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Abstract

Structural ageing and material deterioration require infrastructure managers to continuously seek for improved solutions for bridge condition management. In the last two decades, vehicle-assisted bridge monitoring has emerged among researchers and engineers as a promising tool to support visual inspections, being a cost-efficient alternative to direct Structural Health Monitoring systems. In this work, the authors present a sparse-autoencoder-based damage detection methodology which exploits the vertical acceleration of train’s leading bogie to assess bridge health condition. The bridge under analysis in this work is a Warren truss bridge, whose FE model was designed based on the technical drawings of an actual structure, which belongs to the Italian railway line, and then validated through dynamic measurements. Raw bogie vertical accelerations are preprocessed through Continuous Wavelet Transform, allowing for the extraction of a specific frequency region of interest, governed by the modular configuration of the bridge as well as the forward speed of the convoy. Starting from the average curve of the computed wavelet coefficients, bridge health status is assessed through the use of a sparse autoencoder exploiting multiple train transits and two different damage indices. First of all the Hotelling’s statistic computed at the latent space level, and, secondly, the batch mean absolute reconstruction error. Different damage scenarios and intensities are tested in this work, considering the partial failure of stringers or cross-girder members due to corrosion, modelled as material mass loss and stiffness reduction. A large set of simulations allowed for testing the robustness of the methodology against operational variables, such as travelling speed, geometrical track irregularity evolution, and different weights of the convoy. Globally, promising damage detection performances were obtained when considering batches of 40 trains, even in presence of measurement noise and speed estimation inaccuracies.

Keywords

structural health monitoring drive-by damage detection truss bridge railway infrastructure continuous wavelet transform sparse autoencoder

Introduction

Bridges and viaducts are critical assets for land transportation, given their role for social activities and economic growth. Continuously assuring their safe operation represents a task of major importance for infrastructure managers, especially in a context characterised by an increasing number of bridges which is reaching or overcoming their original design lives.^1–8 Historically, monitoring of highway and railway bridges has relied on visual inspections.⁹ Despite being standardised procedures, visual inspections are characterised by limitations,¹⁰ such as qualitative outputs,¹¹ relatively high costs, safety and accessibility problems and they are subject to operators’ skills and experience.^12,13 In this framework, direct Structural Health Monitoring (SHM) systems came into play as key tools to complement visual inspections, drawing an increasing attention both in the academic and industrial worlds.

Requiring the deployment of sensors on the monitored bridge structure,^14,15 direct SHM systems are inherently characterised by low flexibility, null movability, high installation costs, and safety issues for the operators.^16,17 To overcome some of the drawbacks concerning direct SHM systems, indirect SHM systems has recently emerged as an attractive solution^16–20 after the publication of the pioneering work by Yang et al.²¹ Indirect systems make use of instrumented vehicles as both bridge exciters and vibration response receivers. A modification of the structural behaviour of the bridge deck, due to the onset of a damage, will lead to a change in the dynamic interaction between the vehicle and the bridge itself, that will reflect in the vehicle dynamics.²² Even called drive-by approaches, they potentially offer higher cost-effectiveness to their users: being one train capable to monitor several bridges along its daily route, they require the installation of less sensors, no (or almost null) traffic disruption, and higher flexibility. According to Malekjafarian et al.,¹⁷ drive-by monitoring can be divided into two main families: modal identification^23–25 and bridge condition monitoring, where the latter can be modal-based^26,27 or nonmodal-based.^28,29 The focus of the present work is to introduce a bridge condition monitoring approach which does not rely on the computation of any modal parameter of the bridge structure from the vehicle response. Despite their well-known relationship with bridge health condition, modal parameters are severely affected by environmental and operational effects,^30–36 making them not always reliable for condition monitoring.¹⁷

In the last years, a significant effort has been carried out by academics and researchers in the development of effective nonmodal-based drive-by approaches. Critical and exhaustive discussions about main obtained results, successes and limitations are provided by comprehensive reviews, such as References 17, 20, and 37. Nonmodal-based drive-by approaches basically rely on damage indices which are not related to bridge modal parameters. There exists a number of signal processing tools that have been proposed in the years. In particular, time-frequency domain tools have achieved a great success and development. Wavelet Transform has been used to obtain time-frequency representation of vehicle response signal, allowing for better resolution capabilities than the Short Time Fourier Transform.²⁸ In McGetrick and Kim,³⁸ the authors developed a bridge damage detection and localisation approach based on Morlet wavelet transform, focusing on a sensitivity analysis deriving the main influencing parameters affecting the success of the approach. Hester and González³⁹ proposed a wavelet based methodology which relies on the use of the Mexican hat wavelet applied to vehicle acceleration to detect bridge damage. The authors tested the effectiveness of the methodology using a simply supported beam model for the bridge and two distinct travelling road vehicle models (i.e. point load and 2D 2-axle sprung model). Fitzgerald et al.²⁸ proposed a promising drive-by approach to detect scour damage in railway bridges, which uses mean wavelet coefficients obtained applying a complex Morlet CWT to bogie vertical accelerations coming from multiple train transits. In this work, the authors computed the moduli of the wavelet coefficients, organised in time-frequency matrices, then averaged over the considered number of train passages and then summed along frequency direction, in a range defined from 0.5 to 15 Hz. More recently, as also proposed by Hester and González,³⁹ Demirlioglu and Erduran²⁴ discusses the use of CWT coefficients, obtained from road vehicle residual acceleration using Mexican hat wavelet, exploiting the use of wavelet coefficients in a band which is sufficiently far from the first bridge frequency and excludes driving frequency. Compared to Hester and González,³⁹ this methodology showed promising performances without the need of a prior-knowledge of a baseline reference.

In our work, CWT is used as a pre-processing tool, adopting a complex Morlet wavelet, tuned as in Fitzgerald et al.²⁸ The adopted vehicle consists in a 3D train multibody model, composed of eight coaches, travelling on a 3D Warren truss bridge. The map of the moduli of the wavelet coefficients obtained from leading bogie vertical acceleration and interpolated in the space is sectioned in a range which varies with the vehicle travelling speed. This frequency band of interest is determined also by the modular configuration of the bridge under analysis. This range is shown to reflect damages occurring to bridge modules. The curve built by averaging the wavelet coefficients moduli in the frequency band of interest is the input for the second step of the methodology consisting in the use of a deep sparse autoencoder (SAE), where the latter represents a well-known branch among machine learning approaches. Nowadays, machine learning techniques have been facing a growing adoption in the SHM field,^40,41 given their capability to extract damage-sensitive features and patterns, as well as managing and successfully treating large amount of data, typical in continuous monitoring.⁴² In this context, three main families can be distinguished, namely supervised, unsupervised, and reinforcement learning approaches.^43–45 The difference consists in the need of labelled datasets in the neural network (NN) training process. In fact, drive-by supervised approaches^46,47 require the use of labelled datasets referring both to the healthy and damaged structure. This aspect is of major and practical relevance when dealing with large civil structures, such as bridges, for which comprehensive data sets are often unavailable.⁴⁸

In the last few years, a number of authors have started exploring the use of autoencoders as a promising unsupervised learning strategy for enhancing the effectiveness of drive-by approaches. Nevertheless, as observed by de Souza et al.,³⁶ most of these researches are related to road applications. Liu et al.⁴⁹ showed that stacked autoencoders markedly outperformed principal component analysis (PCA) in extracting representative features from vehicle accelerations spectra. The authors, in Sarwar and Cantero,⁵⁰ propose a deep autoencoder architecture exploiting multiple road vehicle passages to extract damage-sensitive features to monitor the health condition of a highway bridge. In particular, the chosen damage index (DI) is based on the Kullback-Leibler (KL) divergence, in charge of evaluating changes in the distribution of mean reconstruction errors. Kaur et al.²⁷ exploit an adversarial autoencoder (AAE) to detect and localise damages from the CWT map obtained by processing road vehicle response. Similarly to Calderon Hurtado et al.,⁵¹ the analysed frequency band in the CWT map is set to contain inside the first bridge bending frequency component, in contrast with what was observed by Demirlioglu and Erduran.²⁴ The methodology is tested for a forward speed of 0.55 m/s. Hurtado et al.⁵² presented an adversarial auotencoder (AAE) approach to indirectly detect damages occurring to a highway bridge. To do so, residual acceleration spectrum is used, after filtering and averaging, as an input to the AAE adopted. The methodology showed good results (AAE outperformed stacked autoencoders), but some limitations are observed: the approach is tested numerically considering no changes in terms of vehicle properties and its travelling speed. Like Calderon Hurtado et al.,⁵¹ also in Hurtado et al.⁵² the vehicle is assumed to move with a speed around 2 m/s, which could lead to the need for traffic disruption or limitation, hindering some of the benefits offered by drive-by approaches. The authors in Li et al.⁵³ describe a damage detection methodology which makes use of a deep-autoencoder trained to reconstruct short-time Fourier transform (STFT) of the moving vehicle. The approach, based on the use of squared reconstruction error, showed high accuracy for six different damage scenarios tested on a lab-scaled vehicle-bridge interaction model.

