The Bayesian methodology for the detection of railway ballast damage under a concrete sleeper

Engineering Structures 81 (2014) 289–301

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The Bayesian methodology for the detection of railway ballast damage under a concrete sleeper H.F. Lam a,⇑, Q. Hu a, M.T. Wong a,b a b

Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong Special Administrative Region MTR (Mass Transit Railway) Corporation, Hong Kong

a r t i c l e

i n f o

Article history: Received 25 April 2014 Revised 19 August 2014 Accepted 21 August 2014

Keywords: Bayesian model updating Bayesian model class selection Railway ballast Damage detection Modal identification

a b s t r a c t In this paper, a model-based method is proposed to address the problem of detecting railway ballast damage under a concrete sleeper. The rail–sleeper–ballast system is modelled as a Timoshenko beam on an elastic foundation with two masses representing the two rails. The uncertainties induced by modelling error and measurement noise are the major difficulties for vibration-based damage detection methods, and therefore, a probabilistic approach is adopted in this study for addressing the uncertainty problem. The proposed ballast damage detection methodology is conceptually divided into two phases. In the first phase, the Bayesian model class selection method is used to select the most plausible model class from a list of predefined candidates based on a given set of measurements. In the second phase, Bayesian model updating is adopted to calculate the posterior PDF of uncertain model parameters using the selected model class from the first phase. Damage to the ballast decreases its stiffness in supporting the sleeper, and it can be detected via the marginal posterior PDF of the ballast stiffness at different regions under the sleeper. A segment of full-scale ballasted track was constructed indoors and tested under laboratory conditions to demonstrate and verify the proposed methodology. Discussions related to the limitations of the proposed methodology in real application are given at the end of this paper. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction A ballasted track consists of two major parts: a superstructure and a substructure. In the last few decades, many studies have considered track damage to the superstructure, including rails, fasteners and concrete sleepers, but very few have considered damage to the substructure [40]. A typical railway track substructure is composed of three layers: the ballast, sub-ballast and sub-grade. The ballast is the most crucial layer of the substructure. Its main function is to retain the sleepers and rails in their required positions against forces from different directions (due to trains and environmental factors). With the proliferation of high-speed train networks, railway ballast settlement and degradation due to ballast damage have become common. Undetected ballast damage causes changes in track geometries, resulting in uneven support for sleepers and increasing the chance of rail buckling and train derailment. Chinese Railways (CR) statistics have shown that about 75% of the daily maintenance work on track structures is performed due to railway ballast deformation [52]. Only a few track monitoring methods may be applied to ballast damage detection. The core ⇑ Corresponding author. Tel.: +852 34427303. E-mail address: [email protected] (H.F. Lam). http://dx.doi.org/10.1016/j.engstruct.2014.08.035 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

penetration test (CPT) involves obtaining samples of the substructure and testing them in a laboratory. Information related to the ballast, such as soil type, shear strength and compressibility data, may be estimated based on a series of laboratory tests. Ballast damage may be determined by comparing the friction ratio data of clean and fouled ballast [4]. However, the CPT is a destructive test that cannot be conducted frequently. The properties of ballast may be disturbed when samples are cored onsite. Furthermore, this kind of test is time consuming and may affect normal train operations. Ground penetrating radar (GPR) testing, which adopts ‘horns’ to transmit and receive electromagnetic waves within a given frequency range [43], is widely used to evaluate the layer properties of the road pavement substructure. Because there is usually no clear interface between the clean and fouled ballast (i.e., undamaged and damaged ballast), it is difficult for a GPR test to locate ballast damage [1]. In conducting a GPR test, it is not easy to select a proper frequency range for the electromagnetic waves. Although a high frequency (e.g., higher than 900 MHz) is preferred for increasing the resolution of GPR data, a low frequency (e.g., lower than 900 MHz) is suitable for achieving deep penetration [21]. Therefore, conducting a GPR test to detect ballast damage remains a challenging task [25]. In many countries, railway ballast damage detection often relies on visual inspection. However, it is

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only effective in detecting damage on the surface of the ballasted track. Ballast damage under the sleeper, which is believed to have significant effects on track performance, cannot be observed. Therefore, the development of an appropriate and practical railway ballast damage detection method is critical and would significantly decrease not only the costs involved in substructure maintenance, but also the likelihood of track accidents. Vibration measurement is popularly used in structural health monitoring [17,24,44,45]. Modal parameters such as natural frequencies and mode shapes depend on the physical properties of a structural system (i.e., stiffness and mass). If no change in system mass is assumed, detectable changes in the modal parameters can be treated as indicators of a reduction in the system’s stiffness (i.e., damage). This study defines railway ballast degradation during construction and operation (under repeated train loading) as ballast damage. The degradation of the ballast under the sleeper theoretically decreases the ballast’s stiffness, resulting in weaker support for the sleeper. When the stiffness of the system is altered, the vibration characteristics of the in-situ concrete sleeper change (indicated by changes in the measured natural frequencies and mode shapes of the in-situ sleeper). Ballast damage detection can be treated as an inverse problem, in which the measured modal parameters of the target in-situ sleeper are used to back-calculate the ballast’s stiffness. Kaewunruen and Remennikov [26,27] examined the effects of voids and improper ballast packing under sleepers on the modal parameters of an in-situ sleeper via numerical and experimental case studies. Their analysis showed that the natural frequencies of in-situ sleepers are sensitive to the voids underneath. This result provides strong evidence that the vibration characteristics of an in-situ sleeper can be used in ballast damage detection. The model-based method is popular for structural damage detection. Two main factors determine the success of such a method in detecting ballast damage: (1) a proper model to represent the dynamic characteristics of the rail–sleeper–ballast system and (2) an appropriate method for updating the rail–sleeper–ballast model based on a given set of measured vibration data [33]. In this study, the literature related to the modelling of concrete sleepers, ballast and in-situ sleepers is reviewed along with model updating methods. The theory of a beam on a Winkler elastic foundation has been around for almost one and a half centuries, and is considered the classical approach in railroad track analysis [22,53]. The Timoshenko beam element was first used to model the concrete sleeper in 1985 [19]. In 1995, Grassie proposed a simple two-dimensional dynamic model for calculating the modal parameters of a sleeper. This model was experimentally verified by 12 types of sleeper under a free-free support condition [20]. Dahlberg analytically modelled the dynamic behaviour of an in-situ concrete sleeper as a vibrating Timoshenko beam on an elastic foundation [14]. He compared the analysis results of the in-situ sleeper to those of the free-free sleeper, and concluded that the effect of ballast in the vibration characteristics of a rail–sleeper–ballast system was only significant for lower modes. This conclusion is consistent with the analysis results in [23,33]. The literature related to model updating has shown that the sensitivity-based [12,13] and optimal matrix [5,6] updating methods are the representative classical methods. These deterministic model updating methods laid the foundation for the development of the probabilistic model updating methods, which not only identify uncertain model parameters (via the calculation of the most probable values) but also quantify the uncertainties associated with model updating results by calculating the posterior probability density function (PDF) of the uncertain model parameters. Lam and colleagues presented a deterministic vibration-based ballast damage detection method and verified the method by both numerical and experimental case studies [33]. They concluded that

