencies that can obscure damage-induced changes [21,22,23]. To mitigate these effects, damage indicators must be formulated to quantify and eliminate environmental influences [24,25]. When environmental inputs are known, regression models such as auto-regressive models, polynomial regression, and Gaussian process regression can be employed to establish a quantitative relationship between the modal frequency and environmental conditions [26,27,28]. In cases where environmental inputs are unknown, principal component analysis (PCA) has been widely applied to analyze the correlation between different modal frequencies and construct damage-sensitive indicators that are robust to environmental variations [29,30]. Further advancements in PCA-based techniques, such as nonlinear PCA [31], robust PCA [32], and subspace PCA [33], have enhanced damage detection accuracy. For capturing nonlinear relationships between modal frequencies, cointegration analysis (CA) provides an effective means to establish regression models that consider long-term frequency interactions [34,35]. Additionally, Gaussian mixture models (GMM) have been employed to extract statistical features from modal frequency data, enabling damage identification through residual analysis and probabilistic estimation [36]. The estimation of GMM parameters typically relies on expectation maximization (EM) algorithms or Bayesian optimization techniques [37,38,39]. The accuracy and stability of the GMM model are highly dependent on the careful selection of its initial parameters [40,41].

Most of the aforementioned methods are based on theoretical estimation models, which rely on distinguishing frequency variations between healthy and damaged states. However, due to the influence of environmental variations and structural degradation, the relationship among different modal frequencies often exhibits significant nonlinearity, making it challenging to construct precise theoretical models. In recent years, the rapid advancement of deep learning techniques has provided new opportunities for modeling such complex nonlinear relationships [42,43,44]. Convolutional neural networks (CNN) have demonstrated unique advantages in extracting nonlinear features by leveraging convolutional kernels and pooling layers to capture correlations in multi-source time-series data [45,46]. For example, Khodabandehloud et al. directly input vibration signals in matrix form into CNN and successfully validated their feasibility for damage detection using a scaled highway bridge model [47]. Similarly, Liu et al. employed transfer functions to establish relationships between damage-sensitive features before feeding them into a one-dimensional CNN model, demonstrating high detection accuracy in benchmark experiments [48]. Traditional recurrent neural networks (RNN) have been employed for time-series modeling due to their ability to capture temporal dependencies. However, their performance is often hindered by the vanishing gradient problem when handling long sequences. Long short-term memory (LSTM) networks resolve this problem by employing gating mechanisms, such as the input gate, forget gate, and output gate, to effectively manage temporal dependencies. [49,50]. Recent studies have demonstrated that combining CNN and LSTM can leverage their respective strengths, leading to improved performance in damage detection [51,52].

In this study, a novel deep learning-based structural damage identification method is proposed, integrating CNN, BiLSTM, and attention mechanisms to enhance damage detection accuracy. The model extracts nonlinear spatial-temporal features from multi-order frequency data, while the attention mechanism refines feature weighting to improve the estimation precision. To achieve reliable damage identification, a probabilistic modeling approach is introduced, leveraging probability density functions or cumulative distribution functions to establish confidence intervals and define damage detection thresholds based on statistical characteristics of estimated modal frequencies. The proposed framework is validated through both numerical simulations and real-world monitoring data from the Z24 bridge, demonstrating high damage detection accuracy even in the presence of environmental variations. These results confirm the robustness and applicability of the method for structural health monitoring. The remainder of this paper is organized as follows: Section 2 details the proposed deep learning-based damage identification method, including its architectural components and probabilistic modeling framework. Section 3 presents the experimental validation using a mass-spring system and the Z24 bridge dataset. Section 4 presents the conclusions.

2. Damage Identification Method

Building upon the CNN-LSTM framework, this study integrates BiLSTM and an attention mechanism to enhance the model’s capability in capturing nonlinear relationships among multi-order frequency data. The CNN module extracts spatial correlations from high-dimensional frequency data, while the BiLSTM component models bidirectional temporal dependencies in modal frequency sequences. Unlike standard LSTMs, BiLSTM leverages both past and future information to provide a more comprehensive representation of frequency variations. Additionally, the attention mechanism dynamically assigns importance weights to different modal frequencies, improving the model’s ability to focus on the most informative frequency components for damage detection. By combining these three elements, the proposed method effectively captures both spatial and temporal correlations in structural health monitoring data, mitigating the impact of environmental noise and enhancing sensitivity to localized damage. Subsequently, a damage-sensitive indicator is formulated based on the statistical characteristics of errors between estimated and actual values in the validation set. A confidence interval is then established according to a predefined confidence level, serving as the threshold for damage identification. Finally, during the testing phase, structural health is assessed by evaluating whether the estimated values, derived from both healthy and damaged datasets, fall within the confidence interval. Deviations beyond this range indicate potential structural damage.

