Deep learning-driven thermal response estimation for an in-service cable stayed bridge

Abstract

In order to enhance the accuracy of thermal response estimation, this study proposes the utilization of a deep learning-driven method, specifically the Long Short-Term Memory (LSTM) network, for predicting temperature-induced strains and deflections in the Queensferry Crossing, UK. Initially, an examination of the temperature fields of the bridge is conducted based on monitored temperature data. Subsequently, the LSTM network is introduced as a tool to estimate thermal response utilizing temperature field information. The study proceeds to discuss the results of temperature-induced response estimation, including strains and deflections, obtained through the LSTM network, with a comparative analysis against a linear regression model. It is observed that the temperature discrepancies among steel components within the bridge are lesser in comparison to those of concrete components. The LSTM network exhibits higher estimation accuracy than the linear model, manifesting in an improvement of mean square error from 4.45 to 0.89 for strain estimation. In addition, the LSTM network can partially address time-lag effects. However, the estimation performance of the LSTM network is influenced by the structural response type and the resolution of employed sensors. Specifically, the correlation coefficient (R) associated with strain estimation is measured at 0.980 using the LSTM network, whereas R for deflection estimation stands at 0.825. Generally, local structural responses, such as strains, outperform global responses like deflections. Furthermore, sensors with higher resolutions yield enhanced estimation accuracy.

Keywords

Cable stayed bridges structural health monitoring thermal response deep learning temperature fields

Introduction

Thermal action represents a significant load factor in the operation of bridges, alongside vehicle loads and wind loads (AASHTO, 2020). The substantial seasonal and daily temperature differentials give rise to considerable structural responses, particularly in statically indeterminate bridge structures (Zhou et al., 2020b). For instance, the Tsing Ma Bridge in Hong Kong exhibits an annual variation range of temperature-induced longitudinal deck displacements of approximately 500 mm (Xu et al., 2010). Similarly, the 3rd Nanjing Yangtze River Bridge in China experiences a seasonal fluctuation range of cable forces of approximately 80 kN (Ren et al., 2019). Furthermore, the Sutong Cable-Stayed Bridge in China encounters a daily range of temperature-induced girder top flange strains of approximately 200με (Wang et al., 2019).

Understanding the thermal behavior of bridges is crucial for various purposes such as damage detection, early warning systems, anomaly detection, and condition assessment based on structural health monitoring (SHM) measurements. For example, Kostic and Gul (2017) developed a temperature-compensated damage detection method utilizing vibration and temperature data for a footbridge. Gong et al. (2023) established a mapping model correlating temperatures with strains to provide early warnings of structural deterioration or damage for a four-span prestressed concrete continuous box girder bridge. Ren et al. (2022) utilized relationships between temperatures and girder end displacements to identify anomalies for a suspension bridge in China. Furthermore, Zhou et al. (2021) proposed a temperature effect-separated multilevel structural condition assessment approach employing deflection measurements for a prestressed concrete girder bridge. These studies highlight the importance of comprehending bridge thermal behavior in facilitating effective damage detection, early warning systems, anomaly detection, and condition assessment through SHM measurements.

The structural temperature field serves as the cornerstone for discussions on thermal behavior in bridges. With advancements in simulating and sensing technologies, a wealth of research findings concerning bridge temperature fields has emerged. Utilizing temperature measurements from SHM systems, researchers have uncovered bridge temperature distributions, with particular emphasis on girder temperature fields (Elshoura and Okeil., 2022; Fan et al., 2022; Tao et al., 2021; Wang et al., 2021; Xu et al., 2019; Zhang et al., 2022), tower temperature distributions (Xia et al., 2012), and main cable temperature fields (Battista et al., 2015). Nevertheless, temperature fields derived from field temperature measurements are inherently limited by finite sensors and thus fail to capture the entire temperature distribution. Consequently, various simulation methods are employed to extrapolate discrete sensing points to encompass the entire structure through heat-transfer analysis. For example, Shan et al. (2023) utilized heat-transfer analysis and field data to construct global 3D temperature distributions for the Hong Kong‒Zhuhai‒Macao Bridge. The temperature field of composite box girder bridge with corrugated steel webs is simulated and verified by using field monitored temperature data (Cai et al., 2024). Similarly, the temperature distribution of an arch bridge was derived based on measured air temperature, solar radiation, and wind speed (Xia et al., 2022; Zhou et al., 2022). Moreover, temperature fields of flat steel box girders were predicted using in situ environmental parameters and machine learning techniques for the Sutong Bridge (Wang et al., 2022). In addition, the deep learning method is explored to predict bridge temperatures based on shared meteorological data (Zhou et al., 2024).