the use of model updating to detect railway ballast damage is feasible. In Lam’s work, the ballast under a sleeper was divided into six regions (the ballast stiffness in a region is a constant) for the purpose of ballast damage detection. However, no evidence was produced to support the selection of this class of model (i.e., the six-region model class). A model class that divides the ballast into three or eight regions may perform better than a six-region model class. Furthermore, ballast comprises a group of rocks with irregular shapes. There is considerable uncertainty associated with the material properties of ballast. The deterministic approach adopted in [23,33] could not handle this uncertainty problem. The ballast damage detection methodology proposed in the current study adopted the Bayesian model class selection method [2] in identifying the most plausible class of models by calculating the probability of the model class conditional on a given set of measurement. The application of the Bayesian model class selection method in structural model updating of civil engineering structures is not new [37,51]. However, its application for the damage detection of railway ballast is completely new. The original formulation of the Bayesian statistical system identification framework [8] was developed using measured time domain data. It was extended to use measured modal parameters [7,11]. For model updating using measured natural frequencies and mode shapes, modal matching is essential. Recently, the Bayesian model updating method using modal data was extended to be applicable without modal matching [50]. The proposed ballast damage detection methodology follows the approach in Ref. [7] in calculating the posterior PDF of uncertain model parameters utilising a set of measured modal parameters. In this paper, a segment of indoor full-scale ballasted track was constructed according to the specification from MTR [38]. The advantage of building the test panel indoors is that environmental factors such as temperature and humidity can be controlled. From the literature [48,49], the change in temperature will alter the natural frequencies of a structural system. At the development stage of the ballast damage detection methodology, the environmental factors must be kept consistent during vibration measurement to avoid the damage detection result based on the changes in modal parameters be contaminated by the effects of environmental changes. The measured modal parameters from the full-scale ballasted track were employed to demonstrate the procedures and verify the proposed methodology. A study about the marginal posterior PDF of ballast stiffness under the sleeper was carried out to provide valuable information about the uncertainties associated with the ballast damage detection results.

2. Proposed ballast damage detection methodology The three main components of an in-situ sleeper are two rails, the sleeper and the underlying ballast, as shown in Fig. 1(a). The rails carry the vertical load from the train and distribute it to the sleepers. The sleepers are embedded into the ballast and the load is transferred through the ballast to the foundation. The ballast is tightly tamped around the sleepers to keep the track precisely levelled and aligned. Appropriate modelling of the rail–sleeper–ballast system is the key component of a model-based ballast damage detection method. The in-situ sleeper can be treated as a beam on an elastic foundation, which has many applications in the civil and mechanical engineering fields. The most popular modelling method involves representing the elastic foundation as a series of closely spaced linear springs, as shown in Fig. 1(b) [47]. To calculate the vibration of the in-situ sleeper under train loading, the ballast can be considered as a series of springs and dampers [31]. In this study, the in-situ sleeper is modelled not to predict the time-domain responses of the system under train loading,

H.F. Lam et al. / Engineering Structures 81 (2014) 289–301

Fig. 1. (a) A typical ballasted track structure. (b) Beam on an elastic foundation. (c) Modelling of the rail–sleeper–ballast system.

but to detect ballast damage using the measured modal parameters. Therefore, the ballast is modelled as distributed springs with a given stiffness per unit length along the sleeper. In reference to previous studies [14,19], the concrete sleeper is modelled as a Timoshenko beam. As the cross-sectional area of a sleeper usually varies along the sleeper, establishing an analytical model is complicated. Finite element methods have usually been adopted for the research in this area. For example, Kaewunruen and Remennikov modelled the ballast as a Winker foundation using the finite element package STRAND7 [32]. Their analysis results were in good agreement with their experimental measurements. Lam and colleagues modelled the rail–sleeper–ballast system using a self-developed finite element code in MATLAB [33]. Following this idea, this study implements a Timoshenko beam on an elastic foundation, based on a finite element method.