The work presented so far, exploiting autoencoders, are all referred to road vehicles. Nevertheless, there also exists some very recent works that concern railway bridges. Fernandes et al.⁴⁸ propose a deep autoencoder framework which processes raw vertical acceleration responses from a 2D train-track-bridge interaction model accounting for various operational conditions. Conversely to previously presented works, the investigated damage scenarios is represented by scour, modelled as support stiffness decrease, instead of the reduction of deck flexural stiffness due to cross-section degradation. The damage index is based on KL divergence to quantify changes in reconstruction error distributions, similarly to what was proposed in Sarwar and Cantero.⁵⁰ High accuracy was obtained for low intensity damages considering batches of 50 trains, accounting for variabilities in terms of track irregularity profile, speed, and vehicle parameters as well as measurement noise. A drive-by approach for high-speed railways is presented in Souza et al.^36,54: the authors use sparse autoencoders (SAEs) and Mel-frequency cepstral coefficients (MFCCs) to detect two different damage typologies. One affecting end supports and the other, more common in literature, represented by a reduction of beam elements flexural stiffness. Good performances were observed with high robustness to EOVs, considering 3D train-track-bridge simulations. Most of the autoencoder-based damage detection and/or localisation were tested on simply supported beams modelling concrete bridges. In general, Warren truss structures are less investigated in the related literature, despite their spread and importance worldwide, as well as their average age. In this context, Bragança et al.⁵⁵ tested the effectiveness of using freight train accelerations to detect damages affecting side diagonals and lower chords elements of a Warren truss bridge. Accelerations are collected in eight points of the freight wagon: from them, Wavelet scattering coefficients are computed. Then, eight autoencoders are used to reconstruct these coefficients, and the reconstruction errors obtained, once normalised, are fused to generate a highly sensitive damage indicator.

As mentioned before, this paper proposes a novel damage detection approach, based on sparse auto-encoder (SAE) and CWT transform. Compared to the Fast Fourier Transform (FFT), the CWT allows for tracking the temporal behaviour of the harmonic components of interest (related to the bridge module’s length and train travelling speed) and, as stated in Fitzgerald et al.,²⁸ demonstrates superior performance compared to the Short Time Fourier Transform (STFT), enabling a multi-resolution analysis of the input signal.⁵⁶ Moreover, as stated by the authors in de Souza et al.,³⁶ SAE owns a significant advantage with respect to “traditional” autoencoders (AEs), which consists on their higher capability to learn sparse and concise feature representation of the input data,^57,58 that can be markedly useful in damage detection context, especially when dealing with large amount of input data. These features, discussed more in the detail in dedicated sections, guided our choice in the implementation of the proposed methodology. According to the presented state of the art, main innovations, differences, and contributions can be resumed as follows:

compared to Calderon Hurtado et al.,⁵¹ the rationale behind the definition of the frequency band of interest is totally different, given the different bridge archetype considered in this work. In fact, in this case, the frequency band is chosen given the travelling speed of the rail vehicle and bridge modular geometry;

compared to references 36, 51, 52, and 55, this methodology requires the deployment of only one sensor on the leading bogie of the first coach, which may represent a remarkable advantage in terms of costs, in view of a real-world implementation of the onboard set-up;

differently from de Souza et al.,³⁶ the case study is represented by a regional commuter railway line and the bridge is a Warren truss bridge, while compared to Bragança et al.⁵⁵ larger travelling speeds are considered, being the train adopted a passenger one instead of a freight vehicle;

instead of using the KL distance on the reconstruction error at the exit of the autoencoder,⁵⁰ in this work, we introduce the adoption of the Hotelling’s distance, computed at the latent space level, which, according to the authors best knowledge, is not present in the related drive-by literature.

The simulations are performed with a 3D TTB software, considering the 3D model of a Warren truss bridge dynamically validated with real tests, and a set of train models whose characteristics reflect real trains travelling on the actual bridge. This is done considering the variability of different EOVs (as also did in other works, such as de Souza et al.³⁶). The present paper is organised as follows: first, the train-track-bridge models are described in detail. Next, the signal processing methodology is outlined, including the CWT pre-processing and the use of a sparse autoencoder. Following this, the results are presented and critically analysed. Finally, conclusions are provided.

Simulation software

The adopted simulation software, called ADTreS, is a non-commercial software developed, in the years, within the Department of Mechanical Engineering, at Politecnico di Milano.⁵⁹ A comparison between numerical and experimental data, in terms of bridge vertical deflection during train passage, is presented in Bruni et al.⁶⁰ Concerning rail vehicle dynamics, comparisons between numerical results, obtained through ADTreS, and experimental measurements are shown in Bruni et al.⁶¹ and Alfi and Bruni.⁶² In literature, several other works exploiting this software can be found, for example, References 63 –65. The program allows for:

Multibody modelling for each vehicle of the train;

Finite element approach for modelling bridges and track structures;

Time integration of the equations of motion, to account for transient phenomena and problem’s non-linearities.

Its main components are now described in detail.

Train and track models

The rail vehicle (see Figure 1) is modelled using a multibody approach, following a standard configuration that includes a carbody supported by two bogies and four wheelsets. In this study, the carbody, bogies, and wheelsets are treated as rigid bodies. A three-dimensional model is applied to each coach of the train, with the primary and secondary suspensions represented by linear springs and dampers. The motion of the coach is described by a total of 35 degrees of freedom, with each component (wheelsets, bogies, and carbody) contributing five degrees of freedom: two translations and three rotations. The longitudinal degree of freedom is excluded, as the train is assumed to travel at a constant forward speed.

Figure 1.

Coach 3D scheme and its degrees of freedom. From top to bottom: side, bottom, and rear views.

The train configuration used in this work represents a rail vehicle with 32 axles, travelling over the analysed structure. It consists of eight coaches, each 24.9 m long, with a wheelbase of 2.85 m and a distance of 17.4 m between the bogie centres of mass. The modelled train is a numerical twin of a real passenger train that regularly operated on the bridge. This specific train typology was also identified by the Weigh-in-Motion (WIM) system installed near the bridge.⁶⁶ The bridge features a ballastless track system, as shown in Figure 2, with wooden sleepers considered infinitely rigid in the model. Rail pads are modelled through 3D springs, whose vertical stiffness ( $k_{p, v}$ in Figure 2) is assumed to be 350 MN/m, while the lateral is equal to 67 MN/m. This spring connects the rail nodes to the beam elements that form the bridge’s stringers. Outside the bridge, however, sections of track with ballast have been modelled. In particular, ballast mass is neglected, while its vertical stiffness is set equal to 120 MN/m ( $k_{b, v}$ in Figure 2). Outside the bridge concrete sleepers are considered, modelled as concentrated masses of 315 kg ( $M_{s}$ in Figure 2): rail pads properties are assumed to be equivalent to the one used on the bridge. Sleepers spacing is set equal to 0.6 m, both on and outside the bridge, while the rail weights 60 kg/m. The initial ballasted section, in particular, is designed to prevent artificial transients in the vertical accelerations of the vehicle.

Figure 2.

Schematic representation of the track models, with main mechanical properties. Lateral view. Please notice that proportions are not respected in this scheme.

The mechanical properties assumed for the two types of track, modelled in this work, are comparable to the values proposed and collected in various studies available in the literature, such as References 67 –71.

