Fastener partial looseness identification of the ballastless track by a time-domain Bayesian approach

Abstract

The fastener system is an important component of the ballastless track, and the fastener looseness threatens the safety of the railway track. This paper proposes a Bayesian model updating method integrated with a Python–Abaqus interface framework to identify the fastener partial looseness of the ballastless track via vibration data. By following the technical standards of Chinese Railway Industry, a laboratory scaled ballastless track model was constructed for the demonstration and verification of the proposed methodology. Not only the damage location can be accurately identified, but also the damage severity can be evaluated. In addition, the associated posterior uncertainties of the identified results can be quantified by calculating the posterior probability density functions of all uncertain model parameters, which can provide valuable information to engineers for ballastless track system damage detection.

Keywords

Time-domain Bayesian model updating looseness identification fastener ballastless track

Introduction

The fastener system in ballastless track is an important component to guarantee the track geometries and safety. With the rapid development of the high-speed railways, the rail structures are suffering from the reciprocating heavy train loads, which may cause the elastic fasteners to become loose/failed. And the loose/failed fasteners will lead to a large variation in the lateral wheel/rail forces and increase the vibration of rails (Zhou and Shen, 2013; Weng et al., 2008) and reduce the stability of the rail track. What’s more, the fastener looseness will lead to the failure of the adjacent ones (Yuan et al., 2021). Therefore, if the loose fasteners are not detected and repaired in time, the track structure will be further damaged and the safety of the track structure will be threatened (Xu et al., 2017).

At present, regular visual inspection is a common method for the detection of the track fastener damage, especially in a country such as China with the world’s largest high speed railway network. However, visual inspection can only detect the apparent condition of the fasteners, it is difficult to discriminate the slight damage such as fastener partial looseness and clip slight shifting, which is no longer adapted to the development of railway tracks (Aytekin et al., 2015). A variety of loosening detection methods have arisen with the evolution of sensor, signal processing and computer vision technology (Huang et al., 2022), among which the machine vision and artificial neural networks based inspection methods were the major methods utilized for railway fastener detection. Gibert et al. (2017) put forward a new algorithm integrating a multitask learning framework with multiple detectors to detect the defects of railway ties and fasteners, the promising detection performance showed that the proposed approach can improve the detecting accuracy. While this approach is limited to identify invisible defects of the structure. Mao et al. (2018) proposed a promising high-speed railway fastener inspection approach utilizing the structured light sensors to identify the damaged fasteners with a high accuracy over 99.80%. While this method can only effectively identify the location of loose fasteners, but cannot effectively identify the severity of fastener looseness, what’s more, the 3-D point clouds from the structured light sensors are easily affected by some unrelated parts next to the rail track and the cost involved is high. Therefore, the visual-based identification methods can hardly detect the fastener looseness or the elastic clip damage in the early stage, a practical railway fastener looseness detection is significantly demanding nowadays.

Recently, the research on structural damage identification by solving engineering inverse problems has made great progress (Xia et al., 2008; Esfandiari et al., 2010; Zhu et al., 2013). However due to the influence of measurement noise and the limited number of sensors as well as the complexities of boundary conditions, the errors related to constructing a theoretical model of a structure are inevitable, which may lead to the non-convergence in model updating process and high uncertainties of damage identification results (Bakhary et al., 2007). Compared with the deterministic damage identification method, the probabilistic method can better deal with the uncertainties of the damage identification process (Saito et al., 2005). Beck and Katafygiotis (1998) have done pioneering work in the field of Bayesian model updating. Some researchers (Lam et al., 2014, 2018; Hu et al., 2018) proposed a ballast damage detection method based on Bayesian model updating for the first time in railway track. Noted that the mentioned works employ the modal-domain data to identify the uncertain model parameters. For the raw data is in time-domain (e.g., from accelerometers), an extra modal identification process is necessary to transform the time-domain data to modal-domain data, which may be time-consuming and easily cause probable loss of information (Lam et al., 2017). In order to avoid this additional process, a fastener partial looseness identification method based on the time-domain Bayesian model updating integrated with a Python–Abaqus interface framework is proposed for the first time in the realm of structural damage detection in the railway track.