where Nmax is the maximum number of model classes to be considered (see Fig. 2). The question becomes which model class should be used to represent the real system. One may initially select the model class that can provide the best fit for the measured modal parameters. However, this is generally not true because a more complicated model class usually can provide a better fit. If only the discrepancy between the measured and model-predicted modal parameters is used in the selection of an appropriate model class, the most complicated model class should be selected. In the proposed methodology, the Bayesian model class selection method [2,37,51] is implemented to select the most plausible model class for a given set of measurements, and thus to determine the value of N. For a given model class Mj, the ballast is divided into j equal regions. For each region, a dimensionless scaling factor hi, for i = 1, 2, . . ., j, is used to scale the nominal ballast stiffness kb. Therefore, the ballast stiffness at the ith region is hikb. These j scaling factors are considered as uncertain model parameters (i.e., minimisation variables) in model updating. Due to the aging effect, the Young’s modulus of the sleeper may also vary, and another scaling factor hE is used to scale its nominal value. The two rails are treated as two individual uncertain masses (i.e., the left and right masses) denoted by mL and mR, respectively [23,33]. Their values can be calculated by mL = hLmr and mR ¼ hR mr , where mr is the nominal value of the rail mass and hL and hR are the scaling factors for the left and right masses, respectively. As a result, j + 3 uncertain model parameters are considered in the model updating using the Mj model class, and are grouped into an uncertain model parameter vector hj = [h1, h2, . . ., hj, hE, hL, hR]. In this study, the nominal value of Young’s modulus and density of the concrete sleeper are 38  109 N/m2 and 2200 kg/m3, respectively, and the nominal value of the ballast stiffness and rail mass are 150  107 N/m2 and 42 kg, respectively. The finite element method is used to model the rail–sleeper– ballast system in this study. To capture the variations in the

2.1. Rail–Sleeper–ballast system modelling In this study, a rail–sleeper–ballast system is modelled as a Timoshenko beam, including two additional masses, on an elastic foundation with an equivalent ballast stiffness kb, as shown in Fig. 1(c). When the ballast is undamaged, the variation of its stiffness under the sleeper should be very small, and a uniform stiffness of kb should capture the behaviour of the ballast. When the ballast is damaged, its stiffness at certain area(s) under the sleeper has a value lower than kb. To capture this stiffness distribution, the ballast under the sleeper is divided into N (equal to or larger than 1) regions of equal widths, and the ballast stiffness is assumed to be a constant within a given region. It is difficult to determine the value of N for a given set of measurements. Different model classes are used to represent situations involving different numbers of ballast regions. The model class M1 represents the situation with only one region (i.e., uniform ballast stiffness distribution), and M2 represents the situation with two ballast regions (i.e., left and right). In general, there are Mj model classes for j = 1, . . ., Nmax,

291

Fig. 2. Different class of models for ballast damage detection.

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cross-section area and to ensure that the computer model’s degrees of freedom (DOFs) match the measured DOFs, the sleeper is divided into 48 elements. The formulations for classical beams on elastic springs can be found in [28]. For a typical 2D beam element, the axial load and deformation are neglected, and therefore only the vertical and rotational DOFs are considered per node. Owing to the thickness of the sleeper, the effect of shear deformation cannot be neglected, especially in the vibrations of higher modes [33]. Therefore, the Timoshenko beam formulation is selected in this study, and the element stiffness matrix k can be expressed as follows [28]:

2

12w1 6 6w L EI 6 3 k¼ 6 ð1 þ uÞL3 4 12w2

6w3 L ð4 þ uÞw5 L

6w4 L

ð2  uÞw6 L

12w2 2

6w4 L

6w4 L 12w1

2

6w3 L

3

6w4 L

ð2  uÞw6 L 7 7 7 5 6w3 L 2

2

ð4 þ uÞw5 L

12EI

for

j ¼ 1; 2; . . . ; N max

ð6Þ

where ^ hj represents the most probable model parameters in the model class Mj and the compositions of ^ h1 , ^ h2 , . . ., ^ hj , ^ hE , ^ hL , hj are ^ ^ hR . Nj is the number of uncertain parameters in ^ hj , and is equal to j + 3 in this study. H j ð^ hj Þ is the Hessian matrix of the function g(hj) (see Eq. (7)) evaluated at ^ hj . It can be calculated using the finite difference method, where the function g(hj) is given as follows:

ð7Þ

The evidence in Eq. (6) consists of two factors: the likelihood factor

where E, I and L are the Young’s modulus, second moment of area and length of the element, respectively. The effects of shear flexibility are described by the non-dimensional parameter u:

ð2Þ

GA0 L2

Nj  1 pðDM j Þ  pðDj^hj ; M j Þð2pÞ 2 pð^hj jM j ÞjH j ð^hj Þj2

  gðhj Þ ¼  ln pðhj jM j ÞpðDjhj ; M j Þ ð1Þ

u¼

the formulation given in [2,34,35], the probability of a model class conditional on the set of measurements is proportional to the ‘evidence’ of the model class, which can be expressed as follows:

where G is the shear modulus and A0 is the corresponding shear area, which is smaller than the cross-section area. The coefficients wi for i = 1–6 can be expressed as follows:

Nj

^j jM j ÞjH j ðh ^j Þj12 . In genpðDj^ hj :; M j Þ, and the Ockham factor ð2pÞ 2 pðh eral, a model class with higher complexity (i.e., with more uncertain model parameters) can fit the measurement better, resulting in a higher likelihood value. According to [2], the value of the Ockham factor decreases when the number of uncertain model parameters increases. Therefore, it can be treated as a penalty of the complexity of the model class. The balance between these two factors allows one to select the most plausible model class, which is just complex enough to fit the given measurement. In this study, the class of models to be selected is the one with the highest value of evidence among all of the considered model classes.