2.1. CNN Layer

The CNN layer consisted of convolution operations, batch normalization, nonlinear activation functions (e.g., ReLU), and pooling layers, which worked together to extract spatial features from high-dimensional data efficiently. Raw intrinsic frequency data underwent standardization preprocessing to eliminate dimensional inconsistencies and improve the model convergence. The CNN layer first applied convolutional filters to capture local frequency correlations and extract key features related to potential structural damage. Batch normalization stabilized the data distribution, mitigating the gradient vanishing, while the nonlinear activation functions enhanced the model’s ability to represent complex feature relationships. Next, pooling layers (e.g., max pooling or average pooling) performed downsampling to reduce the feature dimensionality while preserving damage-sensitive information. Finally, the extracted features were flattened and passed to fully connected layers, which structured the data for subsequent processing in the BiLSTM layer for temporal sequence modeling.

2.2. BiLSTM Layer

During structural operation, performance degradation, damage, and sensor errors contribute to significant nonlinear relationships among modal frequencies of different orders. Additionally, environmental temperature variations and insufficient excitation modes lead to varying sensitivities of different modal frequencies to structural changes. Traditional linear regression methods struggle to provide accurate estimations due to discrepancies between theoretical assumptions and real-world conditions. To address this challenge, an LSTM architecture was introduced to handle the nonlinear dependencies caused by structural variations. This framework effectively leveraged time-series characteristics, enhancing the modeling capability for structural dynamic evolution. The LSTM included three key components: the forget gate, input gate, and output gate, responsible for filtering historical information, updating new inputs, and extracting relevant memory features, respectively. These gates operated based on the earlier hidden state along with the current input, computing weight values between 0 and 1, following the calculation process below:

$$I_t=\varphi \left(W_i\cdot \left[X_t,H_{t-1}\right]+b_i\right)$$

(1)

$$F_t=\varphi \left(W_f\cdot \left[X_t,H_{t-1}\right]+b_f\right)$$

(2)

$$O_t=\varphi \left(W_o\cdot \left[X_t,H_{t-1}\right]+b_o\right),$$

(3)

where $I_t$, $F_t$, $O_t$ represent the input gate, forget gate, and output gate, respectively, at time step t. $\varphi$ denotes the activation function, $X_t$ represents the input vector, and $H_{t-1}$ is the hidden state from the earlier time step. $W_i$, $W_f$, and $W_o$ are the weight parameters associated with the three gates, while $b_i$, $b_f$, and $b_o$ are the corresponding bias vectors. To further capture the relationships between different modal frequency data, the BiLSTM network was introduced [53,54]. As an extension of the standard LSTM, BiLSTM leveraged both forward and backward LSTM layers, allowing it to extract modal frequency correlations induced by structural variations from both temporal directions. This bidirectional learning mechanism enhanced the utilization of feature data, improving the model’s generalization ability and estimation accuracy. The forward and backward hidden state updates follow these rules:

$${\overrightarrow{H}}_t=\varphi \left({\overrightarrow{W}}_f\cdot \left[X_t,{\overrightarrow{H}}_{t-1}\right]+{\overrightarrow{b}}_f\right)$$

(4)

$${\overleftarrow{H}}_t=\varphi \left({\overleftarrow{W}}_f\cdot \left[X_t,{\overleftarrow{H}}_{t-1}\right]+{\overleftarrow{b}}_f\right),$$

(5)

where ${\overrightarrow{H}}_t$ and ${\overleftarrow{H}}_t$ represent the forward and backward hidden states at time step t, respectively. ${\overrightarrow{H}}_{t-1}$ and ${\overleftarrow{H}}_{t-1}$ denote the forward and backward hidden states from the previous and next time steps, respectively. ${\overrightarrow{b}}_f$ and ${\overleftarrow{b}}_f$ are the corresponding bias vectors used in the computation.

