suspension design research paper

Structural characterization of a suspension bridge by mapping the temperature effects on strain response based on neural network models

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• Published: 23 October 2024

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• Fabiana N. Miranda   ORCID: orcid.org/0000-0002-4792-8397 1 , 2 ,
• Juan Mata   ORCID: orcid.org/0000-0002-8889-7945 2   na1 ,
• João Pedro Santos   ORCID: orcid.org/0000-0003-4960-9653 3   na1 &
• Xavier Romão   ORCID: orcid.org/0000-0002-2372-6440 4   na1  

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Mapping the structural responses based on main loads to characterize signature of complex structures with high-dimensional features is a determinant factor for structural health monitoring (SHM). Current technological advances contribute to the optimization of data analysis, aiming to make the process less demanding in terms of time and computational demand. Machine learning (ML) models became popular due to their capacity to estimate structural behaviour based on the measurements gathered by the SHM systems. This work proposes a methodology supported by Neural Networks (NN) for the characterization and prediction of the structural behaviour based on thermal loads and structural responses. By comparing the observed values and predicted outcomes from the NN, it is possible to identify measuring errors, new trends or pattern variations that need further assessment. A sensitivity analysis is also proposed to confirm the model robustness and to characterize the influence of the temperature on the structural responses. The case study is the 25 de Abril’s bridge, located in Lisbon, Portugal.

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1 Introduction

The occurrence of structural damage is intrinsic to civil engineering structures, and it is likely to propagate due to environmental or mechanical factors. In [ 43 ], the authors define damage as a change in a structural system that affects the current or future performance concerning both, structural safety and serviceability. Thus, while damage occurrence implies a structural change, the occurrence of this change may not necessarily imply damage that compromises the structural performance. Likewise, damage is not always an indicator of a complete failure, but can signal the occurrence of a deterioration of the functionality [ 3 ]. As mentioned by [ 15 ], the scientific community has adopted a damage identification hierarchy as a sequence of different levels of knowledge about damage and how it affects the structure. In this sense, damage identification can be divided into three areas: damage detection, damage diagnosis, and damage prognosis. Furthermore, characterizing the nature of any identified variation as a one-off event or something related to a trend in the data is particularly relevant. In case the variation corresponds to a trend, it is important to characterize the relationship between the observed structural responses to verify the presence of any anomalous behaviour. The monitoring of a structure to detect variations and assess the life cycle of the structure consists of a powerful tool for damage identification and performance evaluation of engineering structures. The process involves the observation and evaluation of a structure over time using periodically sampled measurements from a sensing system. Several civil engineering structures have been provided with structural health monitoring (SHM) systems in the last decades, and, according to [ 50 ], those systems are implemented to ensure structural and operational safety by means of (near) real-time information that can help to detect anomalies in loading and response at an early stage.

SHM techniques can follow inverse or forward approaches for detecting damage. The former, also called model updating, are based on the fitting of analytical or numerical responses to experimental data with the objective of inferring structural quantities that cannot be directly measured. Forward SHM approaches rely, instead, on pattern recognition techniques for extracting sensitive information from acquired data on-site [ 45 ]. Forward approaches, also called data-driven approaches, do not require the development of numerical or analytical models to be fitted with in situ data. Those techniques aim at extracting sensitive information from the measured time series, using statistical learning methods [ 47 ]. Because SHM systems are operated in harsh and noisy environments, the occurrence of abnormal data is inevitable. The large variation in extracted features from massive SHM data causes conventional detection techniques to perform poorly. In addition, detection techniques that are focused on a single type of variation frequently miss other types of changes in the structure [ 5 ].

Structural condition assessment based on changes in the structural response has been practiced in a qualitative manner for decades. Sensing techniques were initially proposed by [ 19 ] to detect structural change or degradation using intelligent monitoring systems that can allow structures to operate at the margin of safety without expended long periods of inspection. The advances in various branches of technology, such as sensing instrumentation, signal acquisition and transmission, data processing and analysis, and numerical simulation and modeling, are now allowing the development of strategies that benefit from the technological precision for accurately evaluating the structural health of civil structures using real-time monitored data [ 10 ].

In particular, several improvements in computational power and advancements in chip and sensor technology have enabled the use of Machine Learning (ML) techniques in engineering applications. According to [ 3 ], those techniques are focused on the development of intelligent algorithms capable of acquiring knowledge automatically from the available data with the objective of providing machines with a human-like ability to learn. However, the formulation of reliable and efficient methodologies for data analysis and signal processing is still a challenge for SHM [ 4 ]. With the recent technological advances, a relatively large number of sensors and sensor networks can now measure large volumes of response data. In this sense, data-driven ML techniques have been proposed by different authors as an efficient way to assess the structural health condition of host infrastructures [ 3 , 5 , 25 , 33 , 44 ]. However, even considering the great advantages of ML tools applied to the assessment of structures, any tool, must be validated by experts in structural behaviour.

The present study proposes a structural behaviour characterization methodology based on neural networks, model validation, and residual analysis for any variation identification in a specific zone of a structure. Since most physical relations between environmental or operational actions and structural responses are linear [ 42 ], applying regression models is common in SHM data analysis. In particular, Multiple Linear Regression (MLR) is currently used to interpret SHM data of operational structures, such as bridges worldwide, due to its capacity to explain the conditional distribution of responses and predictors. However, MLR has limitations when explaining the nonlinearity of the structural behaviour. For this type of problem, NNs have a better generalization capacity and can accurately represent the observed patterns of structural behaviour. MLR models use empirical functions to quantify the effect of the independent variables on the structural response. In NN models, the relationship between variables is established by the architecture and the synaptic weights of the network by taking advantage of the parallel processing of input variables. To enhance this capacity of NN, the authors propose the development of three models, whose integrated use will allow the specialist to validate the quality of each model in representing the observed structural behaviour as well as the database that is used, thereby increasing the understanding of the structural behaviour and the confidence in decisions regarding structural safety. NNs have demonstrated flexibility and superior predictive capacity, particularly in predicting structural behaviour for extreme temperatures [ 31 ]. This is in stark contrast to the performance of MLR models. By adopting NN models, as proposed in this work, specialists can expect better model performance and residuals with a lower standard deviation, further underscoring the advantages of NN models.

