Bayesian void detection of the CA mortar layer in the slab track considering temperature effect

Abstract

This paper proposes a void detection method for the cement-emulsified asphalt (CA) mortar layer of the slab track structure utilizing Markov chain Monte Carlo-based Bayesian model updating technique considering the temperature effect. The mechanical properties of CA mortar are susceptible to the temperature variations, which is one of the main challenges for the CA void detection based on Bayesian model updating. Nearly a half year temperature monitoring on the CA mortar layer was conducted and comprehensive numerical and experimental case studies were carried out to demonstrate the feasibility and applicability of the proposed Bayesian CA void detection method considering the temperature effect. The uncertain model parameters can be evaluated from the Monte Carlo discrete samples by using kernel density estimation, and the associated uncertainties of the uncertain model parameters can be quantitatively described by calculating the posterior probability density functions of the model parameters of the CA mortar layer. Positive investigations can be obtained from both the numerical and experimental cases, which indicates that the proposed method is robust to detect the CA void.

Keywords

Bayesian model updating Markov chain Monte Carlo temperature effect void detection slab track

Introduction

With the rapid development of the public transportation system, high-speed railways have become an integral component of a region’s economic growth. The slab track system has become a widely favored choice for high-speed railways around the world in terms of its stability, durability, and ease of maintenance (Gautier, 2015; Matias and Ferreira, 2020). The slab track comprises rails, fasteners, the prefabricated slab, the cement-emulsified asphalt (CA) mortar layer and the concrete base (Zhu and Cai, 2014). During the long-term service of the slab track system, the degradation of the track system is inevitable due to the repeated high-speed train loads, temperature fluctuations and other environmental factors, and particularly the void in the CA mortar layer become increasingly evident over time (Yu et al., 2019; Zhu et al., 2014). If timely remedial work is not performed, this degradation will adversely affect the track’s suffered force and distortion and even threaten the safety of railway line (Dong et al., 2022; Ren et al., 2020). In addition, the CA mortar void often evades detection by visual inspection owing to its concealed spatial characteristics (Tian et al., 2018; Yu et al., 2018). Therefore, a practical method for detecting the CA mortar void is necessary, which can provide a chance for timely structural maintenance of the slab track system and ensure the track reliability and durability (Alkayem et al., 2018; Xin et al., 2022).

Over the past few decades, vibration-based methods have been widely utilized for structural health monitoring and structural damage detection (Boscato et al., 2019; Ni et al., 2017; Wang et al., 2023; Yan et al., 2007). And it is quite complicated to apply the methods to detect the damage of civil engineering structures due to their large dimensions and the related uncertainties induced from modeling error, measurement noise, and changing environmental conditions (Xia et al., 2006; Zhu et al., 2016). Structural dynamic characteristics are susceptible to the changes in environmental conditions, particularly temperature variations (Huang et al., 2018; Zabel et al., 2010). In certain instances, the alterations of structural dynamic characteristics resulted from structural damage may be obscured by those due to temperature variations, which makes the detection of structural damage intractable (Huang et al., 2021). Some efforts have been made to correlate structural dynamic characteristics with ambient temperature (Ding and Li, 2011; Moser and Moaveni, 2011; Peeters et al., 2000). These studies can effectively eliminate the temperature effect on the accuracy of structural damage detection results. For instance, Sohn et al. (1998) established a linear adaptive model which can be utilized to distinguish between the change of modal parameters caused by temperature variations and those arisen from the structural damage. The applicability of the model was verified by using the vibration data of the Alamosa Canyon Bridge in the state of New Mexico. Yan et al. (2005a, 2005b) proposed to take environmental parameters as embedded variables and adopted principal component analysis (PCA) method to analyze structural vibration features, so as to detect structural damage. Balmes et al. (2008) proposed a nonparametric damage detection algorithm, which eliminates temperature effect by averaging data sets recorded under different and unknown temperature conditions. The feasibility of the proposed algorithm was validated by the case studies of a simulated bridge deck model and a laboratory test-case made of a clamped beam.