2.3. Attention Layer

During structural operation, long-term modal frequency data analysis involves complex spatiotemporal correlations, making it significantly more challenging than traditional time-series estimation. On one hand, temperature fluctuations and structural damage introduce dynamic dependencies between consecutive data points. On the other hand, different modal frequency orders exhibit intrinsic nonlinear interactions governed by structural dynamics. To enhance the modeling capability of deep learning in capturing multi-source data dependencies, this study incorporated an attention mechanism. The attention mechanism dynamically assigned importance weights to different modal frequency components, improving feature representation and enhancing damage-sensitive analysis. Given a query vector $q$ and m key-value pairs $\left(k_1,v_1\right),\cdots ,\left(k_m,v_m\right)$, the attention aggregation function can be expressed as a weighted sum of the key-value pairs related to the query [55]:

$$f\left(q,\left(k_1,v_1\right),\dots ,\left(k_m,v_m\right)\right)={\displaystyle \sum }_{i=1}^m\alpha \left(q,k_i\right)v_i,$$

(6)

where $\alpha \left(q,k_i\right)$ represents the attention weight obtained by computing the relevance between the query and key-value vectors using an attention scoring function, which is subsequently processed through a softmax operation to normalize the weights

$$\alpha \left(q,k_i\right)={\displaystyle \frac{\exp \left(S\left(q,k_i\right)\right)}{{{\displaystyle \sum }}_{j=1}^m\exp \left(S\left(q,k_j\right)\right)}},$$

(7)

where $S\left(q,k_i\right)$ represents the attention scoring function between the query and key-value vectors. Different attention scoring functions result in varying attention aggregation operations. When employing the self-attention mechanism, $q$, $k$, and $v$ are derived from the same input sequence, ensuring that the output dimension of the self-attention layer remains consistent with the hidden unit size of the preceding BiLSTM layer.

2.4. Damage Indicator

After obtaining the estimated modal frequencies from the deep learning model, the statistical characteristics of the difference between the estimated and actual values were analyzed to assess the uncertainty in frequency estimation. Typically, by examining the statistical distribution of the actual and estimated values in the validation set, a confidence interval corresponding to a predefined confidence level $1-\alpha$ was established. The damage assessment was then conducted by checking whether the estimated values in the test set fell within this interval. If an estimated value exceeded the confidence interval, it indicated that the model, trained on the validation set, was less applicable to the testing data. This deviation was likely caused by changes in the modal frequency correlations due to structural damage, leading to a decrease in the estimation accuracy. We assume the sequence length of the modal frequency data in the validation set was $n$, with the actual observed modal frequency sequence represented as $f=\left\{f^{\left(1\right)},f^{\left(2\right)},\cdots ,f^{\left(n\right)}\right\}$ and the estimated sequence from the deep learning model as $\hat{f}=\left\{{\hat{f}}^{\left(1\right)},{\hat{f}}^{\left(2\right)},\cdots ,{\hat{f}}^{\left(n\right)}\right\}$. Here, superscripts denote the modal frequency sequences to distinguish them from the subscripted mode indices. The mean $\mu$ and variance ${\sigma }^2$ of the difference sequence between estimated and actual values are given by

$$\mu ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left(f^{\left(i\right)}-{\hat{f}}^{\left(i\right)}\right)}$$

(8)

$${\sigma }^2={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(f^{\left(i\right)}-{\hat{f}}^{\left(i\right)}-\mu \right)}^2}.$$

(9)

Next, the statistical properties of the estimated data were utilized to define the confidence interval based on a given confidence level $1-\alpha$. If the difference sequence followed a Gaussian distribution, the upper and lower bounds of the confidence interval was determined using the following equation:

$$Q_{\alpha /2}=Z\left(\alpha /2\right)\cdot \sigma$$

(10)

$$Q_{1-\alpha /2}=Z\left(1-\alpha /2\right)\cdot \sigma ,$$

(11)

where $Q_{\alpha /2}$ and $Q_{1-\alpha /2}$ represent the lower and upper bounds of the confidence interval at the confidence level, which can also be interpreted as the quantiles ($\alpha /2$ and $1-\alpha /2$) of the difference sequence. $Z\left(\alpha /2\right)$ and $Z\left(1-\alpha /2\right)$ denote the corresponding quantiles ($\alpha /2$ and $1-\alpha /2$) of the standard normal distribution. In practice, if the difference sequence does not follow a Gaussian distribution, kernel density estimation (KDE) is often used to derive the cumulative distribution function (CDF) of the difference sequence. The quantile values can then be back-calculated to determine the confidence interval limits:

$$Q_{\alpha /2}=F^{-1}\left(\alpha /2\right)$$

(12)