The case study addressed in this research is the 25 de Abril bridge over the Tejo River in Lisbon, Portugal, which is monitored by an SHM system capable of measuring different physical quantities corresponding to fast loads, slow loads, and structural responses. The prediction model is built with data, provided by the Portuguese National Laboratory of Civil Engineering (LNEC), gathered by the SHM system installed in the bridge, considering 5 years of data with an hourly frequency of measurements. This dataset allows to understand the governing thermal behaviour of the bridge and to establish a baseline for the structural characterization.

The implementation of moving average charts and the assessment of the standard deviation of the residuals of the models provide sensitivity measures about the results of the analysis that can be used for verifying and validating the predictions.

2 Theoretical background: machine learning approaches in structural health monitoring

The transition from traditional methods to ML represented an improvement in: (i) sensitive feature extraction from the monitored responses [ 12 , 26 , 34 ], (ii) characterizing the pattern of the structural responses [ 12 , 35 , 39 ], and (iii) classifying the extracted features of civil engineering structures [ 8 , 13 , 21 ].

In [ 52 ], the authors state that SHM is a multilevel and multi-faceted method capable of dealing with data acquisition from single and multiple sources, and further data processing. To have a more robust and confident decision, multiple sensors are installed in the structure to be monitored, and thus, data fusion becomes an indispensable step. ML-based approaches applied to SHM data require extracting damage-sensitive features that later will be used as the input of the ML model. According to [?], robustness refers to the ability of a system, model, or entity to maintain a stable and reliable performance across a broad spectrum of conditions, variations, or challenges, demonstrating resilience and adaptability in the face of uncertainties or unexpected changes.

Lately, SHM technology reflect the variations of structural parameters with the change of structural responses. Many field monitoring data indicate that structural responses are not only related to the characteristics of the structure, but also to the environmental conditions, such as humidity, wind, and, most importantly, temperature [ 18 ]. Current procedures in bridge SHM have a considerable number of limitations and prevailing uncertainties due to non-stationary variability sources associated with operational and environmental conditions. Varying operational conditions include live loads such as traffic loads, speed of operation, and changing excitation sources. Varying environmental conditions are manifestations of weather in the form of temperature, humidity, wind, rainfall, and snow. Note that there are also other sources of variability in the measurement exist at the data acquisition stage such as instrumentation, random noise, estimation methods, and vibration source [ 15 ]. According to [ 6 ], thermal loads due to temperature changes are an important factor in the serviceability limit state design of bridges. Daily and seasonal temperature changes result in thermal expansion and contraction cycles, which bridge designs need to cater for. The influence of ambient temperature will interfere with the estimate of the effects of other factors. To make a more accurate assessment, it is necessary to forecast and separate the temperature-contributions parts of the overall responses [ 18 ] Two main SHM approaches based on ML models are cited by Farral and Worden in [ 14 ] for response modeling of structures: i) the protective approach, which refers to the case when damage-sensitive features are used to identify impending failure, that leads to the adoption of changes in the normal operation of the system to avoid catastrophic failure and; ii) the predictive approach, that, instead, is applied to the cases in which trends in data are identified to predict when the damage will reach a critical level. In this latter case, cost-effective maintenance planning is needed, and ML strategies are useful for this purpose. To identify such patterns, a learning process through the implementation of algorithms is proposed. Such a process can follow a supervised or an unsupervised learning [ 14 ] as well as a self-supervised learning [ 29 ], being the supervised learning the most used approach.

Data-based models based on ML methods learn from past observed behaviour. Consequently, the obtained models are valid for the “domain/range” of the observed response and the corresponding loads used for the model development. From the different ML models, artificial neural networks (ANN) have caught the attention of the scientific community since the 1990s [ 19 ], due to their ability to learn the pattern of structural behaviour in large infrastructures with a good capacity for generalization [ 1 , 13 , 27 , 31 , 32 , 40 , 46 , 48 ]. An ANN model computes a function of the inputs by propagating the computed values from the input neurons to the output neurons using different weights as an intermediate parameter [ 1 ]. A Multilayer Perceptron (MLP) is a supervised feed-forward neural network that has neurons arranged in layers and is the most widely used model for cognitive tasks such as pattern recognition and function approximation [ 31 ]. The specific architecture of feed-forward networks assumes that all nodes in one layer are connected to those of the next layer, and the input layer transmits the data to the output layer going through a set of hidden layers that perform computations that include refining the weights between neurons over many input–output pairs to provide more accurate predictions. Weights after training contain meaningful information, whereas before training, they are random and have no meaning. The back-propagation delta learning rule algorithm, one of the most famous training algorithms for the MLP [ 7 , 38 ], is a gradient descent technique intended to minimize the error or cost function, which adjusts the weights by a small amount at a time. The cost function C, usually considered, is defined by the mean square error.

MLP architecture where x represent the input layer, l the hidden layer, and L the output layer, adapted from [ 31 ]

The MLP (Fig.  1 ) learns by an iterative process of a weight adjustment that enables the correct learning using the training data, so that, in a testing phase, it predicts the unknown data. Those weights w are located in each connection between the input layer x with N neurons to the hidden layer l with Q neurons. The first layer receives the inputs, and the last layer produces the outputs. The middle layer is called a hidden layer, and, within it, the information is constantly feed-forwarded from one layer to the next one having two values associated: the input value and the weight. In mathematics and programming, weights are showed in a matrix format W , where the number of columns contains the input dimensions ( N ) and the number of rows are the output dimensions ( Q ). Another weight associated with the network is the bias b . Bias is attached to every layer in the network, except for the input layer. Activation functions are applied to each of the neurons at the hidden layer and in the output layer. The activation function will map the linear combinations of the inputs and weights to the following layer, in the case presented in Fig.  1 from the input layer to the hidden layer and, in a further step, to the final output of the network, L .