The structure of this paper is outlined below. The modeling of the ballastless track structure is firstly introduced, afterwards the Bayesian approach is proposed and developed for fastener stiffness identification. Then the proposed approach is demonstrated and validated by using the experimental case studies. It turns out that the proposed approach is feasible to properly detect the fastener partial looseness and quantify the uncertainties involved in the identification process simultaneously.

Proposed ballastless track fastener looseness detection method

Modeling of a representative ballastless track system

The ballastless track structure is composed of the rails, fasteners, and the substructure as exhibited in Figure 1(a). The heavy train load was directly supported by the rails, and transformed to the substructure through the fasteners. Fasteners are significant to fix the rails, maintain the gauge distance and reduce the dynamics between the wheel and rail (Xu et al., 2013). A three-dimensional finite element (FE) model (as exhibited in Figure 1(b)) of a scaled ballastless track model in the laboratory (as shown in Figure 4) was established by using the commercial software Abaqus. The substructure (including the prefabricated slab, the cement-emulsified asphalt (CA) mortar layer and the concrete base) was established by using linear hexahedral elements (C3D8R), and tie contact was utilized to simulate the interface contact between the substructure of the ballastless track. The elastic foundation was considered in this study. As the purpose of this study is to investigate the feasibility of the proposed Bayesian model updating method integrated with a Python–Abaqus interface framework to identify the fastener partial looseness of the ballastless track, only half of the track was considered in this study. In the FE model, the length of the track system was set to 1.6 m (along the rails) and the width was set to 1.4 m (perpendicular to the rails), and the dynamic system is assumed to be classically damped and the damping ratio of the track system was assumed to be 8%, which was estimated by modal analysis of the measured impact hammer test. The fasteners were simulated by using the linear elastic spring elements, and the masses of the fasteners are ignorable compared to that of the whole vibration system. WJ-8 type fastener with the gauge apron plate (seen in Figure 4) was considered and the transverse and vertical stiffness of the fastener system were simulated by using spring sets. The spring stiffness reduction was considered to describe the looseness of the fastener in this study. In order to detect the fastener looseness, the scaled model (Hu et al., 2021) with the fastener system was utilized in this study, and more details about the values of the stiffness of the prefabricated slab, CA mortar layer and concrete base can be referred to Hu et al. (2021).

Figure 1.

The ballastless track structure. (a) Schematic of a representative ballastless track system. (b) A three-dimensional finite element model of the scaled ballastless track system.

The parameters which are likely to change by local conditions, load patterns, or damage should be considered as uncertain parameters (Lam et al., 2020; Adeagbo et al., 2021, 2022). The loose fastener only can influence the fastener stiffness a lot, and the values of the model parameters of the substructure (including the prefabricated slab, CA mortar layer and concrete base as well as the foundation) would not be changed too much, which can be referred to Hu et al. (2021), therefore, only the scaling factors of the fastener stiffness will be treated as the uncertain model parameters in this study. Impact force (as shown in Figure 2) was utilized to calculate the dynamic acceleration responses and was obtained from the impact hammer test on the scaled track model in the laboratory with the value of approximately 20,000N. The linear perturbation analysis was employed to calculate the dynamic responses of the FE model under impact excitations. Fastener looseness leads to the fastening system less stiff, which changes the dynamic characteristics of the track system (Zhan et al., 2020), therefore, fastener looseness identification can be treated as an inverse problem, in which the measured vibrational data is utilized in the process of back-calculating the fastener stiffness. In this study, only the vertical vibration of the track model and the uncertain model parameters of the fastening system were considered in the following analysis.

Figure 2.

A typical impact excitation.

Bayesian model updating

The main objective of Bayesian probabilistic approach is to calculate the posterior probability density function (PDF) of the uncertain model parameters $θ$ conditional on the measured data $D$ and the given model class $M$ (Beck and Yuen, 2014), as expressed in the following

p (θ | D, M) = c p (D | θ, M) p (θ | M)

(1)

where

c

is a normalization constant such that integrating the right-hand side of equation (1) is equal to unity,

p (θ | M)

is the prior PDF of the uncertain model parameters, a non-informative prior is employed to ensure that the posterior PDF is only controlled by the set of measured data. And the likelihood factor

p (D | θ, M)

is the dominant factor to reflect the contribution of the measured data

D

in establishing the posterior PDF.