$$Q_{1-\alpha /2}=F^{-1}\left(1-\alpha /2\right),$$

(13)

where $F^{-1}\left(\cdot \right)$ represents the cumulative distribution function of the difference sequence. Ultimately, the confidence interval limits $Q_{\alpha /2}$ and $Q_{1-\alpha /2}$ are used as thresholds for damage identification. It is important to note that the confidence level $1-\alpha$ is typically set to 95% based on hypothesis testing principles. However, in this study, the confidence level was not fixed across different case studies. To minimize the false alarm rate in the test set (keeping it below 5%), the confidence level was adjusted within the range of [0.01, 0.05]. The method adapts to different structural and environmental conditions, ensuring a more reliable and practical damage detection framework.

2.5. Damage Detection Procedure

The proposed damage identification method consisted of two main components: modal frequency estimation based on deep learning and damage identification based on a probabilistic model. First, multi-order modal frequency data underwent preprocessing, where the dataset was divided into training, validation, and testing sets. The testing set included both healthy and damaged states, ensuring a comprehensive evaluation of the model’s identification capability. Next, the CNN-BiLSTM-attention model was employed to extract spatiotemporal features from multi-source data, generating modal frequency estimations at different structural states. Finally, the statistical characteristics of the difference between the estimated and actual values in the validation set were analyzed to construct a confidence interval. Based on the confidence interval limits, damage identification thresholds were established. During testing, if the estimated values deviated beyond this range, structural damage was indicated. The detailed workflow is illustrated in Figure 1.

3. Case Studies

3.1. Mass-Spring System Numerical Simulation

As shown in Figure 2, a four-degrees-of-freedom (4-DOF) mass-spring system was employed to validate the effectiveness of the proposed method. The structural configuration and boundary conditions of the system can be observed in the figure. To simulate the influence of environmental variations, particularly temperature fluctuations, on the modal frequencies, real-world temperature data from an actual structure were incorporated into the mass-spring system. This allowed for the generation of long-term modal frequency observations under varying environmental conditions. Each node had a mass of 2 kg, and the spring stiffness ($k_1$, $k_2$, $k_3$, $k_4$, and $k_5$) between nodes was determined using the following equation [56,57]:

$$k_1=k_2=k_4=k_5=\left\{\begin{array}{ccc}-0.15\times T+8& \mathrm{if}& T<0\\ -0.05\times T+8& \mathrm{if}& T\ge 0\end{array}\right.$$

(14)

$$k_3=\left\{\begin{array}{ccc}-0.15\times T+10& \mathrm{if}& T<0\\ -0.25\times T+10& \mathrm{if}& T\ge 0\end{array}\right.,$$

(15)

where $T$ represents the temperature.

Figure 3 presents the long-term modal frequency sequences obtained from the mass-spring system, corresponding to the four modal orders of the structure. To simulate measurement noise, 2% Gaussian white noise was added to the system’s output data. Each sequence contained 8760 samples, with the green vertical dashed line marking the boundary between the undamaged and damaged states. The first 7320 samples represented the undamaged structure, while the remaining 1440 samples corresponded to the damaged structure. Damage was introduced by reducing the stiffness values to simulate degradation, with three levels of stiffness reduction set at 10%, 20%, and 30%, corresponding to sample ranges 7321–7800, 7801–8280, and 8281–8760, respectively. The brown vertical dashed lines indicate dataset partitioning, with the first 5880 samples used for training, the next 960 for validation, and the final 1920 for testing. The training and validation sets consisted exclusively of undamaged structure data, while the testing set included both undamaged samples (6841–7320) and damaged samples (7321–8760) across the three damage conditions. To mitigate data fluctuations and enhance modal frequency estimation stability, a moving average smoothing technique was applied. The testing set evaluation aimed to ensure that under undamaged conditions, the estimated modal frequencies aligned with the statistical characteristics of the validation set, demonstrating the deep learning model’s regression capability and accuracy, while under damaged conditions, significant deviations from the validation set confirmed the model’s effectiveness in identifying structural damage and capturing nonlinear correlations among different modal frequencies.