For regression problems, such as those found in the framework of SHM, the activation functions must be differentiable, so that any non linear model built from them can be derived with respect to the weights [ 7 ]. In this case, in the activation function from the input layer to the hidden layer, f ( x ), a logistic sigmoid is adopted (Eq.  1 ), and in the activation function from the hidden layer to the output layer, g ( x ), a linear function is adopted [ 42 ]

As mentioned before, ANNs, and particularly MLPs, are widely used for pattern recognition and variation identification tasks. Applications for damage prediction and variation identification can be found in civil engineering such as in the case of [ 24 ] where the authors consider the difference of ratios of the mode shape components from damaged and undamaged finite element models as the input to the neural networks of the ANN. Other examples include [ 46 ] where the authors use NN to predict seismic damage in reinforced concrete bridges, and [ 37 ] where the authors developed an autoencoder capable of reconstructing the contamination dynamics over real and artificial water distribution networks. In all these examples, ANNs are seen to be versatile systems capable of changing their structure based on inner or outer information, thus being able to be rearranged to obtain different patterns in data or to model the multiple relationships between inputs and outputs. This is one of the reasons why ANNs have been used to solve many engineering problems over the past few years.

As referred by Figueiredo [ 16 ], non-stationary sources of variability present in the structures response can be associated with operational and environmental conditions and may lead to variations in structural behaviour. Varying operational conditions include live loads (e.g., traffic loads on bridges), speed of operation, and changing excitation sources. Varying environmental conditions include thermal effects, wind-loading, and moisture content [ 16 ]. These variations can be equal to or greater than the ones that occur due to damage and it is fundamental to interpret them to prevent undesirable consequences [ 13 ].

In structural dynamics and earthquake engineering problems, neural networks have experienced great advancements, which can be considered for determining the static and dynamic parameters of structures [ 46 ]. In the particular case of bridges, response measurements comprise the effects of several types of loads including vehicular traffic and ambient conditions. According to [ 22 ], an important step in the interpretation of these measurements involves characterizing the influences of the individual load components on the collected measurements. Long-term monitoring studies have illustrated that daily and seasonal temperature variations have a large influence on the structural response of bridges. In [ 20 ], the authors confirmed that environmental effects cause changes in physical parameters, thereby inducing changes in modal parameters, which may yield a false indication of damage when applying vibration-based damage detection algorithms. In [ 9 ], the authors observed that temperature-induced stresses on long-span bridges create responses that are very difficult to model due to unexpectedly high levels of stress. In [ 23 ], by considering that the daily periodic air temperature change causes the characteristic global thermal deformation of a long-span cable-stayed bridge, the authors proposed a damage-sensitive feature extraction method through ARIMA models from Global Positioning System (GPS) measurements of those deformations, assuming that they are sensitive to changes on the global structural properties.

From these examples, the design of methodologies to improve damage identification including environmental and operational conditions can be seen to be an area of active research interest, motivated by the fact that there are an increasing number of new civil engineering, aeronautics, or astronautics designs that include increasingly more complex structures subjected to variable environmental and operational conditions that need to be assessed [ 2 ].

Despite the design and maintenance codes and methodologies that are available, civil engineering structures deteriorate over time and structural health is affected by operational and environmental factors, including normal load conditions, current and future environments, and expected hazards during the lifetime [ 17 ]. All these factors are variables with uncertainties, and to ensure the safety and serviceability of the structure, reliable structural health assessment, and continuous monitoring are essential.

3 Proposed methodology

Real-time monitoring can be effective for structural safety control activities and to facilitate decision-making regarding interventions in the monitored structure. However, real-time diagnosis is usually sensitive to environmental and operational effects. In this sense, the acquired data need to be processed, so that pattern recognition techniques can be employed.

The main aim of this study is to define a baseline for the observed response pattern in a suspension bridge. For this purpose, the analysis based on ANN models is carried out considering first the response at one point of a beam at the middle section of the bridge (see Sect.  4 ) as a function of temperatures measured in the same section. A second analysis is then performed considering the responses at four points of the beam’s stiffening truss at that section (see Sect.  4 ) as a function of all the loads and, finally, a third analysis is performed considering the response at one point of the beam as a function of the responses in the other three beams (see Sect.  4 ). Using these different models involving different inputs is also strategic to identify false positives. This approach facilitates their detection, therefore increasing the overall robustness of the model.

The ANN models are developed with data gathered by the structural monitoring system during a time period where the structural behaviour is considered to be consistent with normal conditions. Interpretations about the structural behaviour are inferred from the model results and from the residual analysis, which describes everything that was not explained by the prediction model at first and contains information about errors, from the measurements or another unknown effect. For the residual analysis, a moving average with a window of a week and a timestep of 1 h is suggested in the current research. The standard deviation of the residuals is proposed to define a smoothed pattern behaviour to overcome the inherent randomness introduced by the traffic.

Two aspects are fundamental to support the informed structural assessment proposed herein; (i) the redundancy of the models, which are able to provide a general overview and a local detailed diagnosis about the structural behaviour and its evolution, and (ii) the validation of the analysis based on the accuracy and suitability of the models to represent the structural behaviour pattern through a sensitivity analysis [ 32 ] that is able to describe the learned pattern (Fig.  2 ).

The main steps of the proposed methodology, namely related to the prediction model, the residual analysis, and the sensitivity analysis, are presented ahead.

Proposed methodology for the observed structural behaviour characterization based on ANN models

Prediction model By following a data-driven approach, the development of numerical or analytical models to be fitted with in situ data may not be required in a day-to-day basis. However, numerical models are very important for verification of damage scenarios that may be identified. Data-driven models rely on the discipline of ML or, more specifically, on the pattern recognition aspects of ML. The idea is that one can learn relationships from data, namely to recognize the relationship between the structural response and the temperatures, and also between the observed responses at the section under study. For this purpose, three different analysis models are proposed, based on the relationship between main loads (L) and structural responses (R) as referred in Table  1 . In this sense, when the pattern recognition is performed during a time period that corresponds to a situation where the structure behaved normally, it will be possible to identify behaviour patterns that are different from the learned ones when new data are collected. Changes in the observed pattern behaviour can be identified, and experts, based their knowledge and in a deep analysis, may decide if the changes observed are related to a potential damage scenario or to another type of scenario. In this context, the proposed models provide the ability to verify if the observed pattern is maintained, and if not, a timely detection is possible. Multilayer Perceptron (MLP) Neural Network (NN) models have been adopted due to their capacity for pattern recognition. The fully connected architecture of the MLP can consider several outputs (structural responses) in the same model. As such, the model can learn the response pattern considering all the responses at the same time, providing predictions that consider a broader observed pattern. For the learning strategy, the learning data must be divided into three subsets: the training, the cross-validation, and the testing sets. The training subset is used to build the learning model. The MLP might use different hyperparameters for the learning rate, and the same subset is used to build the model. The cross-validation subset is used for model selection and parameter tuning to select the best choice for the architecture and for the parameters of the NN model [ 1 , 31 ]. The cross-validation subset should be viewed as a strategic solution to find the NN with a better generalization capacity to avoid over fitting. Finally, the testing subset is used to test the accuracy of the final (tuned) model at the end of the process. In cases of complex and large available data sets, the biggest percentage of data must be dedicated to the training and cross-validation subsets. The root-mean-squared error (RMSE) is used herein to assess the quality of the predictions and is defined as follows:

where \(x_i\) describe the observed values and \(\hat{x}\) the predicted values. The resulting deviations between the observed and the predicted behaviour must be assessed with model validation. To address this issue, three different prediction models are developed herein that relate: (i) the structural responses (R), one at a time, to evaluate each response as a function of the main loads (L), (ii) several simultaneous structural responses as a function of the main loads, and (iii) structural responses as a function of other structural responses in the same section (see Table 1 ). The set of variables that remain after the outcome of the models, known as residuals, include the variance of the data that is not explained by the predictions. Based on the analysis of these residuals, it is possible to detect variations that can point to a trend or a change in the structural behaviour.

Residual analysis Model deviations are generally systematic, producing consistent predictions under identical conditions. The suitability of a model depends on the objective and can be evaluated using different parameters such as the coefficient of determination or the standard deviation of the residuals. The residuals include different contributions, such as measurement uncertainties, errors, and factors unexplained by the model. Although the pattern representing the behaviour of residuals is random, this pattern is expected to remain the same if the behaviour of the structure does not change. Therefore, the occurrence of any new or distinct pattern in the residuals suggests a situation that requires reassessment. In light of this, the detection of the deviation between the observed and the predicted data that is not identified by the prediction model is one of the main objectives of the residual analysis. To reduce the intrinsic randomness of the data and identify as early as possible any deviation over time, a moving average analysis with a time window of 1 week and a time step of 1 h is proposed. A moving average control chart of the residuals ( m . a . r .) is a type of memory control chart based on an unweighted moving average of the residuals r , that smooths out the data and allows the analysis of the weekly evolution of the moving average. The m . a . r . is defined as follows [ 28 ]:

where w is the width of the moving average at time i . For periods \(i < w\) , the moving average width must be smaller than this value. For these periods, the average of all observations up to period i defines the moving average. Four different m . a . r ., related to 6, 12, 24, and 168 records, corresponding to a quarter of a day, a half of a day, an entire day, and an entire week are potential options to be considered in this type of study. However, for clarity of visualization, only the results of a m . a . r . with a time window of 1 week and a time step of 1 h are shown in this work. To describe the dispersion of residuals, the standard deviation of the m . a . r . ( \(sd_{m.a.r.,i}\) ) is also presented in order to understand how the residuals vary or how they are dispersed along time. The \(sd_{m.a.r.,i}\) is defined as follows:

where \(({m.a.r.}_{j} - \bar{m.a.r.}_{j \, to \, i})\) is the mean of the moving average between times i to j. Once the m . a . r . and the \(sd_{m.a.r.,i}\) are established, it is possible to define a baseline of the structural behaviour under normal conditions. The m . a . r . allows the identification of extreme values and trends along a time period. The \(sd_{m.a.r.}\) , on the other hand, allows the identification of any increment of variability and/or randomness along the same time period that might suggest a decrease of stiffness in the section under study. Based on the residual analysis, it is possible to identify any novel behaviour on the data that has not been explained by the model itself. In addition, residuals contain information related to errors, from the measurements and the model, and other unknown effects. In the particular case of the proposed study, these effects can be indirectly related to the influence of traffic and train loads. The explanatory capacity of the residual analysis strengthens the accuracy of the baseline for the characterization of the structural behaviour.

Sensitivity analysis The importance of improving the understanding of the performance and inner working of the ML techniques is relevant in the SHM field, since decisions regarding the structure under study must be grounded on robust conclusions. As such, the definition of an accurate baseline for the structural behaviour that can be used as a reference point in future studies needs to be validated and verified. In [ 41 ], the authors define a sensitivity analysis as the study of how the uncertainty in the output of a model can be assigned to the uncertainty sources in the model inputs. Different sensitivity analysis methods are proposed in the literature [ 11 , 30 , 32 , 41 ]. To characterize the effects of the environmental loads on the structural responses and the robustness of the predictions, the proposed sensitivity analysis involves the evaluation of the predicted outputs evolution with respect to each of the inputs at the section under study according to the models presented before. The quality of the correlations between the quantities would confirm the reliability of the predictions, i.e., a good correlation implies a good prediction. According to [ 32 ], this exercise would resemble the observed behaviour on the real system and can be used as a strategy for the validation and verification of the model.

4 Case study: the 25 de Abril bridge

The case study chosen for application of the proposed methodology is the 25 de Abril suspension bridge located in the estuary of the Tejo river connecting the cities of Lisbon and Almada, Portugal, (Fig.  3 ). The bridge has a total length of 2277.5 m, with three suspended spans: the central suspended span has 1012.9 m and the two laterals suspended spans have 483.4 m each. The North access is made through a prestressed concrete viaduct comprising 13 spans while the South span is made by a railway tunnel and road access. The bridge deck consists of a suspended stiffening truss with 1344 vertical hangers that, at the same time, are suspended by four principal cables supported at piers P2 and P5 and at pylons P3 and P4 (Fig.  4 ). The cross-section of the deck’s stiffening truss supports six car lanes at the upper chord level, while, supporting a double electrified track railway at the lower chord level.

The 25 de Abril bridge

Schematic representation of the monitoring system of the 25 de Abril bridge [ 36 ]

The structural monitoring system comprises eight types of sensors able to acquire data that allow characterizing the structural behaviour of the bridge, namely its displacements, rotations, strains, stresses; and the imposed loads, such as traffic and environmental loads, Fig.  4 .