As the model class $M$ is determined in this study, hence the symbol $M$ can be dropped in the following formulations. Let the dynamic data $D$ consist of the measured time histories at N_d degrees of freedom (DOFs) with N_s discrete data points. For a large number of available data points and for a relatively small of uncertain structural parameters, the updated probability distribution for these model parameters using Bayes’ theorem is sharply peaked at isolated optimal values (Beck and Katafygiotis, 1998) and the updated (posterior) PDF for the uncertain model parameter vector $θ$ given a large amount of data may be approximated accurately by a Gaussian distribution (Beck and Yuen, 2004). Therefore, it is assumed that the prediction errors at each time step and at each DOF follow independent and identical Gaussian distributions. Then the posterior PDF of the uncertain model parameters can be given by (Lam et al., 2017; Beck and Yuen, 2004):

p (θ | D) = c_{1} p (θ) {(2 π)}^{- N_{s} N_{d} / 2} {σ_{η}}^{- N_{s} N_{d}} \exp (- \frac{N_{s} N_{d}}{2 σ_{η}^{2}} J (θ))

(2)

where

c_{1}

is a normalizing constant,

{σ_{η}}^{2}

is the variance of the prediction error. The objective function is given by:

J (θ) = \frac{1}{N_{d} N_{s}} {\sum_{j = 1}^{N_{S}} ‖ \hat{x} (j) - x (j, θ) ‖}^{2}

(3)

where

\hat{x} (j)

is the measured acceleration at jth time step,

x (j, θ)

is the calculated acceleration based on the uncertain model parameter vector, and

‖ \cdot ‖

denotes the Euclidean norm of a vector. The most probable uncertain model parameters

\hat{θ}

can be obtained by maximizing

p (θ | D)

in equation (2). If the number of data point, N_dN_s, is large, to maximize the posterior PDF in equation (2) is equivalent to minimize the objective function in equation (3). Then the most probable value of the prediction error variance in

\hat{θ}

can be calculated by

{\hat{σ}}_{η}^{2} = \min J (θ)

. In globally identifiable case (Beck and Katafygiotis, 1998), the posterior PDF can be well approximated by a Gaussian distribution with the mean

\hat{θ}

and covariance matrix equals to the inverse of the Hessian of

- \ln [p (θ | D)]

\hat{θ}

. Instead of focusing only on the optimal model identified by model updating, the Bayesian approach aims on calculating the marginal posterior PDFs of the uncertain model parameters, which can provide valuable information on the uncertainties associated with the identification results.

Fastener looseness identification methodology

An overview of the proposed fastener partial looseness identification method of the ballastless track system based on a time-domain Bayesian model updating method integrated with a Python–Abaqus interface framework is schematically illustrated in Figure 3. The FEM of the ballastless track system was established in Abaqus, which is commonly used by engineers for structural analysis and calculation. And the measured impact force from the laboratory impact hammer test was put into the FE model. The linear perturbation analysis was adopted for the dynamic time domain analysis in this study to obtain the vibrational responses of the structure under impact excitation. Impact hammer tests were carried out on the target scaled ballastless track model in laboratory (as shown in Figure 4) to obtain the time domain responses (e.g., acceleration responses at different DOFs). A newly developed Python–Abaqus interface framework was firstly introduced for railway fastener partial looseness identification. Secondary development of the Abaqus model based on Python scripts was utilized to ensure the seamless interaction between the Bayesian model updating algorithm and FEM such that the read and write process of model-predicted time domain responses with different groups of uncertain model parameters can be easily implemented. After the Bayesian model updating was completed, the most probable model parameter vector $\hat{θ}$ can be obtained by following the proposed time domain Bayesian method, as a result, the stiffness of the fasteners can be estimated and the location and severity of the railway fastener partial looseness can be identified, and the uncertainties involved in the identification process can be quantified by calculating their corresponding coefficients of variation (COVs) and posterior PDFs.

Figure 3.

Flowchart of the proposed partial fastener looseness identification method.