w1 ¼

1 ðkLÞ2 w½sinhðkLÞ coshðkLÞ þ sinðkLÞ cosðkLÞ 3

ð3aÞ

w2 ¼

1 ðkLÞ2 w½sinðkLÞ coshðkLÞ þ sinhðkLÞ cosðkLÞ 3

ð3bÞ

w3 ¼

1 2 2 ðkLÞw½sinh ðkLÞ þ sin ðkLÞ 3

ð3cÞ

w4 ¼

2 ðkLÞw sinðkLÞsinhðkLÞ 3

ð3dÞ

The main objective of the Bayesian method is to calculate the posterior PDF of the uncertain model parameters hN (denoted as h in the following formulation to simplify the expressions) for a given set of measured data D, and a given class of models MN, which is selected via Bayesian model class selection. The most probable model ^ h can be obtained by maximising the posterior PDF, which is shown below:

1 w½sinhðkLÞ coshðkLÞ  sinðkLÞ cosðkLÞ 2

ð3eÞ

  1 pðhjDÞ ¼ c exp  JðhÞ 2

w5 ¼

w6 ¼ w½sinðkLÞ coshðkLÞ  sinhðkLÞ cosðkLÞ

ð3fÞ

where

w¼

kL 2

2

sinh ðkLÞ  sin ðkLÞ

JðhÞ ¼

" Nm X Ns X ^ 2r;n  x2r ðhÞÞ2 ðx r¼1 n¼1



kb 4EI

14 ð5Þ

The consistent mass matrix [39] is used in the dynamic analysis. The element stiffness and mass matrices are used to assemble the system stiffness and mass matrices. The modal parameters, such as the natural frequencies and mode shapes of the system, can be calculated by solving the corresponding eigenvalue problem. 2.2. Bayesian model class selection In the first phase of the proposed methodology, the Bayesian model class selection method is used to select the most plausible model class among Nmax given model classes. This is done by calculating the probability of each model class conditional on the set of measurements: p(Mj|D) for j = 1 to Nmax. The model class with the highest probability is the most plausible model class. By following

ð8Þ

where c is a normalising constant such that the integration of the PDF over the parameter space of interest is equal to unity. To maximise the posterior PDF in Eq. (8) is equivalent to minimising the positive definite measure-of-fit function, J(h), [46]:

ð4Þ

and

k¼

2.3. Bayesian model updating

e2r

#

^ r;n w ^ T ÞCu ðhÞ uTr ðhÞCT ðI  w r;n r þ 2 dr kCur ðhÞk2

ð9Þ

where Nm is the number of modes to be considered in the model updating process and Ns is the number of sets of identified modal parameters. In general, many impulses are recorded in an impact hammer test. A set of independent modal parameters can be identified from the time-domain responses of each impulse, resulting in many sets of identified modal parameters. e2r is the (sample) variance of the squared circular natural frequencies obtained from the ^ r;n are the rth natural frequency and mode ^ r;n and w measured data. x shape of the nth set of measurements, respectively. xr ðhÞ is the circular natural frequency of the rth mode calculated from a given model h. The selection matrix C consists of only 1 and 0, and picks the observed DOFs from the model-predicted mode shapes to match the measured DOFs. jj  jj is the Euclidean norm. ur ðhÞ is the calculated mode shape of the rth mode. Note that both Cur ðhÞ ^ r;n are normalised to have unit norms in this study. The sample and w variance of the measured mode shapes d2r can be expressed as follows [46]:

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d2r ¼

Ns ^ r;n  w  r jj2 1X jjw ^ r;n jj2 Ns n¼1 jjw

ð10Þ

 r is the average mode shape of the rth mode from the Ns where w datasets. Owing to space limitations, the expression of the posterior PDF of uncertain parameters is not presented in this paper. Interested readers should consult [46]. The J function in Eq. (9) is the discrepancy between the measured and model-predicted modal parameters. The first and second terms correspond with the discrepancy in natural frequencies and mode shapes, respectively. It is clear from Eq. (9) that 1/er2 and 1/dr2 can be treated as the weighting factor for natural frequency and mode shape of the rth mode, respectively. These weighting factors show the relatively importance of the measured quantity in the model updating process. By following the Bayesian approach, the weighting factors are inversely proportional to the variance of the measured quantity. As a result, the higher the uncertainty, the smaller the weighting factor will be. This result is consistent to the inverse measurement variance method reported in Ref. [18]. Instead of focusing only on the optimal model identified by minimising the J function in Eq. (9), the proposed methodology aims on calculating the marginal posterior PDF of ballast stiffness at different regions under the concrete sleeper for the purpose of ballast damage detection. In the identifiable cases [30], the posterior PDF can be well approximated by a multivariable Gaussian distribution. Then, the ballast damage condition can be described by the most probable values of the identified ballast stiffness together with the corresponding coefficient of variation (COV) or standard derivation. 2.4. Ballast damage detection An overview of the proposed methodology for ballast damage detection based on the Bayesian approach is schematically