Noted that most of the existing researches utilized the modal-domain data to detect structural damage (Adeagbo et al., 2021; Ding and Li, 2011; Moser and Moaveni, 2011; Peeters et al., 2000). Since the raw measured vibration data is in time-domain, an additional procedure is needed to convert it to the frequency-domain/ modal-domain. This transformation process may result in the loss of valuable information from the raw data (Lam et al., 2017). To circumvent this additional transformation, the time-domain vibration responses were directly utilized in the model updating process to detect the CA void considering the temperature effect in this study. Furthermore, a probabilistic approach was taken to cope with the uncertainties induced from the modeling error and measurement noise (Hu et al., 2018; Ni and Zhang, 2021). Lam et al. (2014) presented a novel ballast damage detection method based on Bayesian approach, utilizing multiple sets of in situ sleeper modal parameters. A key advantage of this approach is its ability to quantify the uncertainties of the damage detection results by estimating the marginal posterior PDFs. Recently, the Bayesian model updating method based on the Markov chain Monte Carlo (MCMC) algorithm has been put into the railway substructure damage detection (Adeagbo et al., 2021, 2022; Lam et al., 2017, 2018, 2021). As there were uncertainties and randomness involved in calculating the CA mortar’s elastic modulus for the concealed void detection, Hu and Shen (2023) proposed to employ the MCMC-based Bayesian model updating method incorporated with a newly developed model class selection algorithm to deal with these uncertainties. This approach not only can successfully locate the void and estimate the corresponding void extent in the CA mortar layer, but also can quantify the uncertainties associated with the void detection results.

However, the CA mortar possesses strong temperature sensitivity due to its organic-inorganic composite material consisting of Portland cement, emulsified asphalt, palletized sand, water and admixtures (Fu et al., 2015). Temperature changes can alter the mechanical properties of the material, which may conceal minor damage in the CA mortar layer and affect the accuracy of void detection. In current researches on void detection of the CA mortar layer in slab tracks (Hu et al., 2021; Hu and Shen, 2023), the impact of temperature is often overlooked, which increases the uncertainty in the void detection results. Given the widespread application of high-speed railway systems across different climatic regions, it is necessary to consider the effect of temperature variations on the elastic modulus of the CA mortar layer in the void detection. Recently, a number of researches have been dedicated in investigating the effect of temperature variations on the mechanical properties of CA mortar. For example, Li et al. (2020) examined the effect of temperature changes on the viscoelastic-plastic mechanical properties of CA composite cement, and found that the deformation of viscosity and plasticity of CA composite cement increased with the increase of temperature. Zhang and Wang (2011) studied the compressive strength of CA mortar under different test temperatures (18∼60°C), and an approximately exponential relationship between the compressive strength of CA mortar and temperature was obtained based on the compressive strength test results. Notably, CA mortar exhibits obvious variations in its mechanical properties under different temperature conditions (Qin et al., 2023; Wang et al., 2010, 2019; Zeng et al., 2020), which lay a solid foundation for the void detection considering the correlation between the elastic modulus of the CA mortar and temperature variations.

This study aims to explore the viability of a time-domain MCMC-based Bayesian model updating method with considering the temperature effect for identifying the void in the CA mortar layer of the slab track. A scaled model of the slab track was constructed and placed outside the laboratory in order to consider the effect of ambient temperature in detecting the void in the CA mortar layer. The temperature of the CA mortar layer was monitored and recorded periodically for nearly a half year. And comprehensive numerical case studies and impact hammer tests on the scaled slab track model were conducted to investigate the feasibility and applicability of the proposed methodology.