Bridge FE model

The finite element (FE) model under analysis represents a section of an existing structure located on a regional railway line in northern Italy. The structure, which remains in service, is a Warren truss railway bridge designed in 1946. The bridge consists of two parallel structures, one for each travel direction, both supported by a common pier. These structures include two spans of different lengths: the longer span crosses the riverbed, while the shorter span extends over the floodplain. In this work, only the longer span of the bridge was modelled, to reduce the overall computational burden of the simulation: shorter span modelling was in fact discarded, as well as the central pier. Their contributions for the present work scope were deemed to be irrelevant. Figure 3 illustrates the finite element (FE) model of the bridge under investigation, where d is the single triangular module length, equal to 8.64 m.

Figure 3.

3D FE model of the truss bridge under analysis (H = 7.70 m, W = 5.25 m, and d = 8.64 m).

The structure is characterised by ideal hinge and roller boundary conditions, with a ballastless track. Euler-beam elements were used to create the model, each with six degrees of freedom per node. This modelling choice is based on the fact that the beams in question are slender. Moreover, the elements of this structure predominantly work under axial forces. Since the FEM model reflects the centrelines of the beams, the moments at the nodes are not significant. Therefore, it is possible to model the connections between the bridge elements either using hinges or rigid connections, expecting no significant changes. The geometrical properties of the model were derived from previous field surveys and technical drawings related to the long span of the bridge. The steel material in the model is assigned a Young’s Modulus of 200 GPa, a Poisson’s ratio of 0.3, and a density of 7850 kg/m³. The density was artificially increased by 15% to account for elements not included in the calculation of the net cross-sectional area of each truss member, such as bolts and connecting plates. Additionally, concentrated masses were applied to the bridge deck to represent non-structural components like timber sleepers, track plates, handrails, and footbridges along the sides of the bridge. Proportional damping is assumed in this work (i.e. Rayleigh model^72,73): $α$ and $β$ coefficients were set to have damping ratios coherent with the indications provided by British Standard.⁷⁴ The model was calibrated and validated mainly through dynamic measurements,⁷⁵ performed thanks to a permanent monitoring system, installed on the actual bridge. In fact, in the framework of a project conducted with the infrastructure manager,⁷⁶ the actual bridge was equipped with a permanent monitoring system, put into service in July 2022. Among the installed sensors, velocimeters were chosen to monitor the evolution of structure modal parameters in time. Their positioning is schematically depicted with blue circles in Figure 3, outlined together with the measurement axes (blue arrows). Modal parameters extraction has been carried out through an automatic SSI-COV based Operational Modal Analysis (OMA), whose details are reported in Argentino et al.⁷⁷ The numerical frequencies and mode shapes are compared with the experimental counterparts, measured in a time interval of almost 2 years. The following nomenclature has been adopted to refer to different mode shapes and associated natural frequencies. Precisely, modes labelled with “V” have a dominant vertical component, while the ones with “L” have a dominant lateral component. The second character is represented by the order of the mode shape. Finally, a letter is used to distinguish modes with same predominant component and order. Median values for natural frequencies are collected in Table 1, showing that the model agrees with the actual structure, which is in line with what was shown in Bernardini et al.^75,78 Figure 4 shows the result of the Modal Assurance Criterion (MAC) matrix obtained by comparing numerical and experimental mode shapes for the first seven modes of the presented structure in the points outlined by blue circles in Figure 3. Further details concerning the permanent monitoring system can be found in Benedetti et al.,⁷⁶ while specifics of the OMA algorithm are contained in Argentino et al.⁷⁷

Table 1.

Comparison between experimental and numerical frequencies for the first seven mode shapes of the bridge.

Mode	Exp. Freq. (Hz)	Num. Freq. (Hz)	$Δ %$
L1A	2.24	2.35	4.9
V1	4.18	4.28	2.4
L1B	4.42	4.31	−2.5
L2A	5.00	5.03	0.6
L2B	8.10	8.07	−0.4
L3A	8.71	9.07	4.1
L3B	11.32	11.8	4.2

Figure 4.

MAC matrix for the first seven modes of the bridge under analysis.

Train-track-bridge dynamic interaction

As mentioned before, the simulation of train-track-bridge (TTB) dynamic interaction is conducted using ADTreS.⁵⁹ This software facilitates finite element (FE) modelling of track and bridge structures, offering flexibility in selecting different FE element types, such as Euler beam, plate, or bar elements. As mentioned before, the rail vehicle is instead modelled using a multibody approach.

Structure and train are modelled as two different sub-systems, coupled by contact forces. Structure equations of motion can be written in the following way:

[M_{s}] {\underline{x}}_{s} + [C_{s}] {\underline{x}}_{s} + [K_{s}] {\underline{x}}_{s} = {\underline{F}}_{s v} ({\underline{x}}_{s}, {\underline{x}}_{s}, \dots, {\underline{x}}_{i}, {\underline{x}}_{i}, \dots, t)

(1)

where $[M_{s}]$ , $[C_{s}]$ , and $[K_{s}]$ are the mass, damping, and stiffness matrices of the structure, and ${\underline{x}}_{s}$ is the vector containing the structure’s degrees of freedom. ${\underline{x}}_{sv}$ is the vector containing generalised contact forces, which are defined according to the structure and the vehicle’s motion.

Considering the generic i-th vehicle, the following equation can then be written:

[M_{v i}] {\underline{x}}_{i} + [C_{v i}] {\underline{x}}_{i} + [K_{v i}] {\underline{x}}_{i} = {\underline{F}}_{v i} ({\underline{x}}_{s}, {\underline{x}}_{s}, \dots, {\underline{x}}_{i}, {\underline{x}}_{i}, \dots, t)

(2)

where $[M_{vi}]$ , $[C_{vi}]$ , and $[K_{vi}]$ stand for the mass, damping, and stiffness matrices of the i-th vehicle, and ${\underline{x}}_{i}$ is the vector containing the vehicle’s degrees of freedom. Finally, ${\underline{F}}_{vi}$ represents the column vector which collects the generalised contact forces.

Therefore, the coupling between the train and the track-bridge structures is achieved through contact forces exchanged at the wheel-rail interface, where a multi-Hertzian⁷⁹ wheel-rail contact model is applied. Prior to simulation, using either measured or theoretical wheel and rail profiles, the necessary parameters for calculating the contact forces are stored in a contact table.⁶³ As reported in Alfi and Bruni,⁶² after having computed wheel-rail relative displacements and velocities in all potential contact points, it is possible to enter the contact table with the relative wheel-rail displacement and the longitudinal position. Key contact parameters, stored in the contact table, include the contact angle, wheel’s rolling radius, local curvature radii of the wheel and rail profiles.^62,63,65

Considering the m-th wheel, the normal component of the contact force in the q-th potential contact, is computed according to the following set of equations:

{\begin{matrix} N_{m, q} = 0, & δ_{m, q} \leq 0 \\ N_{m, q} = C_{H_{m, q}} \cdot δ_{m, q}^{\frac{3}{2}}, & δ_{m, q} > 0 \end{matrix}

(3)

where $C_{H_{m, q}}$ is the Hertzian constant obtained from the local curvature radii of the wheel and rail profiles (using the Hertzian formulae), stored in the contact table, while $δ_{m, q}$ represents the elastic wheel-rail penetration. The latter depends on wheel-rail relative displacements in vertical and transversal direction, considering the local rail reference system, and it is computed according to the formulation presented in Alfi and Bruni.⁶² The first equation in system 3 refers to the case in which the contact is not active, while the second equation is valid for active contact points.

Once longitudinal and lateral creepages are calculated according to the expression in Alfi and Bruni,⁶² longitudinal and tangential contact forces are derived using the Shen, Hedrick, and Elkins formulation.⁸⁰ Then, lagrangian components of the contact forces on wheelset and track free coordinates are evaluated.^60–62 ADTreS supports multiple simultaneous wheel-rail contact points, with the option to include both new and worn wheel-rail profiles and to account for track geometrical irregularities. Time integration is carried out using the Newmark integration method, with a modified time-step procedure to handle the nonlinearity of the contact problem,^62,65,73 as schematically depicted in Figure 5.

Figure 5.

Time-integration scheme to handle contact non-linearities.