293

illustrated in the flowchart in Fig. 3. Impact hammer tests are carried out on the target in-situ sleeper, which is possibly damaged, to obtain the time-domain data (e.g., acceleration responses at different DOFs). In this study, the MODE-ID method [3] is adopted to identify the modal parameters from the measured time-domain responses. The set of measurements D can then be used for the model class selection and model updating following the Bayesian approach. By dividing the ballast into different regions, different model classes of the rail–sleeper–ballast system are formed (as discussed in Section 2.1). By following the Bayesian model class selection method (as discussed in Section 2.2), the most plausible class of models can be selected depending on the set of measurements D. By maximising the posterior PDF of the set of uncertain model parameters, conditional on the ‘most plausible’ model class and the set of measurements, the ‘most probable’ model can be identified according to the Bayesian model updating method (as discussed in Section 2.3). The ballast stiffness value of an undamaged system is required to detect ballast damage, and is referred to as the ‘reference line’ (the value is generally near unity). In this study, this reference line is determined using the set of measurements from the undamaged system. The reference ballast stiffness can be calculated using the Bayesian model class selection and updating methods. In real-world application, a set of measurements for a reference system is usually not available. Comprehensive field measurements could be carried out, and a database of modal parameters (and corresponding identified ballast stiffness values) could be established for different types of ballasted track systems under different configurations. The reference line of a particular ballasted track could then be obtained from this database. By comparing the marginal posterior PDF of ballast stiffness at different regions of the target sleeper with the reference line (see the schematic illustration in Fig. 4), the ballast damage location and the corresponding damage extent can be detected. In Fig. 4,

Fig. 3. Flowchart of the proposed ballast damage detection methodology.

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H.F. Lam et al. / Engineering Structures 81 (2014) 289–301

Fig. 4. A schematic figure illustrating the idea of ballast damage detection.

the solid line shows the identified ballast stiffness (the mean calculated from the marginal posterior PDF). The two dashed lines show the upper and lower bounds of one standard deviation from the mean. The three-region model class M3 is used in this illustrative example, and the probability of having a 40% reduction in ballast stiffness under the middle of the sleeper (a scaling factor of 0.6) is very high. 3. Experimental verification using full-scale tests To illustrate the procedures and verify the proposed Bayesian ballast damage detection methodology, a full-scale in-door test panel consisting of eight in-situ sleepers was built in the basement of a factory building in Kowloon Bay, Hong Kong (see Fig. 5). The ballasted track in the test panel was constructed according to Hong Kong MTR specifications [38]. Two UIC60 rails were fastened to the concrete sleepers, with a rail mass of 60 kg/m. The centre-to-centre spacing of the sleepers was 700 mm, and the thickness of the granite ballast layer was 350 mm. The ballast was made of crushed rock aggregate at a size of 50 mm (ballast of this size is considered normal). 3.1. Experiment description In this study, ballast degradation is defined as railway ballast damage, which leads to a reduction in ballast size, resulting in a drop in the ballast’s packing level and stiffness in support of the concrete sleeper. An undamaged case was first considered in the test panel, where the ballast under all eight in-situ sleepers was in good condition and normal in size (50 mm). After conducting a series of comprehensive impact hammer tests on the undamaged

Fig. 5. The full-scale in-door test panel in Kowloon Bay, Hong Kong.

cases, artificial damage was simulated under the left-hand side of the sleeper (two thirds of its length) by replacing the normal-sized ballast (50 mm) with a small-sized ballast (15 mm). This was performed by two groups of labourers. One group lifted the two rails and concrete sleepers, and the other group replaced the ballast at the desired region. The rails and sleepers were then released back into their original positions. In fact, the ballast under the sleeper was disturbed when the rails and sleepers were lifted. Therefore, it was impossible to guarantee no change to the ballast under the right-hand side of the sleepers. The equipment used in the impact hammer tests is summarised in Fig. 6. The target sleeper was excited by an impact hammer equipped with different tips, as shown in Fig. 6(b). The head tips had different textures and colours. Softer tips were more suitable for exciting the lower modes (e.g., modes 1 and 2 of the in-situ sleeper), and stiffer tips were more appropriate for exciting the higher modes (e.g., modes 3–5). The measurement duration was set to 20 s with a sampling frequency of 6400 Hz, and approximately 10 impulses were recorded in each measurement. Eleven KISTELER-8776A50M3 accelerometers (with sensitivity levels of 100 mV/g; see Fig. 6(a)) were installed on the top surface of the concrete sleeper along the central line. All of the sensors were installed vertically upward to capture the vertical vibration of the system. The measured vibration responses from the 11 sensors were transferred through cables with special coating (see Fig. 6(c)) and digitised using four NI-9234 modules. These modules were connected to a Compact DAQ-9178 chassis (see Fig. 6(d)) that could handle a maximum of 16 channels simultaneously. 3.2. Modal identification Fig. 7(a) shows a typical measured time-domain response at channel 6 under a given impulse. The responses at the other channels and impulses were similar. The time-domain responses were transformed to the frequency domain via Fast Fourier Transform (FFT). The power spectral density was calculated and is shown in Fig. 7(b). Some peaks appeared in the spectrum, indicating frequencies where the amplitude of the vibration was large [15]. The peaks in the spectrum were used as the initial trials of natural frequencies in the modal identification using MODE-ID [3]. The first four modes were identified with high accuracy, and were used in the model updating and ballast damage detection. The first four measured natural frequencies for both the undamaged and damaged cases are identified and summarised in Table 1. The natural frequencies of all of the modes decreased in the damaged case. Especially, the first two modes have relatively large reductions of 34.45% and 36.23%, indicating that these two modes were sensitive to the simulated ballast damage. The measured mode shapes of the undamaged and damaged cases are presented in Fig. 8. The solid lines with circles in each figure represent the mode shapes, and the dashed lines represent the un-deformed shapes of the sleeper. The circles in the figure indicate the sensor locations. As shown in the figure, the first mode was mainly the rigid body rotational mode, and the second mode was mainly the rigid body translational mode with a small bending of the sleeper. Furthermore, modes 3 and 4 were the first and second bending modes of the sleeper, respectively. This result is consistent with those in [14,33]. Depending on the ballast stiffness distribution, the natural frequencies of the first two modes were sometimes very close to each other, increasing the difficulty of modal identification. Modes 1 and 2 mainly involved the vibration of the ballast. They were sensitive to the properties of the ballast, and therefore essential for ballast damage detection. Modes 3 and 4 mainly involved the vibration of the sleeper, and were important in the model updating of the stiffness of the sleeper (e.g., the Young’s modulus). Fig. 8 shows that the mode shapes of modes 3 and 4