Proposed CA mortar void detection methodology

Slab track modelling with temperature sensitivity

A three-dimensional finite element (FE) model of the slab track structure was built in ABAQUS as shown in Figure 1(a). Linear hexahedral elements (C3D8R) were employed to simulate the substructure including the prefabricated slab, CA mortar layer, and concrete base. As illustrated in Figure 1(b), the FE model featured predominantly uniform mesh distribution with 50 mm hexahedral elements, while around the T-shaped protrusion, the mesh was more intensive. The loading condition involved a vertically applied dynamic concentrated force in the region shown in Figure 1(b), with its magnitude derived from data collected during the impact hammer test (described in subsequent sections). Assuming that the interface between the asphalt mortar layer and other structural components was fully bonded with no relative sliding or separation phenomena, and tie contact was employed in the substructure interface (Romanoschi and Metcalf, 2001; Smith, 2009). The boundary condition was defined by applying an elastic foundation interaction at the bottom surface of the concrete base. The soil reaction coefficient was utilized to scale the stiffness of the elastic foundation, and the nominal stiffness of the elastic foundation was set to $K_{B} = 2 \times 10^{6} kN / m^{3}$ . Additionally, the nominal elastic moduli of the prefabricated slab and concrete base were set to $K_{G} = 52 GPa$ and $K_{D} = 39 GPa$ , respectively. The computational model employed a linear perturbation analysis to calculate the time-domain response of the slab track structure under dynamic loading. Notably, the model validation has been conducted in the previous work (Hu et al., 2021). In this study, the initial parameter settings for prefabricated slab, concrete base and elastic foundation were adopted from previously updated results.

In order to detect the CA void, the CA mortar was evenly divided into several regions for a given model class. The void in a given region was simulated by the reduction in the corresponding elastic modulus in this study. Notably, the elastic modulus of CA mortar is susceptible to temperature variations. Therefore, to establish a reasonable relationship between the elastic modulus of the CA mortar and temperature is crucial for accurately detecting the CA void. Kong et al. (2014) had investigated the temperature sensitivity of CA mortar with different $A / C$ ratios (mass ratio of asphalt cement), as shown in Figure 2. The elastic modulus of CA mortar with a higher $A / C$ ratio exhibited a greater temperature sensitivity. Specifically, when the temperature increased 1°C, the reduction ratio of the elastic modulus of CA mortar was almost in the range of 0.29%∼3.24%, which was much larger than that of the concrete (Comité Euro-International du béton, 1993; Xia et al., 2011). Therefore, only the temperature effect of the CA mortar layer was considered in the FE model in the following analysis.

The elastic modulus of the CA mortar was assumed to vary linearly with temperature in this study, therefore, the CA mortar’s elastic modulus under any given temperature can be calculated by the following equation:

E_{T} = E_{0} [1 - θ_{T} (T - T_{0})]

(1)

where

θ_{T}

is the temperature coefficient of elastic modulus and is set to be 0.0074 in this study (Kong et al., 2014),

T_{0}

is the reference temperature and it is 20°C in this study.

E_{0}

is the elastic modulus of the CA mortar in the FE model at the reference temperature, which is initially suggested to be 900 MPa for a given slab track system.

MCMC-based Bayesian model updating

The main objective of the Bayesian model updating method is to determine the posterior probability density functions (PDFs) of the uncertain model parameters conditional on the given measured data and prior distribution. Following Bayesian theorem, the posterior PDF of the uncertain model parameters is:

p (θ | D) = c p (D | θ) π (θ)

(2)

where

θ

is the uncertain model parameter vector,

D

is the measured time-domain vibration data,

c = 1 / \int p (D | θ) π (θ) d θ

is a normalizing constant,

p (D | θ)

is the likelihood function,

π (θ)

is the prior PDF of

θ

, and uniform distributions were utilized in this study to ensure that the inference fully dependent on the measured data and to simplify the computational process in Bayesian inference (Lam et al., 2015).

To construct the likelihood function, impact hammer tests were conducted in this study to obtain the measured accelerations. During one test, the accelerations at $N_{d}$ observed degrees of freedom (DOF) within a time interval of $t = N_{t} \cdot Δ t$ were collected and denoted as $D = {{\hat{\ddot{x}}}_{1}, {\hat{\ddot{x}}}_{2}, \dots, {\hat{\ddot{x}}}_{N_{t}}} \in R^{N_{d} \times N_{t}}$ . At time $t_{j} = j \cdot Δ t$ , the prediction error between the measured output at the s-th DOF and the corresponding model output can be expressed as:

ε = {\hat{\ddot{x}}}_{j} (s) - {\hat{\ddot{x}}}_{j} (s, θ)