The statistical features obtained from all the quantities measured at the bridge involve hourly maximum and minimum values, hourly medians, and hourly quartiles. The median values are particularly important, since they are expected to be free of the influence of extreme values related to traffic and other fast effects, thus reflecting only slow effects that can be analyzed with higher precision. The data acquired continuously on the 25 de Abril bridge reflects numerous effects generated by simultaneous actions imposed on the structure. For SHM purposes, those data are divided into two subgroups: the imposed loads and the structural responses. The information extraction strategy applied to the bridge is based on time series obtained for certain predefined time-intervals from which the following are extracted: (i) statistical features that can be directly and precisely correlated with the structural behaviour, and (ii) vibration information such as natural frequencies and their corresponding mode shapes and damping ratios. The proposed methodology was applied to the statistical features extracted from the time series in a defined period of time and from the mid-span section of the bridge.

Location of thermometers and strain gauges in section 0

Strains are measured directly through strain gauges in 88 points located in the stiffening truss at the downstream and upstream chords and in the plates that make the sections of the towers. In this study, the strains measured at Section 0 (the mid-part of the central span) are the structural response parameters considered and temperatures are the environmental loads (Fig.  5 ). The data considered in the present study are from a 5-year monitoring period, from 20/04/2016 to 09/11/2021, with hourly measurements and a total of 45000 measurements per physical quantity. It is important to mention that the vehicular traffic loads and the train loads existing on the bridge were not considered in the present study, although they were indirectly included in the residual analysis. Since there are two strain gauges and two thermometers in each of the chords of the selected section of the stiffening truss (Fig.  5 ), the mean value of each pair of the same type of sensors (i.e., strain gauges and thermometers) was considered in order to simplify the analysis. The corresponding mean value are \(\epsilon _A = (\epsilon _1 + \epsilon _2)/2\) , \(\epsilon _B = (\epsilon _5+\epsilon _6)/2\) , \(\epsilon _C = (\epsilon _7+\epsilon _8)/2\) , and \(\epsilon _D = (\epsilon _3+\epsilon _4)/2\) for the strain measurements (in [ \(\mu \epsilon\) ]), and \(T_a = (T_1+T_2)/2\) , \(T_b = (T_5+T_6)/2\) , \(T_c = (T_7+T_8)/2\) and \(T_d = (T_3+T_4)/2\) for temperature measurements (in \([^{\circ }{\hbox {C}}\) ]) (Fig.  5 ). The strains are gathered from the strain gauges installed at the bridge and their time evolution is presented in Fig.  6 . The temperatures’ evolution is presented in Fig.  7 .

Time series of the strain measurements at Section 0, top to bottom A, B, C, D

Time series of the temperature measurements at Section 0, top to bottom A, B, C, D

5 Results and discussion

This section presents the results of the application of the proposed methodology to characterize the structural behaviour of the 25 de Abril Bridge. Three MLP models were developed to assess the evolution of the predicted responses and validate the modeling approach. As shown in Table 2 , the first model (Model 1) considers the influence of temperature on each response separately. This study presents the results obtained for the strains measured in point A, \(\epsilon _A\) . In this scenario, nothing can be inferred from the other responses. The second model (Model 2) considers the influence of all the temperatures on the prediction of all the responses simultaneously. In addition, the third model (Model 3) addresses the relationship among the responses in the section under study. All of the models proposed herein were developed in R [ 49 ] using the nnet package [ 51 ].

For all the cases, the learning dataset was established using 90% of the dataset, while the prediction dataset was defined by the remaining 10% of the data. The learning dataset was then split into three subsets: the first 65% of the data, taken sequentially, are used as the training set, the next 15% are used for the cross-validation, and the remaining 20% are used as the testing set. The training and cross-validation sets can be seen as the learning data, and the test set as the data that allows evaluating the quality of the NN model. The training data set is randomized and, in each iteration, the training performance increases. It stops when the error for the cross-validation set increases, thus ensuring a better generalization [ 31 ]. Regarding the residual analysis, the moving average and its standard deviation with a time period of 7 days ( \(7 \times 24 = 168\) records) are computed for all the cases. The results of the three models proposed are presented in the following sections.

5.1 Model 1: \(\epsilon _A=f(T_a, T_b, T_c, T_d)\)

This model used as input values the environmental loads (temperatures \(T_a\) , \(T_b\) , \(T_c\) and \(T_d\) ), and as output values, each strain at a time of the section under study at a time ( \(\epsilon _A\) , in this study), resulting in an NN of four inputs and one output. Values in the hidden layer were defined based on the input dimensionality due to the complex relationship between input–output and the large-size dataset. A hyperbolic tangent transfer function has been chosen to be the activation function for the hidden layer, and a linear function was chosen for the output layer. The generalized delta rule back-propagation algorithm was used in the training process. The chosen network architecture was the best of all the tested networks with 4–7 neurons at the hidden layer. To find out the optimum result, five initializations of random weights and a maximum of 35000 iterations were performed for each NN architecture. In this case study, the NN model with the best performance, i.e., involving the lowest error for the cross-validation set, was a 4-7-1 MLP (4 inputs—7 neurons in the hidden layers—4 outputs).

The MSE values were calculated to evaluate the quality of the model representation. An MSE value of 19.89 \(\mu \epsilon\) was obtained for the training subset, 19.99 \(\mu \epsilon\) for the cross-validation subset, and 19.28 \(\mu \epsilon\) for the test subset. Given that all the values are in the same range, a good quality and generalization of the model is confirmed. Figure  8 a shows the time series obtained with the predictive model for the target \(\epsilon _A\) . The blue points represent the values obtained with the prediction model, and the black points are the observed measurements. From the plot, it is possible to confirm the good fitting of the model outcome regarding the observed values.