H.F. Lam et al. / Engineering Structures 81 (2014) 289–301

295

Fig. 6. List of equipment for the impact hammer test. (a) KISTELER accelerometer, (b) impact hammer, (c) cables with special coating and (d) four NI-9234 modules installed on a Compact DAQ-9178 chassis.

were almost the same in the undamaged and damaged cases, indicating that these two modes were not very sensitive to the ballast damage. The highly asymmetrical mode shape of mode 2 in the damaged case possibly implies that the ballast stiffness distribution under the sleeper was not uniform (due to ballast damage).

3.3. Bayesian model class selection and model updating 3.3.1. Undamaged case The proposed methodology was tested using the measurements from the undamaged system. The evidence of the model class M1, conditional on the set of measured data, was calculated using Eq. (6). M1 corresponded to the class of models with only one ballast region under the sleeper. The logarithm of the evidence together with the logarithms of the likelihood (showing the matching between the measured and model-predicted modal parameters) and the Ockham factor (showing the penalty of the model class complexity) are given in the first row of Table 2. The logarithms were incorporated into the analysis because the numerical value of the evidence was very large and might have caused computational problems. The model class M2, in which the ballast was divided into two regions, was then tested. The analysis result of Eq. (6) is summarised in the second row of Table 2. It is clear that M2 fit the data slightly better than M1. This was expected, as M2 is more complicated than M1. However, the evidence of M1 was higher than that of M2 due to the penalty of model complexity caused by the Ockham factor. Because the evidence of M1 was higher than that of M2, the model class M1 was selected. The ballast distribution was uniform under the sleeper, implying that there was no ballast damage. The model updating results of the undamaged case is shown in the first row of Table 3. The updated ballast stiffness was very close to the nominal value, with a stiffness scaling factor of 1.04. Therefore, the reference line was at the 1.04

level. The updated Young’s modulus of the sleeper was 21% larger than the nominal value. It must be pointed out that the sleeper employed in this study is a pre-stressed concrete sleeper, and the pre-stress force, which will increase the stiffness of the beam, is not considered in the modelling of the sleeper. This may be the reason for the inconsistence between the identified and nominal Young’s modulus. Consider the updated rail masses on the leftand right-hand sides. Both values were similar and smaller than the nominal value. It can be concluded that the proposed Bayesian methodology successfully detected an absence of ballast damage based on the set of measured modal parameters.

3.3.2. Damaged case The measurements from the damaged system were then used to test the proposed methodology. The model classes M1 and M2 were first considered, and the Bayesian model class selection results are summarised in rows 3 and 4 of Table 2. It is clear that M2 fit much better to the measurement, resulting in a much larger logarithm of likelihood (28.95 for M1 and 42.97 for M2). Because the logarithm of the M2 evidence was larger than that of the M1 evidence, it can be concluded that the ballast stiffness distribution was not uniform under the sleeper, implying ballast damage. To find an ‘appropriate’ model class for modelling the rail–sleeper–ballast system, more complex model classes were tested individually with increasing numbers of ballast regions. The model class M3 was then tested, and the results are summarised in row 5 of Table 2. It is clear that the likelihood and evidence logarithms increased significantly, and that the reduction of the Ockham factor logarithm was very minor. This result shows that the model class M3 fit the measurement much better than all of the previously tested model classes. The model class M4 was then tested, and the results are summarised in row 6 of Table 2. This model class fit the measurement only slightly better than M3 (see the small increase in the

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1

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0

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−1

−1.5 1.56

1.58

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2

Average power spectral density (g /Hz)

(a)

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(b) Fig. 7. (a) A typical measured time-domain response at channel 6. (b) Frequency domain response.

Table 1 Identified natural frequencies of the undamaged and damaged cases. Natural frequency (Hz)

Mode 1

Mode 2

Mode 3

Mode 4

Undamaged Damaged Difference (%)

91.47 59.96 34.45

98.22 62.63 36.23

170.90 157.50 7.84

407.13 405.47 0.41

likelihood logarithm). Because the logarithm of the M3 evidence was higher than that of the M4 evidence, M3 was selected as the most plausible model class. The Bayesian model updating results for the model class M3 using the data from the damaged structure are summarised in row 2 of Table 3. First, the ballast stiffness distribution was considered to detect damage to the ballast. Using the model class M3, the ballast under the sleeper was divided into regions 1, 2 and 3, and the corresponding ballast stiffness values were scaled by h1, h2 and h3, respectively. The identified ballast stiffness factors for the left-hand side and middle regions were 0.66 and 0.43, respectively, representing percentage reductions of about 34% and 57%, respectively. The ballast stiffness on the right-hand side was very close to unity. The identified ballast stiffness distribution is consistent with