The estimation of structural thermal response stands as one of the foremost areas of interest in the analysis of bridge thermal behavior. Broadly, methods for thermal response estimation can be categorized into two classes. The first class is the structural response-only method, which employs signal processing algorithms to isolate thermal components from measured structural response signals. Commonly utilized signal processing methods include wavelet transform (Xu et al., 2020; Zhao et al., 2019), principal component analysis (Zhu et al., 2019), blind source separation (Zhu et al., 2018), among others. This method estimates thermal response based on variation trends or frequency features. However, the separated thermal response may be confounded by other components. For example, while the wavelet transform method isolates thermal response based on its unique 24-h period, the response induced by traffic also exhibits a similar frequency due to morning and evening peak hours, thus leading to a mixing of wavelet-based thermal response with traffic load-induced components. In addition, since temperature data are routinely measured in practice, the response-only method fails to fully capitalize on these data.

The second class is the temperature-response model, which utilizes temperature data to estimate thermal response. Previous approaches in this category include mathematical models (Zhou et al., 2022; Zhu et al., 2024; Zhou & Sun, 2018), regression methods (Ding et al., 2017; Kromanis and Kripakaran, 2017; Wang et al., 2019), finite element models (Tome et al., 2018; Xu et al., 2019), and others. However, conventional temperature-response models often struggle to account for time-lag effects. Recently proposed deep learning methods show promise in addressing time-lag effects, particularly when applied to strain and deflection data. Zhao et al. (2023) used bidirectional long short-term memory regression network to predict temperature-induced strain field by learning distributional feature parameters. Yue et al. (2022a) took advantages of Stack-LSTM-CNN network to estimate temperature-induced deflection, which achieved a ultra-high precision. Yue et al. (2022b) established an LSTM network to capture nonlinear features between temperatures and deflections for a cable-stayed bridge, where the average error of the LSTM digital regression model is 1.4%. Meng et al. (2024) compared the prediction performance of the LSTM, Bi-LSTM and Transformer variant for temperature-induced deflections, where the Transformer-variant network achieved superior prediction performance. In practice, the reliability of estimation using deep learning methods depends not only on the algorithms employed but also on the types of structural response and the resolution of sensors used for measurement. To the best knowledge of the authors, these issues has not been well discussed in previous literatures.

In this study, a deep learning network, specifically the Long Short-Term Memory (LSTM) network, is constructed to estimate temperature-induced strains and deflections for the Queensferry Crossing, located in Scotland, UK—a triple-tower cable-stayed bridge. Initially, the cable-stayed bridge and its SHM system are introduced to provide background information. Subsequently, the temperature field of this iconic structure is analyzed based on field-measured temperature data. Finally, the LSTM network is employed to perform estimations of temperature-induced strains and deflections within the bridge structure.

The Queensferry crossing and its SHM system

The Queensferry Crossing, formerly referred to as the Forth Replacement Crossing, stands as one of the Forth Bridges—Figure 1—serving as crucial arteries within Scotland’s transportation network. This architectural marvel is a triple-tower cable-stayed bridge boasting a main span of 650 m and towering towers standing at 207 m. With a significant investment totaling 1.35 billion pounds, the construction of the Queensferry Crossing was completed, and it was opened to public traffic in August 2017. Situated adjacent to the existing Forth Road Bridge, the Queensferry Crossing spans the Firth of Forth, connecting Edinburgh at South Queensferry to Fife at North Queensferry, serving as a vital link for the M90 motorway. Featuring two lanes in each direction, the bridge facilitates the passage of motor-cycles, cars, and heavy goods vehicles. Meanwhile, public transport, cyclists, and pedestrians utilize the Forth Road Bridge for their commute.