In Fig.  8 a, the residuals corresponding to the same target \(\epsilon _A\) are represented in grey. In red, overlapping the residuals, the evolution of the moving average ( m . a . r .) for a time period of 7 days is represented, whose maximum absolute values are equal to 27.33 \(\mu \epsilon\) and 4.09 \(\mu \epsilon\) for the training and prediction sets, respectively. The plot also shows the evolution of the m . a . r . standard deviation ( \(sd_{m.a.r.}\) ) in blue, whose maximum values are equal to 27.33 \(\mu \epsilon\) and 16.71 \(\mu \epsilon\) for the training and prediction sets, respectively. By comparing the predicted m . a . r . and \(sd_{m.a.r.}\) with the observed values from the learning set, it is possible to notice a considerable reduction on both metrics. This could be an indication that the model is capturing the existing patterns and performing accurately.

As proposed in the methodology, a sensitivity analysis is also carried out to test the robustness of the models and characterize the effects of the inputs (temperatures) on the predicted output (response) of the models (Figure   8 b). For the proposed models, the correlation of the output was analyzed with each input of the NN. The sensitivity analysis for \(\epsilon _A\) through Model 1 was carried out for temperatures measured in \(T_d\) along each month of the period under study to understand the differences among the different temperature seasons, as presented in Figure   8 b. The correlation between \(\epsilon _A\) and \(T_d\) can be seen to be high, confirming the reliability of the prediction.

Model 1. a Measurements, predicted values, m . a . r . and \(sd_{m.a.r.}\) of strains measured in \({\epsilon _A}\) . b Sensitivity analysis at target \({\epsilon _A}\) for temperatures at \(T_d\)

5.2 Model 2: \((\epsilon _A, \epsilon _B, \epsilon _C, \epsilon _D)=f(T_a, T_b, T_c, T_d)\)

This model used the environmental loads (temperatures \(T_a\) , \(T_b\) , \(T_c\) and \(T_d\) ) as input values, and all the responses (strains \(\epsilon _A\) , \(\epsilon _B\) , \(\epsilon _C\) , \(\epsilon _D\) ) of the section under study as output were \(\epsilon _A\) is the reference in this case. The resulting NN model is a four inputs and four outputs model. For this model, the network with the best performance, i.e., involving the lowest error for the cross-validation set, was a 4-7-4 MLP.

The adopted NN model presented fewer errors for the cross-validation set. The MSE values for the training, cross-validation, and testing sets were 21.76 \(\mu \epsilon\) , 22 \(\mu \epsilon\) , and 21.71 \(\mu \epsilon\) , respectively. As for the previous model, Fig.  9 a shows the resulting time series for the target \(\epsilon _A\) where the blue points correspond to the prediction model results, and the black points correspond to the observed measurements. These results confirmed that a model with a good generalization quality is also obtained, in this case.

The bottom plot of Fig.  9 a shows the residuals corresponding to the same target in grey. In red, overlapping the representation of the residuals, it is possible to see an expected m . a . r . close to zero for a time period of 7 days whose maximum absolute value for the learning set is 28.38 \(\mu \epsilon\) and for the prediction set 14.47 \(\mu \epsilon\) . In the same plot, the standard deviation of the m . a . r . is also represented (in blue) with a maximum value for the learning set of 24.09 \(\mu \epsilon\) and the prediction set of 15.14 \(\mu \epsilon\) . By comparing the m . a . r and the \(sd_{m.a.r.}\) , it is possible to notice, as with Model 1, a reduction of the m . a . r . and \(sd_{m.a.r.}\) from the learning to the prediction set as evidence of the model accuracy.

The sensitivity analysis for target \(\epsilon _A\) based on the variations observed in \(T_d\) per month are presented in Fig.  9 b. These results also confirm the good correlation of the predicted response with the input.

Model 2. a Measurements, predicted values, m . a . r . and \(sd_{m.a.r.}\) of strains measured in \({\epsilon _A}\) . b Sensitivity analysis at target \({\epsilon _A}\) for temperatures at \(T_d\)

5.3 Model 3: \(\epsilon _A=f(\epsilon _B, \epsilon _C, \epsilon _D)\)

This model uses as input and output values the responses (strains) of the section under study, resulting in an NN of three inputs ( \(\epsilon _B\) , \(\epsilon _C\) and \(\epsilon _D\) ) with one output ( \(\epsilon _A\) ). For this model, the network with the best performance, i.e., involving the lowest error for the cross-validation set, was a 3-5-1 MLP. For the chosen model, the MSE values obtained were 7.04 \(\mu \epsilon\) for the training set, 7.09 \(\mu \epsilon\) for the cross-validation set, and the same value, 7.09 \(\mu \epsilon\) , for the testing set. Since the values are in the same range, the good generalization of the model is confirmed. MSE values of Model 3 are lower than those obtained with Models 1 and 2, showing the best performance among the three models. This can be related to the greater correlation between the variables under study.

Figure  10 shows, in the top plot, the time series of the predictive model corresponding to target \(\epsilon _A\) . The same plot also represents the points obtained by the prediction model (in blue), overlapping the points corresponding to the observed measurements (in black). From the graph, it is possible to confirm the good fitting of the model.

In Fig.  10 a, the residuals corresponding to the target \(\epsilon _A\) are also represented in grey. An expected m . a . r . for a time window of 7 days is represented in red overlapping the residuals, and the evolution of \(sd_{m.a.r.}\) is shown in blue. The maximum absolute value of m . a . r . registered for the learning set is 21.14 \(\mu \epsilon\) with a reduction, in the prediction set, to 3.62 \(\mu \epsilon\) . In the case of \(sd_{m.a.r.}\) , the maximum value for the learning set is 11.62 \(\mu \epsilon\) decreasing to 9.62 \(\mu \epsilon\) in the prediction set. Those values, the same as for Model 1, indicate the model’s accuracy in capturing the existing patterns in the data.

Model 3. a Measurements, predicted values, m . a . r . and \(sd_{m.a.r.}\) of strains measured in \({\epsilon _A}\) . b Sensitivity analysis at target \({\epsilon _A}\)

The plots shown in Fig.  10 b correspond to the variation of \(\epsilon _A\) based on the variation of \(\epsilon _D\) , for each month. A good correlation between the two quantities is verified for the cold and warm seasons.