the simulated ballast damage. Other updated model parameters were then individually discussed. The updated Young’s modulus was about 34% higher than the nominal value. As mentioned previously (in the undamaged case), this discrepancy may be very likely caused by the unconsidered pre-stress force in modelling the sleeper. The updated left rail mass was almost zero, and the updated right rail mass was about 59% larger than the nominal value. This might have been caused by the levelling problem of the damaged sleeper. The left-hand side of the sleeper might have been at a slightly lower level than the neighbouring sleepers. As a result, some of the rail mass was distributed to the neighbouring sleepers. The right-hand side of the sleeper might have been slightly higher than the neighbouring sleepers. It must be pointed out that the re-ballasting process for simulating ballast damage is complicated and labour intensive especially when it is implemented in the indoor test panel (heavy machines cannot be used in the relatively small area in the indoor test panel). The relatively poor workmanship in the sleeper levelling in the damaged scenario is very likely the main reason for the large difference between the identified left and right rail masses. Using the ‘most probable’ model (given in Table 3), the modelpredicted natural frequencies and model shapes are calculated and compared with the measured natural frequencies and mode shapes in Table 4 and Fig. 10. Table 4 makes it clear that the matching between the model-predicted and measured natural frequencies were very good for all of the modes. The largest error percentages were about 3% for mode 1 in the undamaged case and about 3% for mode 4 in the damaged case. Fig. 10 indicates that the matching between the model-predicted and measured mode shapes was generally very good, especially for modes 1 and 2 in the damaged case, in which the model-predicted mode shapes almost overlapped with the measured mode shapes. This result indicates the good performance of the model updating process. The marginal posterior PDFs of ballast stiffness at the three regions were calculated (see next section), and the corresponding means and standard derivations can be calculated. The ballast damage detection result can be better presented by plotting the means and standard derivations of ballast stiffness at different regions together with the reference line. The Bayesian model class selection results show that the stiffness distribution in the undamaged case was uniform, with a scaling factor of 1.04 (i.e., a reference line at 1.04, the blue dot-dashed line in Fig. 11). The red solid line represents the mean, and the black dashed lines represent the upper bound and lower bound at one standard derivation from the mean. From the figure, the probability for the ballast in Region 1 to be damaged is very high as the mean value of ballast stiffness is at 0.67 and the standard derivation is relatively small. The probability for the ballast in Region 2 to be damaged is also high but it is not as high as that in Region 1. As the mean value of ballast stiffness at Region 3 is close to unity and the standard derivation is similar to that in Region 2, the probability for Region 3 to be damaged is much lower than that for Region 2 (and so as Region 1). This result is consistent to the simulated ballast damage (at Regions 1 and 2). 3.3.3. Uncertainties associated with the model updating results Unlike the deterministic approach, the Bayesian method provides information about the reliability or uncertainty of damage detection results. The posterior PDF of uncertain model parameters is extremely complicated, and its analytical expression cannot be obtained in most situations. Under the Bayesian statistical framework [9], the model updating problem is considered as globally identifiable when the posterior PDF of uncertain parameters can be well approximated by a multivariable Gaussian distribution. When the amount of information available is reduced or when the number of uncertain model parameters is increased,

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Normalised mode shape

Mode 1: Measured−91.47 Hz

Mode 2: Measured−98.22 Hz

0.2

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location of sleeper (mm)

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(a) Undamaged case

Normalised mode shape

Mode 1: Measured−59.96 Hz

Mode 2: Measured−62.63 Hz

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(b) Damaged case Fig. 8. Identified modal parameters of the in-situ sleeper in (a) undamaged case and (b) damaged case.

the uncertainties associated with the model parameters will increase in general. The transition from identifiable to unidentifiable model updating problem is not simple. To study this transition in more details, Ref. [30] defined the identifiability of order R. In some ‘‘identifiable’’ model updating problems, the uncertainties associated with the identified model parameters are not small. As a result, approximating the posterior PDF using a multivariable Gaussian locally at the optimal model (by central difference method) may result in relatively large error. To have a better understanding on the marginal posterior PDF, a set of regular grid points was generated around the optimal model for the approximation of the marginal posterior PDF in this paper. The calculation of one single point on this posterior PDF required the calculation of

the J function value in Eq. (9). To roughly estimate the computational power required to calculate the posterior PDF in this case, 10 grid points were generated along each uncertain parameter. There were six uncertain model parameters for the M3 model class, requiring 1 million (=106) points. If 0.1 s was required to evaluate a single J function value, it would have taken 1666.7 mins (i.e., about 27.8 hours or 1.2 days) to complete the calculation process. In this study, parallel computing was employed to speed up the calculation. The normalised posterior PDFs of all uncertain model parameters for both the undamaged and damaged cases were calculated (four and six uncertain model parameters for the undamaged and damaged cases, respectively). As it is impossible to plot the four- or

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H.F. Lam et al. / Engineering Structures 81 (2014) 289–301 Table 2 The evidence together with the likelihood and Ockham factors for the different model classes in the undamaged and damaged cases.

4.5 4

Normalised PDF

3.5 3 2.5 2 1.5

Class of models

Logarithm of evidence

Logarithm of likelihood factor

Logarithm of Ockham factor

Undamaged M1 M2

49.15 49.38

35.14 30.01

14.01 19.37

Damaged M1 M2 M3 M4

4.52 16.69 229.58 226.45

28.95 42.97 261.93 262.41

24.43 26.28 32.35 35.96

The model class with the bold value is the most probable model class.

1 0.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Table 3 The most probable values of model parameters in the undamaged and damaged cases.

First region stiffness scaling factor

(a)

Undamaged Damaged

^ h1

^ h2

^ h3

^ hE

^ hL

^ hR

1.04 0.66

– 0.43

– 1.03

1.21 1.34

0.73 0.001

0.68 1.59

1.4

Table 4 Measured and model-predicted natural frequencies (after model updating) in the undamaged and damaged cases.