More in detail, assuming a forward motion of the rail vehicle, with constant speed V, the longitudinal position of the m-th wheel $ξ_{m}$ along the generic k-th finite element can be written:

ξ_{m} = ξ_{m 0} + V \cdot t - L_{k}

(4)

where $ξ_{m 0}$ represents the initial position of the k-th wheel (at initial time instant), while $L_{k}$ is the position of the first node of the k-th beam finite element on which the wheel is running at time t. Once the positions and velocities of contact points at $ξ_{m}$ are defined, it is possible to derive the components of the contact forces exchanged between the track and each wheel (according to the procedure defined before). As previously mentioned, the wheel-rail contact model is based on a preliminary geometrical analysis through pre-computation of geometrical parameters, such as the contact angle or local radii of curvature, expressed as functions of the wheel-rail relative position. Once the contact forces are computed for each activated contact point (multiple contact points can be activated⁶⁵), they are projected onto the global reference system and assembled into the generalised forces vectors,⁸¹ that appear in equations (1) and (2). The latter are integrated in the time domain according to the procedure shown in Figure 5.

At the beginning of the generic j-th time step, the first approximation (p = 1) of the contact point motion is calculated, considering the wheels in the new position. Once the contact forces are calculated, after having computed the generalised contact forces, equations (1) and (2) are integrated. Then, a new approximation (at p = p + 1) of the contact point motion is obtained and the contact forces are calculated again. Therefore, an iterative procedure is performed (green dashed box in Figure 5) within the j-th time step. Once convergence is reached for the contact forces, the time step can be then incremented.⁸¹

For further details on the simulation software, please refer to References 59, 62, and 63.

Damaged scenarios

The simulated damage scenarios, considered in this study, all pertain to corrosion, the primary cause of material degradation in steel structures. Corrosion is a natural electrochemical process in which metals react with their surrounding environment. In steel structures, this process leads to a reduction in the cross-sectional area of the affected elements, compromising their structural integrity and performance.⁸² In this work, corrosion is simply modelled as a reduction of the same quantity of both density and Young modulus to account for flexural and axial stiffness reduction as well as material loss. This modelling choice is equivalent to a reduction of the cross-sectional area, which causes a decrease of the resistance of the chord and its mass. As assumed by Simoncelli et al.⁸² and Xiao et al.,⁸³ this damage affects the whole extension of the flawed element. A set of scenarios is considered in this work, as schematised in Figure 6. Three levels of damage were modelled, namely 15, 25, and 50%, while the damage was applied to two distinct members (one at a time), that is, cross-girders and stringers. In case of damage SD (applied to stringers), both the two stringers are degraded of the same quantity: in this damage scenario, two different locations are investigated (i.e. subscripts 2 and 3). Instead, for damage type CG (applied to cross girder), three positions were considered (i.e. subscripts 1, 2, and 3), to investigate the effect of damage position on the detection algorithm.

Figure 6.

Top view of the bridge deck. Damages scenarios are highlighted in the figure, concerning cross girders and stringers.

Plan of simulations

The methodology presented in this work was tested considering variations in terms of operational variables. To this end, the authors acted on the following features:

train mass;

track irregularity;

travelling speed.

In particular, five different configurations of the same convoy were used, changing carbody masses to account for different passenger traffic levels during the day. A comparison between numerical axle loads for the adopted vehicles and the experimental counterpart consisting of a sample of 268 trains of the same typology is shown in Figure 7. Track irregularity was changed to model its evolution in time, among different vehicle transits. This was done, starting with the generation of one reference longitudinal profile obtained as a random phase spatial realisation of the PSD curve defined by ORE B176,⁸⁴ considering “low level” defects. Then, other four profiles are constructed starting from the reference one. To do so reference amplitudes and phases are increased of a quantity whose maximum depends on the spatial frequency range under analysis: 5% for spatial defects larger than 3 m, and 10% for spatial defects lower than 3 m. This procedure aims to model the differential evolution of track irregularity, which is assumed to evolve faster for lower defects than larger. As mentioned before, train’s mass variations are considered as well, with five different train convoys used for the simulations. Furthermore, the modelled vehicles move with a forward speed ranging from 80 to 130 km/h. Therefore, the simulated scenarios account for a large variability of inputs other than the occurrence of a damage, whose detection is the target of the presented methodology.

Figure 7.

Axles load for the five passenger trains considered in the simulations set (train 1–5). Solid red line describes the mean value for each axle load of the convoy, measured on a population of 268 corresponding real train passages. Blue dashed lines define $\pm 3 σ$ confidence bounds.

Methodology

This section is dedicated to the presentation of all the steps of the developed damage detection methodology, starting from the raw data consisting of vertical bogie accelerations. As highlighted in the introduction, the authors concentrated on exploring the feasibility of using a single sensor, installed on the leading bogie of the train, to evaluate the health condition of a Warren truss bridge. The ability to rely on one single sensor, to detect the presence of structural damage across the bridge deck, presents significant potential in terms of cost-efficiency, offering a highly advantageous alternative when compared to more complex direct monitoring systems. The presented methodology relies on the following main steps:

CWT applied to leading bogie vertical acceleration. To do so, a complex Morlet wavelet is adopted and set according to Fitzgerald et al.²⁸;

Identification of the frequency range of interest, given the dimension of the bridge module (of length d) and the travelling speed, which together define the passing frequency $f_{p}$ ( $V / d$ ). The interval of interest ranges from $0.5 \cdot f_{p}$ to $2 \cdot f_{p}$ ;

Averaging of the absolute values of the obtained wavelet coefficients along frequency dimension, in the band of interest;

Resampling over space dimension of the curve obtained in the previous point;

Adoption of a sparse autencoder (SAE) used to encode and decode the obtained CWT curve;

Presentation of two different damage indices: the first is based on Hotelling’s distance applied to the latent space, while the second is simply the signal reconstruction error measured by MAE.

Each point introduced before is illustrated in Figure 8, and explained in detail in the following sections. Please notice that the bogie vertical acceleration time series, as well as its CWT scalogram, are plotted between two specific time instants. Precisely, the time at which the first axle of the “instrumented bogie” enters the bridge and the time where its second axle exits it. To move from time to space domain, the travelling speed must be considered, assumed to be constant. This means that the spatial window of interest (whose length is $L_{w}$ ), that is, including the bridge, is slightly longer than the bridge span, of a quantity equal to the train wheelbase (i.e. 2.85 m). In other words, the spatial window of interest is long as the bridge plus the wheelbase distance.

Figure 8.

Representation of the damage detection algorithm flow chart.

Signal processing: CWT

For each considered scenario, the obtained vertical bogie acceleration is processed through 1D Continuous Wavelet Transform (CWT).⁸⁵ The latter offers several advantages compared to Fast Fourier Transform⁸⁶ when processing and analysing time series with the purpose to extract damage-sensitive indications. In fact, CWT provides a time-frequency representation of the processed signal, allowing the analysis of its frequency content as well as how it evolves with time. Instead, FFT provides the user only with frequency information without retaining any time localisation capability. Therefore, FFT is generally less effective in the identification of damages which are localised in space (i.e. in time). Moreover, CWT is inherently more suitable for the processing of non-stationary signals and it also allows for multi-resolution analysis. This is possible provided that wavelet atoms can be translated in time and dilated⁸⁵ in shape: this aspect clearly enhances the detection of damages that can manifest their effects at different frequencies.

In this work, a complex Morlet wavelet is considered to perform CWT operation on vertical bogie accelerations. This family of wavelets is described by the following equation:

ψ (t) = \frac{1}{\sqrt{π \cdot B}} \cdot e^{- \frac{t^{2}}{B}} \cdot e^{j 2 π f_{c} t}

(5)

where two input parameters can be clearly distinguished, namely the bandwidth B and the centre frequency $f_{c}$ . While the bandwidth parameter B determines how localised is the wavelet in the frequency domain, $f_{c}$ represents the frequency that characterises its overall behaviour.⁸⁷ Although there are no precise or predefined guidelines for choosing one type of wavelet over another, there exist a number of studies in the literature on damage detection where the authors have opted for the Morlet wavelet,^28,88,89 either real or complex.