Figure 1.

Site plan of the Queensferry Crossing.

A comprehensive network of 2184 sensors has been meticulously devised and deployed across the expanse of the Queensferry Crossing. These sensors encompass various functionalities, including thermometers, corrosion sensors, Global Positioning System (GPS) units, anemometers, and weigh-in-motion systems, among others. The precise placement of these sensors on the Queensferry Crossing is depicted in Figure 2. Among these sensors, 357 thermometers are strategically positioned to capture both structural and ambient temperatures. In addition, 532 static and dynamic strain gauges have been meticulously installed to monitor strains in both concrete and steel components. Furthermore, a network of 19 GPS measuring points has been established to track deformations in the main girders and towers of the bridge. To ensure efficient data management, all monitored data is archived onto the Microsoft Azure server. Subsequently, the data is resampled to a frequency of 1 Hz and stored in cloud storage space. Notably, the sampling rate is intelligently adjusted to match the original frequency of the sensors under extreme environmental conditions, ensuring the integrity and accuracy of the monitored data.

Figure 2.

Sensor layout of the Queensferry Crossing.

Temperature distributions of critical sections, including girder sections, tower sections, and cable sections, are meticulously monitored using a series of Thermal Management Unit (TMU) type sensors, the placement of which is illustrated in Figure 3. TMU sensors are composed of fiber-optic sensors, data acquisition system, control unit and actuators, which are designed to track temperature fluctuations in bridge structures. Specifically, a total of 158 TMU-2 sensors are affixed to steel decks to capture steel deck temperatures, while 20 TMU-3 sensors are embedded into concrete towers to measure temperature data within the towers. Furthermore, 70 TMU-4 sensors are strategically positioned to monitor temperatures across concrete decks, and 56 TMU-5 thermocouples are installed within stay cables to measure internal cable temperatures.

Figure 3.

Thermometer layout of the Queensferry Crossing (a) Thermometer placement of the main girder section, (b) Thermometer placement of the tower section, (c) Thermometer placement of the cable section.

Dynamic strain gauges are employed for monitoring steel deck strains, comprising a combination of single gauges, pairs of gauges (consisting of two T-rosettes), and three-element rosettes. To ensure durability and accuracy, each strain gauge is equipped with a waterproof protection system, and a protective box is securely affixed over the gauge installation to provide mechanical protection. For monitoring girder deflections, a GPS system supplied by Topcon is utilized, comprising receivers, antennas, rover stations, and signal processing and analysis software. This GPS system is meticulously deployed to track and analyze girder deflections with precision and reliability.

Temperature field and thermal response analysis

Temperature fields of the Queensferry Crossing

Based on 1 year of field-measured temperature data collected from July 2020 to June 2021 from various sections of the Queensferry Crossing, temperature distributions are analyzed. The temperature data obtained from soffits, deck chords, concrete decks, bottom chords, tower shell sections, and stay cable sections are presented in Figure 4. Overall, all measured temperature data exhibit a similar variation trend. However, it is noted that the measured temperature data suffer from a data missing problem, with temperature data associated with tower shells experiencing a more pronounced issue. Specifically, approximately 63.79% of the temperature data for tower shells are absent over the 1-year span.

Figure 4.

One-year measured temperature data from the Queensferry Crossing (a) One-year measured deck chord temperatures, (b) One-year measured concrete deck temperatures, (c) One-year measured bottom chord temperatures, (d) One-year measured soffit temperatures, (e) One-year measured tower shell temperatures, (f) One-year measured cable section temperatures.

Furthermore, concrete components, such as concrete decks and concrete towers, exhibit larger temperature differences pertaining to the thickness direction compared to steel components, including soffits, deck chords, bottom chords, and cables. The temperature difference for concrete decks approaches 6°C, while the maximum temperature difference for concrete towers is approximately 4°C. In contrast, temperature differences for steel components consistently remain within 1°C. This disparity in temperature differences is attributed to the high thermal conductivity and thin thickness of steel components, which results in relatively smaller temperature differentials.