Normalised PDF

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(b) 2.2 2

Normalised PDF

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0.6

0.8

1

1.2

1.4

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1.8

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Mode 2

Mode 3

Mode 4

Undamaged Updated Measured Difference (%)

88.46 91.47 3.29

97.07 98.22 1.17

175.66 170.9 2.79

407.55 407.13 0.10

Damaged Updated Measured Difference (%)

60.13 59.96 0.28

62.43 62.63 0.32

156.74 157.50 0.48

419.59 405.47 3.48

distributions are also plotted in Fig. 9 (dashed line) for comparison. In Fig. 9(a), the Gaussian approximation is very good except at the right tail of the distribution. Considering the ballast stiffness at Region 2 in Fig. 9(b), the normalised marginal posterior PDF consists of two peaks. Since one of the two peaks is very small, the Gaussian approximation of this PDF is still not too bad. Finally, in Fig. 9(c), the Gaussian approximation of the normalised marginal posterior PDF of ballast stiffness at Region 3 is acceptable. According to the normalised marginal posterior PDFs of the ballast stiffness at different regions, which were identified using measured data from the full-scale test panel, the model updating problem for ballast damage detection is basically identifiable [29] and the approximation of the marginal posterior PDF by using Gaussian distribution is good except for the ballast stiffness at Region 2. When the class of models becomes more complicated (i.e., with more uncertain model parameters), the model updating problem may become unidentifiable. Under such situation, Markov Chain Monte Carlo (MCMC) simulation [10] may be employed to generate samples in the regions of high probability for the approximation of the posterior PDF of uncertain model parameters.

Third region stiffness scaling factor

(c) Fig. 9. The normalised marginal posterior PDF of ballast stiffness at different regions together with the approximated Gaussian distribution. (a) Region 1. (b) Region 2. (c) Region 3.

six-dimensional PDFs, the normalised marginal posterior PDFs of the stiffness parameter at the three regions in the damaged case are plotted in Fig. 9 (solid line). The approximated Gaussian

4. Discussions and conclusions This study proposes a new ballast damage detection methodology following the Bayesian approach utilising multiple sets of measured modal parameters from an in-situ sleeper. Using the Bayesian model class selection method, if the selected model class shows a uniform ballast stiffness distribution under the sleeper, the system is treated as undamaged. Otherwise, the ballast at

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Mode 1: Updated−88.46 Hz. Measured−91.47 Hz

Mode 2: Updated−97.07 Hz. Measured−98.22 Hz 0

Normalised mode shape

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(b) Damaged case Fig. 10. The matching between the measured and model-predicted modal parameters (after model updating) in different cases.

certain Region(s) under the sleeper may be damaged, and the Bayesian model class selection method helps ‘identify’ an ‘appropriate’ class of models to capture the dynamic behaviour of the system based on a given set of measurements. The posterior PDF of

ballast stiffness at different regions under the sleeper can be calculating by the Bayesian model updating method. The ballast damage can then be detected by comparing the means and standard derivations of ballast stiffness with the reference line. In this study,

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0.8

Putting the test panel indoors meant that environmental factors such as temperature and humidity could be kept at a consistent level in all of the vibration tests. This is very important in the development stage of a ballast damage detection method. However, the effects of temperature and humidity on the modal parameters of the in-situ sleeper must be studied in detail. This is because changes in temperature and humanity will induce changes in modal parameters of the in-situ sleeper.

0.6

Acknowledgements

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Ballast stiffness factor

1

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Region 3

Region 2

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location along sleeper (m)

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 114712 (GRF 9041758)]. The authors would like to express their sincere appreciation to MTR Corporation for the in-kind support in the construction of the full-scale test panel in Kowloon Bay, Hong Kong. The authors would also like to thank Mr. Jiahua Yang and Mr. Huayi Peng for their help in the vibration measurement.

Fig. 11. Damage detection results of the full-scale experimental case study.

References the reference line is obtained via Bayesian model class selection and model updating using a set of measurements from the undamaged system. Most structural damage detection methods require a reference system [16,41,42]. However, it is well known that the set of measurements corresponding with an undamaged (reference) system is usually difficult to obtain. According to the results of the full-scale experimental case study, the ballast stiffness distribution in an undamaged case was uniform under the sleeper, and the ballast stiffness value was very close to the nominal value. As a result, it is proposed to carry out comprehensive impact hammer tests, and form a database of measured modal parameters (including updated ballast stiffness) for different types of railway track system under different configurations. The ballast stiffness values in the database can then be treated as the ‘reference line’ in detecting damage to the ballast in real applications of the proposed methodology. One outstanding advantage of the Bayesian approach is the calculation of the marginal posterior PDFs of uncertain model parameters, which provides measures of the uncertainties associated with the model updating (and damage detection) results. Another contribution of this study is the design and build of an indoor test panel. The train service in Hong Kong is extremely busy, and it is almost impossible to carry out vibration test on real ballasted track systems. The indoor test panel allows the simulation of different ballast damage scenarios and vibration measurement can be carried out without time limit. Several difficulties must be overcome before the proposed methodology can be put into real-world practice. The first involves the modelling of the rail–sleeper–ballast system. The ballast stiffness variation along the concrete sleeper would be continuous in a real-world situation, and the discontinuous ‘jumps’ in ballast stiffness at the interfaces between two regions in the case studies were ‘artificial’. Further research must be carried out for developing a continuous model to describe the distribution of ballast stiffness along a sleeper. In the proposed methodology, it is assumed that the sleeper is undamaged. However, both the sleeper and ballast may be damaged in a real-world situation. Damage to the sleeper would alter the modal parameters of the system [36] and certainly increase the difficulty involved in detecting damage to the ballast. In the experimental case study, modes 1 and 2 were shown to be sensitive to ballast damage, and modes 3 and 4 were sensitive to sleeper damage. By following this direction, the current Bayesian ballast damage detection methodology could be extended to consider possible damage at both ballast and sleeper.

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