For the present study, the authors opted for the same centre frequency and bandwidth used by Fitzgerald et al.,²⁸ given their promising results for the detection of scour defects. Therefore, a bandwidth B of 1 was assumed, together with a centre frequency $f_{c}$ equal to 1.5. For each simulated scenario, the CWT operation was carried out on the vertical acceleration of the leading bogie of the train convoy, by using the Python function cmor, based on Torrence and Compo.⁹⁰ The modules of the obtained coefficients are referred to space (instead of time), where they are resampled, considering a unique spatial step (invariant with the simulated scenarios). Compared to Fitzgerald et al.,²⁸ in this work, the focus is on a frequency range which is not universal once given the train, but depends on two factors, mentioned before: first, the bridge modular geometry, that is, the length of each module composing the truss structure, and, second, the speed at which the train travels during its transit over the monitored structure. Therefore, the passing frequency ( $f_{p} = V / d$ ), whose concept was presented in Bernardini et al.,⁹¹ is the main driver of this step of the methodology herein presented. In particular, the absolute values of the wavelet coefficients (CW) contained in the frequency range going from $0.5 \cdot f_{p}$ to $2.5 \cdot f_{p}$ (see Figure 9(a) and (b)) are considered and averaged along frequency direction, according to:

C W_{m} (ξ) = \frac{1}{N_{f}} \sum_{k = 1}^{N_{f}} CW (ξ)

(6)

where $N_{f}$ is the number of frequencies contained between $0.5 \cdot f_{p}$ and $2.0 \cdot f_{p}$ values, while $ξ$ is the spatial abscissa, once resampled ( $Δ ξ = 0.045 m$ ). After a normalisation of the spatial abscissa, considering $L_{w}$ as the length of the window of interest (length of the bridge plus the wheelbase distance), a curve such as the one reported in Figure 9(c) is obtained, where a scenario characterised by healthy structure and a forward speed of 100 km/h are considered.

Figure 9.

Process to obtain $C W_{m} (ξ)$ : (a) scalogram, (b) $C W_{m}$ curve as a function of time, and (c) $C W_{m}$ as a function of $ξ / L_{w}$ . Healthy structure, train at travelling at 100 km/h.

Autoencoders architecture and preprocessing

Sparse Autoencoder (SAE) is a variation of the conventional autoencoder, specifically designed to learn sparse representations of data, emphasising the extraction of the most distinctive features. The SAE achieves this by introducing a sparsity constraint during training, which encourages the model to encode only the most relevant information while ignoring unnecessary details.⁹² This feature results in a more efficient and focused representation of the data, helping the model distinguish between significant patterns and background noise.

During the training process, differently from a Convolutional autoencoder (CAE), the SAE incorporates a sparsity penalty term $Ω (h)$ in the loss function used for training the network, as defined by:

L (x, g (x)) + Ω (h)

(7)

where x is a vector representing the network inputs, $h = f (x)$ is the output of the encoder (i.e. the latent space), being $f (x)$ and $g (x)$ the functions that respectively represents encoder and decoder. The L function represents the Mean Absolute Error (MAE) employed, while $Ω (h)$ introduces a constraint on the latent layer, encouraging some of the neurones in $h$ to remain inactive (close to zero) for most inputs. This promotes sparsity, meaning that only a few neurones are activated, focusing the model on the most crucial features of the data.⁹²

By enforcing sparsity, the SAE enhances the model’s ability to highlight fundamental characteristics of the data, making it more effective at identifying structural changes or anomalies.²¹ The sparsity constraint reduces the risk of overfitting by preventing all neurons from being activated simultaneously, making the model more effective for unseen data. Moreover, it enforces compact feature representations, making them beneficial for high-dimensional data analysis and reducing computational costs.

In this work, two different neural networks (NNs) are employed, distinguished by the type of input they receive: the first network, hereafter called $SA E_{CW}$ and shown in Figure 10, uses a scaled $C W_{m}$ signal as its input ( $C W_{avg}$ in Figures 10 and 11), while the second one, hereafter called $SA E_{CW, V}$ and shown in Figure 11, incorporates the velocity of the train as an additional input. In this second case, the velocity is not provided directly to the encoder, but, once scaled, is fed to the latent space as suggested in Bono et al.⁹³ The concept of providing the velocity in the latent space is based on its critical role in shaping the signal, and it needs to be appropriately weighted by the network. Concatenating velocity with the $C W_{m}$ signal as a single input causes the velocity information to become entangled with the signal’s dimensional characteristics. Both the two NN structures are adjusted by means of a Bayesian optimisation that is described in detail in the dedicated section. Before being given as input to the autoencoder, $C W_{m}$ signal is preprocessed by means of a z-score transformation, defined by the equation (8):

C W_{avg} = \frac{C W_{m} - μ}{σ}

(8)

where $μ$ and $σ$ respectively represent the mean and standard deviation of the $C W_{m}$ over the training set. This ensures that the distribution of the signals in the training have zero mean and a standard deviation of one. Additionally, in the second network also the velocity is preprocessed with a z-score transformation.

Figure 10.

Schematic representation of the first network without considering train velocity, called $SA E_{CW}$ .

Figure 11.

Schematic representation of the second network, which considers train velocity, called $SA E_{CW, V}$ .

Damage detection: Batch reconstruction error

The reconstruction error used to assess the presence of damage is the Mean Absolute Error (MAE), as defined in equation (9):

MAE (x_{j}, {\hat{x}}_{j}) = \frac{1}{N_{p}} \sum_{i = 1}^{N_{p}} | x_{j, i} - {\hat{x}}_{j, i} |

(9)

where $x_{j, i}$ is the i-th element of the j-th original signal fed to the NN, ${\hat{x}}_{j}$ is the corresponding signal reconstructed by the NN, and $N_{p}$ is the total number of points composing them. To ensure a more robust estimation of the damage, the error is not computed for a single signal. Instead, a mean value is taken over a batch of signals, resulting from a group of train transits. Thus, for a batch of N signals, the damage index $B_{MAE}$ is defined in equation (10):

B_{MAE} = \frac{1}{N} \sum_{j = 1}^{N} MAE (x_{j}, {\hat{x}}_{j})

(10)

This approach ensures that the damage index accounts for multiple signals (multiple runs), providing a more reliable estimate of the damage level. In this work, the original signal $x_{j}$ is represented by $C W_{avg}$ , defined by equation 8.

Damage detection: Hotelling’s $T^{2}$ control chart

By means of Hotelling’s $T^{2}$ control chart, an in-control region is established, and deviations from normal processes, which may indicate damage, are detected when data points fall outside this defined region. In addition to its effectiveness in early detection, $T^{2}$ is a widely accepted tool within the scientific and industrial communities due to its straightforward and intuitive interpretation, making it easy for users to understand and act upon the results. Its versatility and applicability across different fields make $T^{2}$ a powerful tool for identifying and quantifying structural changes in analysed signals, thereby enhancing the quality, efficiency, and reliability of SHM.^94,95 This control chart assesses statistical distance, which is inherently positive, and thus, the in-control region is defined as the interval $[0, UCL]$ . The upper control limit (UCL) represents the statistical distance corresponding to a specific significance level $α$ based on the distribution of data from the training set.

In this study, $T^{2}$ values were computed based on the latent space extracted by the autoencoder model (more specifically it consists in the output of the encoder), enabling the identification and quantification of changes or damage in the analysed signal. These changes were detected by comparing the $T^{2}$ statistic, which measures the distance between a new data point and the sample mean vector. A higher $T^{2}$ value indicates a greater deviation from the mean. This metric relies on the relationship between variables and data dispersion (covariance matrix).

In particular, considering a matrix $H_{NXM}$ containing N observations (corresponding to N train passages) of the M features contained in the latent space of the autoencoder, the j-th $T^{2}$ statistic is defined as following:

\begin{matrix} T_{j}^{2} = r {(\bar{h} - \bar{\bar{h}})}^{T} Σ_{0}^{- 1} (\bar{h} - \bar{\bar{h}}) & j = 1, 2, \dots, N / r \end{matrix}

(11)

where r is an integer parameter referred to as subgroup size that determines the $N / r$ number of $T_{j}^{2}$ that can be computed from the matrix $H_{NXM}$ , $\bar{h}$ is the vector containing the mean values of the latent space for the j-th subgroup containing r observations. Moreover, $\bar{\bar{h}}$ and $Σ_{0}$ are respectively the mean values and the covariance matrix of the latent space statistically estimated in the training period.