Figure 4, Figure 5, Figure 6 and Figure 7 illustrate the estimated modal frequencies for the four mode orders along with the corresponding confidence intervals. In these figures, the black solid line represents the actual observed values, the red dashed line indicates the estimated values obtained from the deep learning model, and the blue-shaded region in the testing set denotes the confidence interval. As seen in subplot (a), the estimated values in the training and validation sets closely matched the actual observations, indicating that the deep learning model achieved high estimation accuracy without significant overfitting. In subplot (b), a clear distinction between undamaged and damaged conditions can be observed in the testing set. Under the undamaged condition (samples 6841–7320), the estimated values largely fell within the confidence interval, demonstrating that the model maintained high estimation accuracy in a healthy structural state and that the test results were statistically consistent with the validation set, further validating the model’s generalization capability and reliability. However, under the damaged condition (samples 7321–8760), most estimated values exceeded the confidence interval, suggesting that structural damage altered the modal frequency correlations, rendering the model, which was trained on undamaged data, less effective for damaged conditions. This indicates that the model successfully captures changes in the modal frequency relationships caused by structural damage and exhibits strong discriminatory capability in damage identification.

By analyzing the proportion of estimated values in the testing set that fell outside the confidence interval, the structural damage identification rates, as shown in

Table 1. Detection rates of structural damage for the mass-spring system.

Estimated Frequency	Damage Detection Rate (%)
	Undamaged	State 1	State 2	State 3
f1	2.9	99.6	100	97.9
f2	0.2	100	100	100
f3	4.8	100	100	100
f4	4.9	68.2	72.6	78.4

, can be determined. Cases where the model incorrectly classified an undamaged state as damaged were considered false alarms, and reducing the false alarm rate was crucial for improving the model reliability. Among the three damage conditions, higher damage identification rates indicated greater model sensitivity to structural damage. As observed in Table 1, damage identification based on the first to third modal frequencies achieved superior performance, exhibiting low false alarm rates and high accuracy. This result not only validated the effectiveness of the proposed method in capturing intrinsic data correlations but also suggested that the first three modal frequencies were more sensitive to the assumed damage conditions. However, the identification accuracy for the fourth modal frequency was relatively lower, with a correct detection rate below 80%, likely due to its reduced sensitivity to the imposed stiffness reduction. Overall, the proposed method demonstrates promising results on the mass-spring system: in undamaged conditions, the estimated values align well with statistical characteristics, confirming the high regression accuracy of the deep learning model, while in damaged conditions, the estimates show significant deviations, highlighting the model’s effectiveness in structural damage identification.

3.2. Z24 Bridge Long-Term Monitoring Data Analysis

To further evaluate the applicability of the proposed method in real-world engineering scenarios, long-term modal frequency data from the Z24 bridge [26,28,33] were utilized for damage identification. Figure 8 provides a schematic of the Z24 bridge, a prestressed concrete bridge located in the Bern region of Switzerland. The bridge was equipped with a monitoring system that recorded long-term structural response data. Before its demolition, the bridge was subjected to controlled damage tests, meaning that the collected monitoring data included both healthy and damaged conditions. Figure 9 presents the four modal frequency time series observed from the Z24 bridge, corresponding to the structure’s four modal orders. Each sequence consisted of 3932 samples, with the green vertical dashed line marking the boundary between the undamaged and damaged states. The first 3470 samples represented the undamaged structure, while the remaining 462 samples corresponded to the damaged structure. The brown vertical dashed lines indicate dataset partitioning: the first 2546 samples were used for training, the next 462 samples for validation, and the final 924 samples for testing. The training and validation sets consisted solely of undamaged structure data, while the testing set included both undamaged samples (3009–3470) and damaged samples (3471–3932). Similar to the mass-spring system case, a moving average smoothing technique was applied to improve the stability of the modal frequency estimation. The evaluation objectives for the testing set in the Z24 bridge case study were as follows: (1) Under undamaged conditions, the estimated modal frequencies should align with the statistical characteristics of the validation set, verifying the deep learning model’s regression capability and estimation accuracy. (2) Under damaged conditions, significant deviations in estimated modal frequencies should be observed, demonstrating the model’s effectiveness in identifying structural damage and capturing the nonlinear correlations across multiple modal orders.