5.4 Comparative analysis

From the comparison of the model results, it was possible to confirm that the three models exhibited a good fitting between the observed values and the predicted ones. Table 3 shows the performance parameters for the learning and the prediction sets per model by means of the RMSE values, maximum values of the residuals ( \(\left| r\right| _{Max}\) ), the average value of the residuals across the whole dataset ( \(E\left( \left| r\right| \right)\) ), maximum m . a . r ., and maximum \(sd_m.a.r.\) values for the three models proposed.

Given that Model 1 and Model 2 considered the influence of the environmental loads to predict the structural responses, it is possible to compare their performance through the parameters shown in Table 3 . Given that Model 3 does not consider directly the influence of the environmental loads, since the model was built considering sensors only related to the measurements of the response, the difference in the results, when compared to the other models, can be related to a higher correlation among the considered quantities.

A comparative analysis between the RMSE values results of Models 1 and 2 for the prediction and the learning sets confirms their good generalization, since these values are in the same range. On the other hand, the maximum value of the residuals \(\left| r\right| _{Max}\) shows a decrease from the learning to the prediction set in Model 1 and Model 2, which confirms the adequacy of the prediction. The results of the residual analysis of Model 1 show no significant error or unknown effect at the response point of \(\epsilon _A\) or in any of the environmental loads. Regarding Model 2, the extra uncertainty added using more sensor measurements for the prediction could explain the higher residual values when compared with Model 1. When analyzing the m . a . r . and \(sd_{m.a.r.}\) values in both models, the difference in results between the learning and the predicted sets are in the same range, confirming the model accuracy of prediction at target \(\epsilon _A\) .

As referred before, Model 3 has a different condition, since it uses different information (only structural responses) compared with the other models. However, it shows a good generalization as suggested by the RMSE results. When considering only response measurements for the prediction model, the predictors were seen to be more stable due to the greater correlation among the quantities. When analysing the residuals, it is possible to see the maximum values of m . a . r . and \(sd_{m.a.r.}\) decreasing from the learning to the prediction sets, confirming the good performance of this model.

To evaluate the performance of the models more comprehensively and comparatively, a Taylor diagram is presented in Fig.  11 . The horizontal and vertical axes of the diagram represent the values of the standard deviation, which are connected to each other by circular dotted lines. The radial lines from the origin show the value of the correlation coefficient between the observed values and the model results and the grey circular continuous lines show the RMSE. Consequently, each model is plotted by means of its standard deviation, RMSE, and correlation coefficient that are the result of the linear relationship between the model results, for the three proposed cases, and the observed values (described as the reference value) for the strain measured at \(\epsilon _A\) . The closest model to the reference value is identified as the most accurate model.

Taylor diagram comparing the three model results at target \({\epsilon _A}\)

From Fig.  11 , it is possible to notice that the three models registered a standard deviation value close to the reference ratio, indicating that all the models accurately captured the variability and trends of the observed values. It is worth noticing that Model 3 (in green) is closer to the reference curve, indicating that the model reproduces the variability of the reference dataset better than the other models. Regarding the correlation coefficient, Model 3 also shows greater correlation compared with the other models. Those differences can be explained by the fact that Model 3 considers just the structural responses and does include with the influence of the temperature in the prediction as Models 1 and 2. Since the correlation of theses models was calculated using the temperatures as inputs, the proximity of the models’ performance in the Taylor diagram reflects this issue. As such, there is no relevant difference between the performance of Models 1 and 2. The three models are also located near the lowest RMSE circle, confirming the good fitting of the models and in agreement with the performance parameters presented in Table 3 .

To interpret the pattern behaviour during the seasonal temperature changes, polar coordinate plots were created. In Fig.  12 a, it is possible to see the evolution of the values of m . a . r . (calculated for a time window of 7 days and time steps of an hour) in blue and \(sd_{m.a.r.}\) in red obtained with Model 1 for the prediction and the learning sets, with separate plots for \(m.a.r. > 0\) and \(m.a.r. < 0\) . For \(m.a.r. < 0\) . The higher values are seen mostly for the winter season, around 15 \(\mu \epsilon\) to 20 \(\mu \epsilon\) . Since there is a low dispersion of the values in that period of time (as seen from the values of \(sd_{m.a.r.}\) ), it is possible to relate those high values and the content of error showed by the residuals with the temperatures, considered as the input of the prediction model. Still, for the case of \(m.a.r. > 0\) , the maximum m . a . r . value is registered in December, around 25 \(\mu \epsilon\) , while \(sd_{m.a.r.}\) shows extreme values in May and June. Regarding the values shown in the prediction set, for \(m.a.r. < 0\) , the highest m . a . r . and \(sd_{m.a.r.}\) values were registered in the month of November.

When analyzing the results obtained with Model 2, the same pattern can be seen in both the learning and the predicted sets (Fig.  13 ) which confirms that the extreme values are time located and due to the recorded temperatures or strains measured at point \(\epsilon _A\) . In the case of Model 3, the polar plots in Fig.  14 show high dispersion in the months of February, April, and October for the learning set. However, all values are concentrated in the central zone, as expected for the prediction set, thus showing that Model 3, which uses structural strain as input and output, performs as well as the other two while having the advantage of not needing to measure temperatures. Literature and practice review allow conclusions about the need to perform temperature measurements for understanding, compensating, and controlling strain and other structural responses. Nevertheless, these results suggest that for carefully chosen measurements, such as strains within the same transversal cross-section, structural responses can only be used for building the models and detecting damage, resulting in more precise, more straightforward, and less expensive SHM practices.

a \(sd_{m.a.r.}\) and m . a . r . for the learning set of Model 1, and b m . a . r . and \(sd_{m.a.r.}\) for the prediction set of Model 1, in a polar coordinate system, by month

a \(sd_{m.a.r.}\) and m . a . r . for the learning set of Model 2, and b m . a . r . and \(sd_{m.a.r.}\) for the prediction set of Model 2, in a polar coordinate system, by month

a \(sd_{m.a.r.}\) and m . a . r . for the learning set of Model 3, and b m . a . r . and \(sd_{m.a.r.}\) for the prediction set of Model 3, in a polar coordinate system, by month

6 Concluding remarks

The methodology proposed in this study takes advantage of the relation among the structural responses from several physical quantities for the identification of patterns and changes of trend in the behaviour of the structure. It focuses in i) the verification of the model’s performance, i.e., validating the quality of the model to represent the relation between the inputs (environmental loads or structural responses) and the outputs (observed structural response), and ii) the definition of a baseline for the characterization of the residuals, based on the adoption of moving average charts of the residuals ( m . a . r ) and the m.a.r. standard deviation ( \(sd_{m.a.r.}\) ). Through the results obtained by the application of the proposed strategy, trends, or patterns in the structural behaviour can be identified for future observations. Following a supervised learning to develop an accurate baseline model capable of representing the structural behaviour of the 25 de Abril bridge, 5 years of hourly measurements gathered from the SHM system installed at the mid-span section bridge were analyzed. Three NN prediction models combining the environmental loads and the structural responses under study in different ways were proposed to predict, more accurately, the general structural behaviour and assess the quantities considered in the study.