Results

Bayesian optimisation

To identify the best combination of hyperparameters, a sensitivity study is conducted involving multiple trials. Optuna⁹⁶ is employed to utilise sampling algorithms like the Tree-structured Parzen Estimator (TPE) for selecting hyperparameter sets to be evaluated. The TPE algorithm is particularly effective for high-dimensional optimisation problems and efficiently navigates complex search spaces. A total of 200 trials are conducted for each analysis, where each selected hyperparameters set is used to train and evaluate the model. An objective function is defined to measure the model’s performance based on the chosen hyperparameters, specifically by minimising the reconstruction error of the NN. As trials progressed, Optuna adjusts its sampling strategy to concentrate on areas of the search space likely to yield optimal hyperparameters. After completing 200 trials for each analysis, Optuna identifies the best hyperparameters combinations. It optimises the following parameters with a given set of possible values:

Number of convolutional filter of $con v_{1}$ layer – start: 16, step: 16, end: 64.

Kernel size of $con v_{1}$ layer: 3 or 5.

Number of convolutional filter of $con v_{2}$ layer – start: 8, step: 8, end: 32.

Kernel size of $con v_{1}$ layer: 3 or 5.

Dimension of the latent space: start: 8, step: 8, end: 64.

Number of convolutional filter of $con v_{3}$ layer – start: 8, step: 8, end: 32.

Kernel size of $con v_{3}$ layer: 3 or 5.

Number of convolutional filter of $con v_{4}$ layer – start: 16, step: 16, end: 64.

Kernel size of $con v_{4}$ layer: 3 or 5.

This optimisation is run for each network by considering ADAM optimiser on a training process of 1000 epochs and implementing a early stopping callback on validation loss of 30 epochs. The best hyperparameters obtained are resumed in Table 2.

Table 2.

Results of the hyperparameter optimisation.

Hyperparameters	$SA E_{CW}$	$SA E_{CW, V}$
$Con v_{1}$ # filters	32	64
$Con v_{1}$ kernel size	3	5
$Con v_{2}$ # filters	16	16
$Con v_{2}$ kernel size	3	3
Dimension of the latent space	16	16
$Con v_{3}$ # filters	16	16
$Con v_{3}$ kernel size	3	3
$Con v_{4}$ # filters	32	64
$Con v_{4}$ kernel size	3	5

Damage detection

The sparse autoencoder (SAE) is trained considering a total number of 760 simulations over the healthy structure, accounting for variability in terms of vehicle mass, speed, and track irregularity. The training set comprises 80% of the healthy bridge simulations, while both the validation and testing sets each account for 10%. The autoencoder is trained to encode the input $C W_{avg}$ curve into a compressed representation, from which the original signal is then reconstructed. In this regard, Figure 12 shows an example of the capability of the trained SAE to reconstruct the original signal. While the latter is plotted in blue, its reconstructed version is drawn with red dashed line. The adopted SAE architecture is the one schematically depicted in Figure 10, that is, $SA E_{CW}$ , while the considered cases refer respectively to healthy structure, in Figure 12(a), and damage scenario $S D_{3, 50 %}$ , in Figure 12(b). For both the two cases, the train travelling speed is set equal to 100 km/h. Comparing the two diagrams, it is possible to clearly notice that the SAE is much more capable of reconstructing the signal when considering a healthy structure than in the dual case considering a damage affecting the stringers in the last module of the bridge (see Figure 6).

Figure 12.

Original $C W_{avg}$ (blue) and signal reconstructed (red) by the autoencoder $SA E_{CW}$ : (a) healthy bridge and (b) damage $S D_{3, 50 %}$ and travelling speed of 100 km/h.

As previously introduced, two architectures were considered for the SAE neural network (NN), where the second accounts for vehicle speed, whose information is directly fed into the latent space (see Figure 11). For this second case, the performance of the $SA E_{CW, V}$ are shown in Figure 13. The analysed scenario is the same plotted in Figure 12. Therefore, the only difference is represented by the architecture of the NN which now also accounts for the train forward speed. From a visual point of view, the capability of the two NNs to reconstruct the signal appears to be similar. As expected, also in this case, the reconstruction appears much more effective when the original signal refers to the healthy structure.

Figure 13.

Original $C W_{avg}$ (blue) and signal reconstructed (red) by the autoencoder $SA E_{CW, V}$ : (a) healthy bridge and (b) damage $S D_{3, 50 %}$ and travelling speed of 100 km/h.

In this work, the objective is to investigate the feasibility of adopting single or multiple train transits to detect damage occurrence across bridge span. It is important to stress the following aspect: for each degraded scenario, simulations are carried out considering operational variables presence (considered also in healthy case), and, before grouping different train transits together, they are automatically shuffled. This means that each batch of train transits, referred to a specific damage scenario, contains a set of bogie vertical accelerations that was randomly extracted from the whole dataset concerning that specific damage. An important aspect to consider for damage detection is the number of train runs that are necessary to detect the onset of a damage. To this end, damage $C G_{3, 15 %}$ is now considered, which is the lightest damage considered for cross-girder members and, therefore, the most difficult to be detected. Figure 14 shows the accuracy performance obtained considering the $B_{MAE}$ (equation (10)) damage index and the $T^{2}$ statistic (equation (11)). Accuracy is defined according to the following equation:

Accuracy % = \frac{TP + TN}{TP + TN + FP + FN} \cdot 100

(12)

where TP refers to the number of true positives, TN to true negatives, while FP and FN are respectively the number of false positives and negatives identified by the presented damage index. In particular, while Figure 14(a) refers to the $SA E_{CW}$ Figure 14(b) regards the outcomes obtained with $SA E_{CW, V}$ .

Figure 14.

Accuracy of the damage detection algorithm as a function of train batch size: (a) $SA E_{CW}$ and (b) $SA E_{CW, V}$ . $B_{MAE}$ performances are plotted in blue, while in red the outcomes referred to $T^{2}$ statistic are shown. Damage scenario $C G_{3, 15 %}$ .

For the considered scenario, the $T^{2}$ damage index is shown to be significantly and consistently more accurate than $B_{MAE}$ , as shown both in Figure 14(a) and (b). Referring to the latter, it is possible to observe that the consideration of train forward velocity in the NN worsens the performance of $B_{MAE}$ for each train batch size: however, also without feeding the latent space with train velocity, insufficient performances are observed with the batch mean absolute error. Furthermore, the consideration of train forward velocity does not markedly affect the performance of the $T^{2}$ distance. The results obtained witness that, $T^{2}$ statistic seems to be much more accurate than the $B_{MAE}$ parameter for low intensity damages affecting cross-girders. According to the results shown in Figure 14, to detect $C G_{3, 15 %}$ damage with a 100% accuracy, batch of 40 trains are required. For smaller groups of travelling trains accuracy performance are not acceptable (even lower than 80%). Therefore, as a first results it was found that accounting for train speed as an input of the NN at the latent space level does not produce significant changes in terms of its performance and that adopting groups of 40 train transits produces accurate outcomes even in presence of relatively mild damages.

Going more in detail with the damage detection indices, Figure 15 shows the $B_{MAE}$ damage index, plotted for all the scenarios simulated in this work considering a damage towards the end of the bridge and batches of 40 trains. Instead, Figure 16 collects $T^{2}$ statistic, for the same scenarios and batch size. It is important to notice that different damage scenarios may have different samples being the number of simulations performed not equal for all the cases and, however, smaller than the training dataset. In Figures 15 and 16, the threshold (dashed blue line) is defined by the $95^{th}$ percentile computed on the training set. A common aspect emerges in these two figures: as expected, the two damage indices increase as the damage intensity grows, regardless the member affected. Table 3 gathers the accuracy performances showed by both the two DIs as a function of damage type and trains batch size, when damages towards the end of the bridge are taken into account (see Figure 6). In green, accuracies larger than or equal to $95 %$ are highlighted, while orange is used when accuracy is within the $81 %$ and $94 %$ . For smaller values, red colour is adopted. For high intensity damages (i.e. 50%) satisfying performances are obtained even considering batches of five trains only for both the DIs presented in this work. For corrosion intensities equal to 15%, the two DIs show some significant differences: in particular, damage scenarios affecting stringers (i.e. $S D_{3, 15 %}$ ) can be better detected considering $B_{MAE}$ (with at least 20 trains transits), which performs worse in case the damage affects cross girder (i.e. $C G_{3, 15 %}$ ). This result strengthens the idea of using two DIs, instead of one, to have a higher robustness in terms of damage detection capabilities. Two distinct damage typologies may produce very different effect on vehicle response, making one of the two DIs unsuitable compared to the other and vice versa.