In summary, while monitoring temperature differences for steel girder sections like soffits, deck chords, bottom chords, and cable sections may not be strongly recommended due to their relatively small temperature difference magnitudes. Conversely, temperature difference measurements for concrete deck and tower sections are strongly advised, with the recommendation to install five or more sensors depending on their thickness. This strategic deployment of sensors will provide more comprehensive and accurate temperature data, facilitating effective monitoring and management of thermal behavior in concrete components of the structure.

Thermal response analysis

In the analysis of thermal response for the Queensferry Crossing, two types of structural responses, namely strains and deflections, are considered. As an illustrative example, the strain gauge affixed to the surface of the inclined member associated with the south mid-span girder section is examined. In addition, for the discussion, deflection data obtained from GPS devices located at the south mid-span are utilized. Furthermore, temperature measurements of the inclined member subjected to the south mid-span are incorporated into this section to provide a comprehensive understanding of the thermal response of the structure.

Figure 5 presents relatively long-term strain, deflection, and temperature data. To facilitate a clearer understanding of the correlations between temperatures, strains, and deflections, Figure 6 depicts temperature-strain and temperature-deflection relationships. Upon analysis of seasonal scale datasets, a notable negative linear correlation between temperatures and strains is observed. Conversely, there appears to be a weak correlation between temperatures and deflections. 1-year temperature and deflection data are used in Figure 6(b), which indicate a global weak correlation between temperatures and deflections. The weak correlation between temperatures and deflections results from the measurement errors and time-lag effects.

Figure 5.

Relatively long-term strains, deflections and temperatures.

Figure 6.

Temperatures against strains and deflections.

In Figure 7, short-term strain, deflection, and temperature data are presented. To represent temperature-induced components, 10-min mean strains are utilized (Yue et al., 2021). A significant negative correlation is observed between temperature and strain measurements. In addition, noticeable time-lag effects are evident between temperature and strain, indicating that the impact of temperature changes on strain does not occur instantaneously. Specifically, when strain reaches its peak, temperature does not necessarily reach its trough simultaneously. The time differences between peaks and troughs of strains and temperatures are defined as time-lag effects in this study. Due to the solar radiation received by the bridge deck, the upper part of the main girder typically heats up faster, leading to a structural response occurring earlier than in the lower part. Meanwhile, 10-min mean deflections monitored by GPS are significantly influenced by high-frequency components (Figure 7(b)). In light of this, wavelet transform is introduced to suppress high-frequency signal components for GPS-measured deflection data (Xu et al., 2020). Similar to strain data, deflection measurements also exhibit time-lag effects. It is noted that the majority of previous thermal response estimation methods fail to account for these time-lag effects.

Figure 7.

Short-term strains, deflections and temperatures.

Indeed, the analysis reveals notable differences in the correlation between temperature and strain compared to temperature and deflection data. For strain data, a negative linear correlation with temperatures is observed consistently across both relatively long-term and short-term measurements. This observation aligns with the understanding that strain, being a local structural response, is particularly sensitive to structural temperatures in adjacent areas. On the other hand, for deflection data, there is a rare correlation with temperatures based on relatively long-term measurements, while a negative linear correlation with temperatures is observed within the short-term window. This discrepancy underscores the complexity of deflection as a global structural response. Unlike strain, deflection is influenced not only by structural temperatures in nearby areas but also by temperature differences across the structure. For example, the same uniform structural temperature of 10°C at midnight in summer and at noon in winter would yield significantly different temperature-induced deflections due to variations in structural temperature differences. This highlights the importance of considering temperature differentials in understanding and predicting girder deflections accurately.

Deep-learning driven thermal response estimation

LSTM network

To accurately estimate temperature-induced structural responses, the utilization of the LSTM network is deemed appropriate due to its inherent capability to recognize time-dependent patterns (Xu et al., 2023a). Given the presence of time-lag effects between temperatures and structural responses, the LSTM network is particularly suited for this task. Hence, in this study, the LSTM network is employed to effectively estimate temperature-induced structural responses, enabling the consideration of temporal dependencies and addressing the inherent time-lag effects in the data.