Model 1 considered temperatures as input and the strain measures in one point of the bridge as output. Model 2 considered the same inputs as Model 1, and the strains measured in four points of the bridge as outputs. Model 3 allows performing a response–response analysis, where the inputs are the strains measured in three points, and the output the remaining strain (from the set of four strains measured in the section of the bridge under study).

Based on a residuals’ moving average and standard deviations with a length of 7 days to allow stable measurements of expected value and dispersion of the model’s performance and of the structural health, it was possible to observe that the three models were accurate and robust. Knowledge of the maximum and minimum residual values is an appropriate reference for identifying specific situations that, in principle, did not occur during the training period, such as a trend in the evolution of residuals (based on the moving average) or a new pattern (based on the increase of dispersion in the moving average of the standard deviation of residuals).

Models 1 and 2 reveal the ability to identify damage by controlling strain, and using temperatures as the input. Their performance accuracy is 5.4% and 5.5%, respectively, for the learning set, and 5% and 5.1%, respectively, for the prediction set. Model 3, instead, reveals better performance using only strain measurements as input and output, a performance accuracy of 3.2% for the learning set and 3.3% for the prediction set. These results show better performance than the other two models, as expected for strain measurements within the same cross-section. Therefore, Model 3 should be used in conjunction/complementary with Models 1 or 2 for better interpretation and validation.

To test the robustness of the prediction models and characterize the influence of the inputs on the outputs of the three models, a sensitivity analysis that correlates each output with each input was performed. The evolution of the responses over a year was analyzed to enhance the interpretation of the model results. This time window accounts for the seasonal change and provides a more comprehensive understanding of the source of the variable ranges where the models struggle to reproduce the structural responses by the time of the year. This analysis provides additional insights on the values of inputs and outputs that are not being correctly modeled, thus providing additional insight for owners when managing safety and analyzing the results.

Considering the analysis results at each stage, it is possible to conclude that the extreme values are not related to structural changes but to higher or lower temperature values related to seasonal changes during the year. As the models are updated with these values, future improvements are expected, and, therefore, lower residuals’ moving averages and standard deviations, more confidence in the results, and higher structural safety levels. The results of Model 3 also suggest that an accurate prediction is possible considering less sensor information and no environmental main loads, just structural responses. Nevertheless, further verification and validation of these assumptions should be carried out in future research.

Data availability

The database used in this study is the property of Infraestruturas Portugal under the management of LNEC and is provided to the authors under confidentiality terms for research purposes. Therefore, any request for data exploration must be made to Infraestruturas Portugal.

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Acknowledgements

The authors acknowledge the Portuguese National Laboratory of Civil Engineering (LNEC), the SAFESUSPENSE project-POCI-01-0145-FEDER-031054 (funded by COMPETE2020, POR Lisboa and FCT), and Infraestruturas Portugal, for providing the necessary information for this study. The Portuguese Foundation for the Science and Technology (FCT) for funding this research under Grant No. PD/BD/150407/2019. The fourth author also acknowledges the financial support of the Base Funding—UIDB/04708/2020 of CONSTRUCT—Instituto de I &D em Estruturas e Construções, funded by national funds through FCT/MCTES (PIDDAC).

Open access funding provided by FCT|FCCN (b-on). This study was funded the Portuguese Foundation for the Science and Technology (FCT) under Grant No. PD/BD/150407/2019. Conflict of Interest: The Portuguese National Laboratory of Civil Engineering (LNEC), the SAFESUSPENSE project-POCI-01-0145-FEDER-031054 (funded by COMPETE2020, POR Lisboa and FCT) and Infraestruturas Portugal, provide the necessary information for this study. The fourth author, Xavier Romão, acknowledges the financial support of the Base Funding— UIDB/04708/2020 of CONSTRUCT—Instituto de I &D em Estruturas e Construções (https://doi.org/10.54499/UIDB/04708/2020), funded by national funds through FCT/MCTES (PIDDAC).

Author information

J. Mata, J.P. Santos, and X. Romão have contributed equally to this work.

Authors and Affiliations

Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, Porto, 4200-465, Portugal

Fabiana N. Miranda

Monitoring Division, Concrete Dams Department, National Laboratory for Civil Engineering, Av. do Brasil 101, Lisbon, 1700-066, Portugal

Fabiana N. Miranda & Juan Mata

Structural Health Monitoring Engineer, Lisbon, Portugal

João Pedro Santos

CONSTRUCT-LESE, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, Porto, 4200-465, Portugal

Xavier Romão

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Contributions

All authors contributed to the study conception and design. Information for analysis was provided by Infraestruturas Portugal, the Portuguese National Laboratory of Civil Engineering (LNEC) and the SAFESUSPENSE project-POCI-01-0145-FEDER-031054 (funded by COMPETE2020, POR Lisboa and FCT). Material preparation and data analysis were performed by Fabiana N. Miranda. The first draft of the manuscript was written by Fabiana N. Miranda and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Fabiana N. Miranda .

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Miranda, F.N., Mata, J., Santos, J.P. et al. Structural characterization of a suspension bridge by mapping the temperature effects on strain response based on neural network models. J Civil Struct Health Monit (2024). https://doi.org/10.1007/s13349-024-00855-0

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Received : 20 June 2023

Accepted : 01 September 2024

Published : 23 October 2024

DOI : https://doi.org/10.1007/s13349-024-00855-0

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