Figure 4.

The scaled model of the ballastless track structure.

Experimental case studies

Experimental setup

To demonstrate the feasibility and practicability of the proposed time-domain Bayesian fastener partial looseness identification methodology, a scaled model of the ballastless track structure (as shown in Figure 4) was established in the laboratory of Huazhong University of Science and Technology, Wuhan. The main structural parameters and configuration dimensions of the ballastless track structure are listed in Table 1. One CHN60 rail was locked by three pairs of WJ-8 type fasteners (labelled by 1#, 2# and 3# along the rail as shown in Figure 4). The space of two fasteners is 0.65 m. The fasteners should be tightened with a torque of 100 Nm to 140 Nm according to the technical standards of Chinese Railway Industry (TB/T 3065.1, 2002). In this work, the standard pretension torque for the intact fastener was set as 120 Nm and the severity of fastener looseness could be defined as the ratio variation of the theoretical installation torque and the intact torque.

Table 1.

Main parameters of the ballastless track structure (Hu et al., 2021; Zhang et al., 2019).

Components	Parameters	Value
Rail	Density	60 kg/m
	Young’s modulus	210 GPa
	Poisson’s ratio	0.2
	Length (L)	1.6 m
Prefabricated slab	Density	2500 kg/m³
	Young’s modulus	36 GPa
	Poisson’s ratio	0.2
	Dimensions (L × W × H)	1.6 m × 1.4 m × 0.2 m
CA mortar	Density	1900 kg/m³
	Young’s modulus	330 MPa
	Poisson’s ratio	0.2
	Dimensions (L × W × H)	1.6 m × 1.4 m × (0.05∼0.15)m
Concrete base	Density	2500 kg/m³
	Young’s modulus	33 GPa
	Poisson’s ratio	0.2
	Dimensions (L × W × H)	1.6 m × 1.4 m × 0.25 m
Foundation	Stiffness	1.97 × 10⁶ kN/m²

Figure 6 shows the measurement equipment used in the impact hammer tests. A torque wrench (shown in Figure 6(a)) was used to set the preload torque of the fastener system. The impact hammer (Type 8210, seen in Figure 6(b)) with a sensitivity of 0.1985 mV/N and maximum force of 22,200 N was used to excite the track structure with different tips, softer tips were more suitable for exciting the lower modes of the system and stiffer tips were more appropriate for exciting the higher modes. And the medium hammer tip with the green color was employed, which was suitable for the current scaled model of the slab track, and the excitation location was shown in Figure 5. Six unidirectional IEPE sensors (with sensitivities around 100 mV/g, seen in Figure 6(c)) were placed on the rail seat near to the fasteners (as shown in Figure 5) to measure the vertical vibration responses of the track structure. The measured vibration signals from the accelerometers were transferred to the data acquisition equipment (Dong-Hua DH5922, seen in Figure 6(d)), and the equipment was connected to a computer with DHDAS dynamic signal acquisition and analysis system. The sampling frequency was set as 1000 Hz in the impact hammer tests and approximately five impulses were recorded for each set of measurements. A typical measured time-domain vibration response from one accelerometer is shown in Figure 7(a), and the fourth impulse is zoomed and shown in Figure 7(b), which was damped to zero in 0.2 sec implying that the damping of the system is relatively high.

Figure 5.

Schematic of the sensor locations and impact location in the impact hammer test.

Figure 6.

Experimental equipment.

Figure 7.

A typical measured time-domain vibration response. (a) A typical set of measured acceleration response. (b) zoomed response of the fourth impulse.

Three damaged scenarios were considered to demonstrate and verify the proposed fastener partial looseness identification method, which are shown in Table 2. Three representative loss ratios of 83.33%, 33.33% and 16.67% were utilized in this study. When the fasteners are set as the torque decreased by 83.33%, namely 20 Nm, and decreased by 33.33%, namely 80 Nm, and decreased by 16.67%, namely 100 Nm, which are partially loosened. Damaged Case 1 is a single damaged case with the damage located at 1# fastener, and Damaged Case 2 investigated a multiple damaged case with the damage located at 1# and 3# fasteners, and Damage Case 3 is a small damage case with the damage located at 2# fastener.