The LSTM neural network, introduced by Hochreiter and Schmidhuber in 1997, represents a departure from conventional artificial neural networks. At the heart of the LSTM network lies the memory block, as depicted in Figure 8. This memory block comprises input and output gates, which serve to regulate the activation of inputs and outputs, respectively. The memory unit within the LSTM network is instrumental in preserving temporal states. Notably, the core of the memory unit features a recurrently self-connected linear unit known as the constant error carousel. This enables multiplicative gates to learn to open and close, thereby circumventing the vanishing error issue by maintaining the network error constant. Moreover, to prevent internal cell values from growing infinitely, the forget gate is incorporated within the memory block. This innovative addition allows the memory block to automatically reset when the information flow becomes outdated, ensuring the stability and efficiency of the LSTM network.

Figure 8.

Basic cell architecture of the hidden layer in the LSTM networks.

The LSTM network cell is described by the following equations:

Forget gate:

f_{t} = σ (W_{f} [h_{t - 1}, x_{t}] + b_{f})

(1)

where

f_{t}

is the activation vectors of the forget gate, σ denotes the Sigmoid function, W_f denotes the model parameters of the input vector and the hidden layer vector for the forget gate, and b_f denotes the model parameters of the input vector and the hidden layer vector for the forget gate.

Input gate:

i_{t} = σ (W_{i} [h_{t - 1}, x_{t}] + b_{i})

(2)

where

i_{t}

is the activation vectors of the input gate, W_i denotes the model parameters of the input vector and the hidden layer vector for the input gate, and b_i denotes the model parameters of the input vector and the hidden layer vector for the input gate.

Candidate memory unit:

\tilde{C_{t}} = \tanh (W_{c} [h_{t - 1}, x_{t}] + b_{c})

(3)

where W_C denotes the model parameters of the input vector and the hidden layer vector for the memory unit, and b_C denotes the model parameters of the input vector and the hidden layer vector for the memory unit.

Current time memory unit:

C_{t} = f_{t} \cdot C_{t - 1} + i_{t} \cdot \tilde{C_{t}}

(4)

where C_t denotes the activation vectors of the memory unit at time t, and ‘·’ denotes the dot production.

Output gate:

o_{t} = σ (W_{o} [h_{t - 1}, x_{t}] + b_{o})

(5)

where o_t is the activation vectors of the onput gate, W_o denotes the model parameters of the input vector and the hidden layer vector for the onput gate, and b_o denotes the model parameters of the input vector and the hidden layer vector for the onput gate.

Output:

h_{t} = o_{t} \cdot \tanh (C_{t})

(6)

The general strategy for training, validation, and testing using the LSTM network is illustrated in Figure 9 (Xu et al., 2023b). The total dataset is partitioned into three sections: the training dataset, the validation dataset, and the test dataset, utilizing a predetermined ratio. Typically, a common split ratio of 0.8:0.1:0.1 is adopted for training, validation, and testing, respectively. However, it’s important to adjust this ratio based on factors such as the total data size and the available graphics processing unit memory of the computing devices being used. Subsequently, three main flows are involved in the strategy to train, validate, and test the network model. This process ensures that the LSTM network is effectively trained on a subset of the data, validated to optimize its performance, and finally tested to evaluate its effectiveness in predicting temperature-induced structural responses.

Figure 9.

The general strategy for training, validation and testing by using LSTM networks.

The training process is crucial for enhancing the performance of the LSTM network. Initially, a training batch is randomly selected from the training dataset to form the network input. Subsequently, the network parameters are iteratively updated based on the loss computed between the estimated and actual values. Upon reaching a predetermined step value, denoted as N, the training for the current batch concludes, and the validation process commences. It is important to note that the training process continues cyclically until either an early stopping criterion is met or the predetermined epoch, denoted as M, is reached. Early stopping is a regularization technique employed to prevent overfitting by halting the training process when the model’s performance on a validation set ceases to improve or even begins to deteriorate. This ensures that the model does not become overly specialized to the training data and maintains its ability to generalize to unseen data.