Figure 15.

$SA E_{CW}$ : $B_{MAE}$ plotted for different damaged scenarios for batch size of 40 trains.

Figure 16.

$SA E_{CW}$ : $T^{2}$ statistic plotted for different damaged scenarios for batch size of 40 trains.

Table 3.

Accuracy performance for the two damage indices, as a function of considered scenario and the batch size. Two damage types and three different intensities are considered. The adopted NN architecture is $SA E_{CW}$ .

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

According to Figure 6, different damage positions were modelled in this work, for both CG and SD damage scenarios. This was done in the attempt to investigate any influence played by damage position over DI magnitude. Focusing on damage affecting cross girders, Figures 17 and 18 report the results in terms of $B_{MAE}$ and $T^{2}$ statistic for three different positions, considering batches of 40 trains. From Figure 17, it emerges that $B_{MAE}$ presents a markedly larger magnitude when the damage affects the last module of the bridge than the other two scenarios. When dealing with $T^{2}$ statistic, DI magnitudes significantly change according to damage position. Therefore, the Hotelling’s distance appears to be more sensitive to damage position than the batch mean absolute error, in the sense that the three curves referring to $C G_{1, 50 %}$ , $C G_{2, 50 %}$ , and $C G_{3, 50 %}$ are well separated. Table 4 collects accuracy performance featuring the two DIs as a function of damage positions considering the two damage typologies and maximum corrosion intensity, with the same colour scheme used for Table 3. Globally, worse performances are obtained when dealing with a damage affecting a cross-girder towards the centre of the bridge span. The same can be observed when dealing with damages affecting stringers, but in this case, this aspect is less clear. Before moving to the next section, it is important to recall that, throughout the whole paper, the colour shadings scheme, adopted for Tables 3 and 4, is used for all the tables collecting the performance, in terms of accuracy, of the presented methodology.

Figure 17.

$SA E_{CW}$ : $B_{MAE}$ considering batches of 40 trains. Same damage, affecting cross girder, with same intensity (50%), but different positions.

Figure 18.

$SA E_{CW}$ : $T^{2}$ statistic considering batches of 40 trains. Same damage, affecting cross girder, with same intensity (50%), but different positions.

Table 4.

Accuracy performance for the two damage indices, as a function of considered scenario and the batch size. Two damage types, different positions, and same intensity are considered. The adopted NN architecture is $SA E_{CW}$ .

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

Robustness to velocity error and measurement noise

To check and investigate damage detection performances in scenarios closer to real-working conditions, two other factors are taken into account in this section. One is the measurement noise, modelled according to the approach employed by de Souza et al.³⁶ and Sarwar and Cantero.⁵⁰ Therefore, a random artificial noise is added to the vertical bogie accelerations, with the following expression:

{\overset{\cdot\cdot}{Z}}_{1, n} = {\overset{\cdot\cdot}{Z}}_{1} + L_{n} \cdot σ ({\overset{\cdot\cdot}{Z}}_{1}) \cdot N (0, 1)

(13)

where $\underset{1}{\overset{\cdot\cdot}{Z}}$ is the vertical bogie acceleration directly obtained from the simulation, while $σ ({\overset{\cdot\cdot}{Z}}_{1})$ represents its standard deviation. Moreover, $N (0, 1)$ is a standard normal distribution vector with zero mean and unit standard deviation, while $L_{n}$ represents noise level and it is expressed as a percentage. In this work, three noise levels were considered, equal to $2 %$ , $5 %$ , and $10 %$ respectively. Being the proposed methodology dependent on train speed estimation (for the choice of the frequency band, see Figure 8), regardless the considered DI and NN architecture, another aspect must be taken into account. In fact, depending on the chosen methodology, forward velocity can be estimated with some inaccuracies. Therefore, some artificial estimation error is added when defining the frequency range for the computation of the $C W_{avg}$ curve (see equation (8)). In particular, the nominal speed was corrupted with a quantity equal to $\pm 3 %$ , randomly extracted from a uniform distribution, resulting into an inaccurate selection of the frequency range of interest (see Figure 8). For the way they are modelled, inaccuracies in vehicle speed estimation lead also to imprecisions related to train positioning, with maximum errors in the range of 10–12 m, which is deemed to be reasonable in real-world working scenarios. Therefore, the selection of the window of interest for the damage detection algorithm is corrupted: in other words, the selected spatial region, on which $C W_{avg}$ is calculated, can be misaligned with the bridge of a maximum quantity about 12 m. In this way, it was possible to model some of the main non-idealities that could affect the method when applied to real-world scenarios and hinder damage detection performances. In particular, Table 5 is the dual of Table 3, but considering now the addition of measurement noise of $2 %$ and speed estimation inaccuracies. The latter cause a global decrease of the performances (larger red region): damages $C G_{3, 15 %}$ , $C G_{3, 25 %}$ , and $S D_{3, 15 %}$ (see Figure 6) are no longer detectable, while the other damages can be all identified considering batches of 40 trains and both the two DIs. There are still cases in which damage is detected for a lower amount of train transits: for instance, for damage $S D_{3, 50 %}$ the detection is accurate also considering batches of five rail vehicles. Instead, in case of milder damage affecting stringers, that is, $S D_{3, 25 %}$ , good performances are obtained also for groups of 20 transits.

Table 5.

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

Table 6 is the dual of Table 4, again, considering the addition of $2 %$ measurement noise and speed estimation inaccuracies. Regardless the position of the damaged elements, with the largest corrosion intensity, the damage is always detectable with at least one damage index, even with batches accounting for less than 40 trains, except for damage $C G_{2, 50 %}$ that cannot be identified. In fact, another consideration can be derived from Table 6: it appears that damages positioned in proximity of the midspan (i.e. $C G_{2, 50 %}$ and $S D_{2, 50 %}$ ) can be identified with lower accuracy than the ones closer to bridge ends. This feature was also observed when dealing with perfect forward speed estimation (see Table 4) and null measurement noise. As mentioned before, Tables 5 and 6 point out a global reduction of the accuracy performances. Moreover, another aspect can be observed: $B_{MAE}$ appears to be more robust than $T^{2}$ index. In particular, it allows for the estimation of damage $S D_{3, 25 %}$ with groups of 20 train transits, while $T^{2}$ does not produce comparable outcomes when dealing with scenario $C G_{3, 25 %}$ (see Figure 6). Therefore, it derives that speed and position estimation inaccuracies appear to affect $T^{2}$ based DI more than $B_{MAE}$ . However, the use of two indices, instead of one, represents a more robust damage detection solution than considering just one single DI. In other words, if an exceed of one index is confirmed by the exceed of the other, more robust information regarding the possible presence of a certain damage in the structure is provided, at no additional implementation and computational costs (i.e. the NN is the same, regardless the DI).

Table 6.

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

For the case featured by a measurement noise of $5 %$ , similar results were obtained, as shown in Tables 7 and 8. For the latter, no significant differences can be observed if compared to Tables 5 and 6 respectively, where same speed inaccuracies are considered, with a 2% measurement noise levels. Finally, a measurement noise of $10 %$ was modelled, according to equation (13). Compared to the previous case, a slight degradation of damage detection accuracy is observed in Tables 9 and 10. By examining Tables 10 and 8, it can be noticed that $T^{2}$ based DI is not sufficiently accurate for $C G_{2, 50 %}$ damage detection even considering batches of 40 trains. However, it appears that increasing measurement noise from $2 %$ to $10 %$ does not play a role as significant as the one played by the inaccuracies in speed estimation, kept constant and equal to $\pm 3 %$ . Precisely, the degradation from ideal conditions (Tables 3 and 4) due to the consideration of a $\pm 3 %$ corruption of train speed and position estimation, accounting for the lowest noise level (i.e. $2 %$ ), is much larger than the one produced by moving from the lowest noise level to its maximum (i.e. $10 %$ ), keeping speed and position errors constant. This can be explained by two aspects: first, the damage feature that is given as input to the SAE. In fact, $C W_{avg}$ curve strongly depends on the chosen frequency band, which is related to the estimated vehicle speed, given the geometry of the train (i.e. module length). Moreover, another aspect deserves attention: inaccuracies in speed estimation leads also to a corruption of train position identification. This means that the window of interest will not coincide exactly with the one obtained with perfect speed estimation, but it will rather be translated of a quantity. The latter will produce a spatial window which is misaligned with the bridge itself, worsening damage detection performances. Outcomes regarding the NN architecture $SA E_{CW, v}$ are not reported for sake of brevity, but, as expected, this NN presents lower accuracy than $SA E_{CW}$ , since it also feeds an inaccurate speed estimate at latent space level.