Table 2.

Health conditions of the fastener system.

Damage scenario	Preload torque of fastener	Damage Severity (%)	Damage Location
Undamaged case	120 N·m	0	—
Damaged Case 1	20 N·m	83.33	1#
Damaged Case 2	80, 20 N·m	33.33, 83.33	1# and 3#
Damaged Case 3	100 N·m	16.67	2#

Undamaged case

The elastic stiffness of the three fasteners is defined by the multiplication of its nondimensional scaling factor, $θ$ , by the nominal stiffness value, $k =$ 120 kN/mm, where $θ = {θ_{1}, θ_{2}, θ_{3}}$ , and is considered as the uncertain model parameter vector in this study.

Firstly, the time-domain Bayesian fastener partial looseness identification methodology was investigated by the measured data from the undamaged case. The Bayesian model updating results (i.e., the most probable values (MPVs) together with the associated posterior uncertainties (COVs)) are shown in Table 3 (Undamaged case). It can be seen that the identified scaling factors of the three fasteners are 1.03, 1.07 and 1.06, and all of them are very close to unity, implying that there is no damage in this experimental case. One advantage of the proposed Bayesian fastener partial looseness identification method is the capability to quantify the posterior uncertainties associated with the identification results. The calculated COV of the 1# fastener is only 4.35%, and the highest COV is from 3# fastener, which is also below 10.50%, indicating that the involved uncertainties of the undamaged case is relatively low.

Table 3.

Bayesian model updating results in undamaged and damaged cases.

		$θ_{1}$	$θ_{2}$	$θ_{3}$
Undamaged case	MPVs	1.03	1.07	1.06
Undamaged case	COVs (%)	4.35	5.24	10.23
Damaged Case 1	MPVs	0.18	1.08	1.07
Damaged Case 1	COVs (%)	4.27	6.85	17.52
Damaged Case 2	MPVs	0.59	1.10	0.19
Damaged Case 2	COVs (%)	7.97	7.18	16.28
Damaged Case 3	MPVs	1.06	0.77	0.95
Damaged Case 3	COVs (%)	7.36	9.20	14.56

Figure 8 shows the matching between the model-predicted and measured time-domain vibrational responses from the six accelerometers in the undamaged case. It is obvious that the matching is relatively good. It is believed that the accuracy of the identified FE model is good enough to help in response prediction of the track structure, which is extremely significant in the process of model updating.

Figure 8.

Matching between the measured and model-predicted responses in the undamaged case.

Damaged cases

Then three damaged cases are utilized to test the proposed Bayesian fastener partial looseness identification methodology. The Bayesian model updating results of the three damaged cases are listed in Table 3. In damaged Case 1, the most probable model parameter vector is $\hat{θ} =$ {0.18, 1.08, 1.07}, it is obvious that the scaling factor of 1# fastener is significantly reduced while the other two scaling factors of 2# and 3# fasteners are very close to unity, it can be concluded that the 1# fastener is damaged. The severity of the fastener looseness can be detected by comparing the stiffness of particular fastener with that from the undamaged state (the reference line) and the damage severity of 1# fastener is 82.00%, which is very close to the simulated value (83.33%). The associated COVs of $θ_{1}$ and $θ_{2}$ are both less than 7%, the COV of $θ_{3}$ is the highest (17.52%), which indicates that the uncertainties of the identified results are still not high. The comparison between the measured and model-predicted time-domain responses in damaged Case 1 is exhibited in Figure 9(a). It can be seen that the matching for all six accelerometers is acceptable in general. Similar analysis was conducted on the Damage Case 3, it can be noted that the scaling factor of 2# fastener is 0.77, which is much less than the unity, and it can be concluded that 2# fastener is damaged and the damage severity is 23%, while the uncertainty of this identified scaling factor is much higher than that of 1# fastener in Damage Case 1, indicating that the identification uncertainty is increasing when the damage severity become smaller.

Figure 9.

Matching between the model-predicted and measured responses in damaged cases. (a) Damaged Case 1. (b) Damaged Case 2.