Figure 10, Figure 11, Figure 12 and Figure 13 illustrate the estimated modal frequency results for the four modal orders, along with the corresponding confidence intervals. In these figures, the black solid line represents the actual observed values, the red dashed line denotes the estimated values from the deep learning model, and the blue-shaded region in the testing set indicates the confidence interval. As shown in subplot (a), the estimated values in the training and validation sets closely align with the actual observations, demonstrating that the deep learning model achieved high estimation accuracy in real-world engineering data analysis without significant overfitting. In subplot (b), a clear distinction between undamaged and damaged conditions is evident in the testing set. Under undamaged conditions (samples 3009–3470), the estimated values predominantly fell within the confidence interval, confirming that the testing set estimates remained statistically consistent with the validation set, further validating the model’s generalization ability and reliability. However, under damaged conditions (samples 3471–3932), most estimated values exceeded the confidence interval, indicating that structural damage altered the correlation among the modal frequency data. This shift reduced the model’s applicability to damaged states, as it was trained exclusively on undamaged data. The observed deviations in the estimated values reflect the impact of the structural damage on the modal frequency variations, confirming that the deep learning model effectively captures damage-induced changes in modal frequency correlations and exhibits strong damage identification capability.

Table 2. Detection rates of structural damage for the Z24 bridge.

Estimated Frequency	Damage Detection Rate (%)
	Undamaged State	Damaged State
f1	5.0	81.4
f2	4.9	98.3
f3	0	98.3
f4	3.4	95.1

presents the damage identification rates for the Z24 bridge, reflecting the proportion of the estimated values in the testing set that fall outside the confidence interval. A higher damage identification rate, while maintaining a low false alarm rate, indicates greater model sensitivity to structural damage. As shown in Table 2, the damage identification based on second- to fourth-order modal frequencies demonstrated superior performance, achieving low false alarm rates and high accuracy. This suggests that the proposed method is highly effective in capturing intrinsic correlations within modal frequency data and that higher-order modal frequencies exhibit greater sensitivity to the assumed damage conditions. However, the first-order modal frequency showed relatively weaker performance, with a correct identification rate of only 81.4%, a trend consistent with the results from the mass-spring system. This indicates that the first-order modal frequency may not be sufficiently sensitive to actual damage in the Z24 bridge. Thus, relying solely on low-order modal frequency data for damage identification may be limited, whereas integrating multiple modal frequencies can significantly enhance the model’s detection capability. The results of two case studies demonstrated that in both numerical and real-world case studies, the model achieved high accuracy in estimating healthy-state modal frequencies and effectively identified damage in most cases. The observed variation in sensitivity across different modal frequencies was primarily due to how the structural damage manifests in modal characteristics. In this method, the low damage identification accuracy for a specific modal frequency may be due to the structural changes not being well reflected in this mode or the nonlinear relationships between this modal frequency and other frequency orders being less affected by the defined damage in this case study. Despite this limitation in one specific modal frequency, the overall results validated that our method effectively captured the internal nonlinear relationships among modal frequencies. Overall, the proposed method achieved promising results in both the mass-spring system and Z24 bridge damage identification tasks, ensuring high regression accuracy while attaining a high damage detection rate. Furthermore, a comparative analysis with PCA, CA, and GMM-based methods for the Z24 bridge damage identification showed that the deep learning-based probabilistic model proposed in this study achieved comparable detection efficiency [35,36,37,38,39,58,59]. In general, while the proposed deep learning approach performs similarly to traditional correlation analysis methods in terms of overall damage identification rates, it demonstrates superior effectiveness in detecting minor damage, highlighting its advantage in capturing subtle structural changes.

4. Conclusions

This study proposed a novel structural damage identification method based on a CNN-BiLSTM-Attention deep learning architecture, leveraging long-term multi-order frequency data for damage detection. The method employed probabilistic modeling to define the confidence intervals and establish damage detection thresholds based on the correlation changes between modal frequencies. The experimental results demonstrated the effectiveness of this approach in accurately distinguishing between healthy and damaged structural states. The key findings are summarized as follows:

(1) The proposed method improves upon traditional CNN-LSTM architectures by incorporating BiLSTM and an attention mechanism, enhancing the model’s ability to capture nonlinear relationships among multiple modal frequencies. The results confirm that the model accurately estimates structural modal frequencies in healthy states, with a false alarm rate below 5% in most cases, validating its high accuracy and generalization capability.

(2) By employing probabilistic modeling techniques using probability density functions or cumulative distribution functions, the method defines confidence intervals for damage detection thresholds. The proposed approach effectively differentiates between healthy and damaged conditions, achieving damage identification accuracies exceeding 95% in most scenarios.

(3) The nonlinear correlation among multi-order natural frequencies serves as a key damage-sensitive indicator. The proposed method maintains high damage identification accuracy under varying environmental and structural conditions. Its data-driven approach eliminates the reliance on explicit feature engineering required by traditional methods, offering a novel solution for structural safety assessment in complex environments.