The validation process serves as a crucial mechanism for evaluating the effectiveness of the trained model. Following each batch training, the trained network model is evaluated using the validation data. Similar to the training process, a validation batch is initially randomly generated from the validation dataset. Subsequently, the loss associated with this batch model is compared with the loss corresponding to the preceding batch model. If an improvement in the loss is observed, indicating better performance, the current batch model is retained; otherwise, the model is discarded. This iterative process aids in mitigating overfitting and ensures the generalizability of the model. Overfitting is a common challenge in machine learning where a model learns the training data too well, resulting in poor performance on unseen data due to its inability to generalize beyond the training set. By regularly evaluating the model’s performance on validation data and adjusting model parameters accordingly, overfitting can be effectively mitigated, ensuring the model’s ability to generalize to new, unseen data.

The testing process is employed to definitively assess the predictive performance of the model. Unlike the training and validation processes, where only subsets of the data are used, the entirety of the test dataset is utilized as the input for the network. The loss computed between the predicted and actual values serves as the metric for evaluating the predictive performance of the trained LSTM model. Given that the test dataset is never incorporated into any training or validation phases, it presents entirely novel data to the network model. Consequently, the performance of the network model, as evaluated using the test dataset, provides a reliable measure of the model’s predictive capabilities in this strategy. This ensures that the model’s performance is evaluated on unseen data, providing a realistic assessment of its ability to generalize beyond the training and validation datasets.

Data preparations

The dataset for the LSTM model has been prepared, utilizing temperatures recorded from various parts of the deck and cable as input to the network model. Given the high thermal conductivity inherent to steel and the slender nature of steel plates, the observed temperature differential was found to be negligible. Consequently, for temperature readings from the soffit, as well as the deck top and bottom chords, a singular channel suffices.

In contrast, for the concrete deck temperature, a relatively large temperature difference across the section was observed due to the low thermal conductivity. This necessitated the inclusion of data from all five sensors located at the same locale. Similarly, for the cable temperatures, readings from both the lower and upper end sensors were incorporated to account for the temperature difference induced by their spatial separation.

Considering the slow variation trend of temperature data, the input data are resampled to 1/600 Hz to optimize computing cost (Yang et al., 2018; Yue et al., 2021; Zhou et al., 2020a; Zhou and Sun, 2018). This resampling strategy ensures that the temporal resolution of the input data remains sufficient for capturing the underlying temperature dynamics while reducing computational resources required for processing.

For the output data, strain and deflection data have different strategies. For strain data, 10-min mean is used as output data since 10-min mean is adequate to indicate the temperature-induced component in strains. Since 10-min mean deflections are insufficient to suppress high frequency noise components, wavelet transform is employed to separate temperature-induced deflections. In this regard, the wavelet-transformed deflections are taken as output data for the LSTM network.

Results and discussions

To account for the time-lag feature, a window size equivalent to 6 hours was implemented. Each case underwent training 10 times to mitigate potential biases in the results. The model architecture comprised a three-layer LSTM structure, with each layer consisting of 32 neurons.

To assess the estimation performance of the LSTM network, two indexes are proposed herein: the correlation coefficient (R) and the mean square error (MSE). The correlation coefficient (R) evaluates the variation trend between the estimated values and the ground truth, while the MSE measures the distance between the estimated values and the ground truth.

Based on the findings depicted in Figure 6(a), a notable negative linear correlation is evident between temperature and strain. To estimate temperature-induced strains, a linear regression analysis is applied. The results of this regression are illustrated in Figure 10, with the linear fitting function represented as follows:

S = - 2.23 T + 56.2

(7)

where S is the variable strain, and T is the temperature.

Figure 10.

Linear fitting for the temperature and strain data.

The linear fitting function is utilized to estimate the strains, with results presented for both relatively long-term (5 months) and short-term (48 hours) perspective in Figure 11. From a long-term viewpoint, the linear fitting yields satisfactory outcomes, exhibiting a high correlation coefficient (R = 0.97) and a mean square error (MSE = 4.45). However, upon closer examination of the short-term time window in Figure 11(b), a noticeable discrepancy between the estimated values and the ground truth is apparent. Furthermore, the linear regression method struggles to adequately resolve the time-lag effects. The reason is that the linear regression model cannot capture nonlinear features between temperatures and structural response.

Figure 11.

Linear fitting results for the strain.