In damaged Case 2, the identified most probable model parameter vector is $\hat{θ} =$ {0.59, 1.10, 0.19} and it is clear that the stiffness scaling factors of 1# and 3# fasteners are reduced, implying that there is damage in 1# and 3# fasteners, and the damage severities are 41.00% and 81.00%, respectively. The highest COV is 16.28% from 3# fastener, indicating that the uncertainties of identification results are not high. Then it can be concluded that the proposed Bayesian fastener partial looseness identification method successfully identifies the fastener partial looseness in this experimental case. With the most probable model, the model-predicted responses can be calculated, and which is compared with the measured responses (shown in Figure 9(b)). The matching for all six accelerometers is in a reasonable manner. In general, the loose fastener leads to a stiffness reduction of the fastener to the rail, resulting in the fatigue fracture of the fastener system (Zhou and Shen, 2013), which will lead to a large variation in the lateral wheel/rail force and increase the vibration (e.g. accelerations) of the rails, and the vibration of the prefabricated slab will decrease a bit correspondingly. As the excitation in the test is on the slab, the reduced amplitudes of the rail are smaller for the damaged track with reduced fastener stiffness, which can be seen from the comparison between Figure 8 and 9.

Uncertainty analysis

Generally, the less the amount of information available from the measurement, the larger the posterior uncertainties associated with the model parameters. Figure 10 shows the normalized marginal PDFs of the model parameters in the damaged cases 1 and 2. It can be seen that the posterior distributions of the scaling factor $θ_{3}$ in the two experimental damaged cases are much flatter than that of $θ_{1}$ and $θ_{2}$ , indicating that the uncertainty associated with the parameter $θ_{3}$ is larger. The reason may be that sensors 5 and 6 are a little far away from the impact location (seen in Figure 5), therefore limit information was extracted from sensors 5 and 6, which leads to a low accuracy and high uncertainty to the model parameter $θ_{3}$ in these two damaged cases.

Figure 10.

The normalized marginal PDFs. (a) Damaged Case 1. (b) Damaged Case 2.

Conclusions

This paper investigates the feasibility study on the identification of the fastener partial looseness of the ballastless track based on time-domain Bayesian model updating integrated with a Python–Abaqus interface framework using measured vibrational responses. A scaled ballastless track model was built in the laboratory for the verification and demonstration of the proposed methodology. The experimental case studies reveal that:

(1) The proposed Bayesian probabilistic approach integrated with the Python–Abaqus interface framework is feasible to identify the fastener partial looseness of the ballastless track by calculating the most probable values (MPVs) of fastener stiffness scaling factors.

(2) Not only the looseness location can be successfully detected, but also the looseness severity can be identified synchronously.

(3) The uncertainties associated with the fastener looseness identification results can be properly quantified by calculating the posterior PDFs (and COVs) of all uncertain model parameters.

(4) The closer the impact location is away from the sensor, the lower the uncertainties and the higher the accuracies of the corresponding identified model parameters are.

It can be concluded that the proposed method has a high potential of application in fastener partial looseness identification of the ballastless track, which can provide help for the detection of similar structural damage and make the health monitoring and early warning of the ballast track fastener system possible.

In this study, only three pairs of fasteners were considered in the experimental case studies, further work should be conducted on the field tests to ensure that the proposed methodology can be put into industrial applications, and not only the accelerations of the slab track, but the receptances of the rail can be utilized in the model updating in future work. It’s believed that the optimal locations and accelerometer(s) and impact are important to successfully detect the fastener partial looseness, which can be studied under the Bayesian framework by calculating the information entropy (Chow et al., 2011; Lam and Adeagbo, 2022). In addition, impact hammer tests were employed to collect vibrational measurement data of the ballastless track structure, for continuous monitoring of a long span of ballastless track, it is proposed to extend the proposed approach to utilize train-induced vibration for the identification of the fastener stiffness. As thousands of kilometers of the ballastless track are constructed of the same specification, a proper fastener looseness identification methodology can be repeatedly utilized along the entire railway line. Therefore, the long-term potential of the proposed approach cannot be underrated.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the This work was fully supported by the grant from the National Natural Science Foundation of China (52178287, 51708242); and Natural Science Foundation of Hubei Province (2020CFA047).

ORCID iD

Qin Hu

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