Table 7.

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

Table 8.

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

Table 9.

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

Table 10.

For accuracy larger or equal to 95% the shading colour is green. For values between 94 and 81 is orange. Finally, for values lower than 81, the colour of the shading is red.

Based on the results shown in this section, a set of considerations can be drawn: the authors have developed a robust damage detection algorithm utilising Continuous Wavelet Transform (CWT) and sparse autoencoder (SAE) NN. Compared to de Souza et al.,³⁶ where the authors use a SAE-based damage detection algorithm, the present paper requires only one single sensing point rather than multiple ones. In fact, the proposed methodology allows for the identification of defects using a single sensor mounted on the leading bogie of the heading coach, which, compared to techniques developed in other studies,^51,52,54 potentially results in significant cost savings.

The study focuses on a steel truss structure, a structural archetype not extensively discussed in the related literature. In fact, according to the authors’ best knowledge of past researches in this field, only the study, proposed by the authors in Bragança et al.,⁵⁵ presents a damage detection technique exploiting CWT and autoencoders applied to a Warren truss bridge. In relation to that work, the approach proposed in this paper involves a passenger train instead of a freight train, requires the use of only one sensor (instead of eight), and is demonstrated to be effective at higher speeds (i.e. up to 130 km/h instead of 60 km/h). Then, compared to other works, such as Fitzgerald et al.,²⁸ the frequency band of interest is chosen based on the structural conformation which is typical of the bridge under analysis as proposed by Bernardini et al..⁹¹ In this study, the authors also explored the use of a Damage Index (DI) based on features contained in the latent space, rather than focusing on the decoder output, differently from what has been done in other works, including References 36, 51, 52, and 55. Finally, under this regard, according to authors’ best knowledge, this work represents the first application of the concept of Hotelling’s distance for latent space features distributions in the field of drive-by bridge monitoring. Further insights are critically discussed in the conclusions section, outlining main achievement and limitations.

Conclusions

In this work, the authors focused on the definition of a damage detection approach based on Sparse Autoencoder (SAE) and wavelet coefficients, which exploits one single measurement point to detect corrosion damages affecting a Warren truss bridge. Vertical accelerations of the first bogie of the leading coach are processed through Continuous Wavelet Transform (CWT) considering complex Morlet as the mother wavelet. The obtained scalogram was then reduced to a portion of interest defined in the frequency range given by the train forward speed and the module length of the bridge. In this frequency band, the absolute values of the obtained wavelet coefficients are averaged and interpolated in space. This curve represents the input to a SAE NN architecture, whose parameters were defined through Bayesian optimisation, trained to learn to decompose and recompose the input signal. Two NN configurations were modelled and trained in this work, not incorporating or incorporating train speed, that was considered at the latent space level. Two different damage indices (DIs) were proposed: one consisting of the batch Mean Absolute Error ( $B_{MAE}$ ) while the other is represented by the $T^{2}$ Hotelling’s statistic. The former is computed at the exit of the decoder, comparing the reconstructed signals with the input one. Instead, the latter is computed at the latent space level, exploiting the latent representation of each input signals. To test the effectiveness of the proposed approach, operational variables effect is accounted for in this work, considering different train speeds in the range 80–130 km/h, different weights (experimentally calibrated) and track geometrical irregularity to accounts for track geometry evolution in time. Two damage typologies were modelled in this work, affecting cross girders or bridge stringers. Three distinct damage intensities were simulated, namely $15$ , $25$ , and $50 %$ and different positions of the flawed truss (see Figure 6).

When dealing with perfect speed and position estimations, the work led to the following main results:

accounting for train speed information at the latent space level does not lead to a more effective SAE NN with improved accuracy performances;

the proposed DIs are found to be both very accurate when dealing with batches of 40 trains and damage intensities equal to $50 %$ ;

considering the lightest modelled lightest modelled damages, $T^{2}$ was shown to be accurate in detecting $C G_{3, 15 %}$ , $B_{MAE}$ turned out to be capable of detecting $S D_{3, 15 %}$ . When noise and inaccuracies are instead considered, $B_{MAE}$ turns out to be more effective in general. Therefore, it follows that using both indices, instead of one, represents a more robust solution in terms of damage detection accuracy;

the presented DIs are sensitive to damage intensity: an increase of damage intensity reflects in a larger magnitude for both the DIs;

when considering damages affecting cross girders, highest DI magnitudes are obtained when the flawed element is closer to the end of the bridge;

T² distance appears to be strongly dependent on the position of the damaged cross girder.

In this work, the authors also accounted for three different levels of measurement noise ( $2$ , $5$ , and $10 %$ ), as well as inaccuracies in vehicle speed estimation ( $\pm 3 %$ ). It is important to stress that errors in vehicle speed estimation also reflect in train position inaccuracies, which represent a critical aspect for the success of the methodology. The following additional conclusions can be drawn:

Globally, methodology performances decrease, still showing promising results, when dealing with 50% corrosion damages;

When keeping speed and position estimation inaccuracies constant (i.e. $\pm 3 %$ ), moving from $2 %$ to $10 %$ does not markedly affect the overall accuracy performances;

B_MAE index appears to be more robust than $T^{2}$ distance. This could be explained by the fact that this index is probably less affected by speed and position estimation errors;

Despite the previous point, using both of the two DIs is deemed to be a robust choice that does not require any additional implementation and/or computational costs.

While promising results are achieved, it is worth mentioning that the presented methodology has some limitations, among which, the most complex to be tackled regards the need for a robust and accurate estimation of the forward speed as well as train position, which is crucial for a proper identification of the spatial window of interest. In addition, while relying on one single sensor has several advantages in terms of costs, it may result less robust, due to lack of redundancy of the measuring solution. Finally, it is worth stressing that since this approach exploits the modular conformation which is typical of Warren truss bridges, another choice on the frequency range of interest must be carried out when dealing with other bridges. Some possibilities can be found in literature, for example, References 24, 27, 28, and 51. However, the obtained outcomes pointed out the promising performances of the presented methodology, which requires only one sensor to be installed on the train, representing a significant advantage in terms of costs, if compared to direct systems and to other drive-by approaches relying on sparse autoencoders. Future outlooks will consist of combining the developed methodology with the adoption of more than one sensor. Moreover, the effect played by module length and speed estimation errors, without any position estimation inaccuracy, will be investigated.

To further explore the potential of sparse autoencoders, future research can focus on integrating sparsity with additional regularisation techniques such as variational methods to enhance performance. Another promising direction is the combination of sparse autoencoders with generative models, such as GANs, to improve latent space representations.

Footnotes

Handling Editor: Chenhui Liang

ORCID iDs

Lorenzo Bernardini

Francesco Morgan Bono

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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.

Data availability statement

The data that support the findings of this study are not openly available due to reasons of sensitivity. Some of the data are available from the corresponding author upon reasonable request.

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Drive-by damage detection based on the use of CWT and sparse autoencoder applied to steel truss railway bridge

Abstract

Keywords

Introduction

Simulation software

Train and track models

Bridge FE model

Train-track-bridge dynamic interaction

Damaged scenarios

Plan of simulations

Methodology

Signal processing: CWT

Autoencoders architecture and preprocessing

Damage detection: Batch reconstruction error

Damage detection: Hotelling’s T 2 control chart

Results

Bayesian optimisation

Damage detection

Robustness to velocity error and measurement noise

Conclusions

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

Data availability statement

References

Damage detection: Hotelling’s $T^{2}$ control chart