Considering the limitations of the linear regression method, the LSTM network is employed to estimate temperature-induced strains, as depicted in Figure 12. From a long-term perspective (30 days), the estimation results demonstrate satisfaction, with a high correlation coefficient (R = 0.98) and significantly reduced mean square error (MSE = 0.89). While the correlation coefficient R does not exhibit a significant improvement compared to linear regression, the remarkable decrease in MSE from 4.45 to 0.89 underscores the LSTM network’s ability to capture finer details of the measurements, thereby narrowing the gap between estimated values and ground truth. Zooming into a short-term time window (72 hours) in Figure 12(b), the LSTM network demonstrates mitigation of time-lag effects to a certain extent. However, notable extreme estimation errors are observed for the initial few samples, denoted as boundary side effects in Figure 12(b). This occurrence is attributed to the LSTM network’s dependence on a sequence of temperature data, wherein limited historical temperature data are available for the initial samples.

Figure 12.

LSTM network estimation results for the strain.

According to Figure 6(b), a weak correlation is observed between temperature and deflection. To estimate temperature-induced deflections, the LSTM network is employed, with the estimation results depicted in Figure 13. Upon comparison with the LSTM estimation results for strain, it is noted that the performance in deflection estimation is slightly inferior. Specifically, the correlation coefficient (R) for deflection estimation is 0.825, whereas R equals 0.98 for strain estimation. Two primary reasons contribute to this observed discrepancy:

(1) Complexity of Deflection Response: Unlike strain, deflection may exhibit a more complex response to temperature variations, potentially involving non-linear relationships or other external factors influencing deflection behavior. This complexity poses challenges for the LSTM network to accurately capture and predict deflection dynamics.

(2) Data Characteristics: The nature and characteristics of the data concerning deflection may differ from those of strain, leading to variations in the LSTM network’s ability to effectively learn and generalize patterns. Factors such as data sparsity, noise, or inadequate representation of relevant features in the dataset could impact the network’s performance in deflection estimation.

Figure 13.

LSTM network estimation results for the deflection.

Despite these challenges, the LSTM network still provides valuable insights into temperature-induced deflections, albeit with a slightly diminished performance compared to strain estimation. Further refinement of the network architecture or exploration of additional data preprocessing techniques may help address these challenges and improve the accuracy of deflection estimation in future studies.

Concluding remarks

The study introduced a novel deep learning-driven methodology utilizing LSTM networks to estimate temperature-induced strains and deflections, leveraging field-measured temperature data from the Queensferry Crossing, a cable-stayed bridge. The discussion delved into the temperature distribution within the bridge structure to provide context for the estimation process. Subsequently, the estimation performance of the LSTM network concerning both strain and deflection was thoroughly analyzed and compared. In summary, the conclusions drawn from this study are as follows:

(1) The temperature differential between steel components tends to be smaller compared to concrete elements. While monitoring temperature variances in steel components such as the girder soffit, deck chord, girder bottom chord, and cable section is not deemed imperative due to their relatively minor temperature discrepancies, it could still be considered given their limited temperature variation magnitudes. Conversely, it is recommended to monitor temperature differences in concrete sections such as the deck and tower sections. This can be achieved by installing five or more sensors, the number depending on the thickness of the concrete sections.

(2) The LSTM network demonstrates higher estimation accuracy compared to linear regression. It excels in capturing measurement intricacies, evident in the significantly smaller mean square error (MSE) between estimated values and ground truth compared to the linear regression method. In addition, the LSTM network exhibits the ability to mitigate time-lag effects to some extent, further enhancing its effectiveness in modeling complex relationships within the data.

(3) The estimation performance of the LSTM network is influenced by both the type of structural response and the resolution of the sensors employed. Typically, local structural responses such as strains exhibit superior performance compared to global responses like deflection. Furthermore, employing sensors with higher resolution tends to yield more accurate estimation results.

In conclusion, this study underscores the potential of deep learning-driven methodologies in enhancing our understanding of structural behavior and facilitating proactive maintenance and management of critical infrastructure assets such as cable-stayed bridges.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 801215 and the University of Edinburgh Data-Driven Innovation Programme, part of the Edinburgh and South East Scotland City Region Deal, and Screening Eagle, AG, Zurich, Switzerland; the National Natural and Science Foundation of China grant No. 52308150.

ORCID iD

Xiang Xu

Data availability statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.*

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