(3)

where

ε

is an independent Gaussian distribution with a zero-mean and variance

φ_{0}^{2}

Therefore, the likelihood function at time $t_{j}$ and s-th DOF can be calculated as:

p ({\hat{\ddot{x}}}_{j} (s) | θ) = \frac{1}{φ_{0} \sqrt{2 π}} \exp [- \frac{1}{2 φ_{0}^{2}} {({\hat{\ddot{x}}}_{j} (s) - {\hat{\ddot{x}}}_{j} (s, θ))}^{2}]

(4)

It’s assumed that the measured acceleration at each time step and each DOF is mutually independent, then the likelihood function considering all sets of measured accelerations $D \in R^{N_{d} \times N_{t}}$ can be expressed as follows:

p (D | θ) = \prod_{s = 1}^{N_{d}} \prod_{j = 1}^{N_{t}} p ({\hat{\ddot{x}}}_{j} (s) | θ) = {(\frac{1}{φ_{0} \sqrt{2 π}})}^{N_{d} N_{t}} \exp [- \frac{N_{d} N_{t}}{2 φ_{0}^{2}} J (θ)]

(5)

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

(6)

where

J (θ)

is the objective function.

Therefore, the joint posterior PDF of the uncertain model parameter vector $θ$ can be written as follows:

p (θ | D) = c_{0} \exp [- \frac{N_{d} N_{t}}{2 φ_{0}^{2}} J (θ)]

(7)

where

c_{0}

absorbs all the constants.

When the available measured data is limited and the number of model parameters is relatively large, the model updating problem may become unidentifiable (Hu and Shen, 2023). Consequently, an enhanced MCMC-based Bayesian model updating method was employed to approximates the posterior PDF by generating samples in multiple levels (Lam et al., 2015). In this study, the whole sampling process was divided into $g_{r}$ levels and the samples were generated using the Metropolis-Hastings algorithm at each sampling level. The initial proposal density was set as the uniform distribution of the uncertain model parameters (Hastings, 1970). As the sampling process transitions between levels, samples are generated using the bridge PDF as the target PDF and the proposal density is the approximation of the bridge PDF constructed by kernel density estimation with samples from the preceding level (Ang et al., 1992; Au and Beck, 1999). Simultaneously, the importance region of the target PDF decreases in a controlled manner, leading the bridge PDF to gradually converge toward the target posterior PDF. Based on the formulation of the posterior PDF exhibited in equation (7), the bridge PDF in the g-th sampling level can be calculated as follows:

p_{g} (θ | D) = c_{g} \exp [- \frac{N_{d} N_{t}}{2 η_{g}^{2}} J (θ)]

(8)

where

η_{g}^{2}

is related to the size of the important region of the bridge PDF in the g-th sampling level,

η_{1}^{2}

is set to 1 initially, and its value is reduced in a controlled manner:

η_{g}^{2} = \frac{1}{A} \cdot η_{g - 1}^{2}

(9)

where

A

is an algorithmic parameter that controls the level-to-level change of the bridge PDF, and it is set as to be 1.2 in this study. A larger A results in a faster reduction of the important region across levels and a shorter computational time.

The posterior PDF of each uncertain model parameter can be determined by utilizing the generated samples in the final sampling level, therefore the marginal posterior PDF of the i-th component in $θ$ can be calculated by kernel density $k (θ)$ (Au and Beck, 1999):

k (θ (i)) = \sum_{h = 1}^{N_{s s}} W^{(h)} N (S_{g_{r}}^{h} (i), C_{g_{r}} (i, i))

(10)

where

N_{s s}

is the number of samples,

g_{r}

is the number of required sampling levels.

W^{(h)}

is the weighting of the h-th sample.

N (S_{g_{r}}^{h} (i), C_{g_{r}} (i, i))

is a Gaussian distribution with the mean of

S_{g_{r}}^{h} (i)

and the variance of

C_{g_{r}} (i, i)

. The advantage is that a multi-dimensional numerical integration is not required. More detailed derivations can be found in the reference Lam et al. (2015).

Flowchart of the proposed CA mortar void detection method with considering the temperature effect

Generally, structural damage may lead to the reduction in structural stiffness, and the change in elastic modulus is one of the crucial factors to affect the structural stiffness (Wang et al., 2014). Therefore, the CA void for a given region is simulated by the reduction of the corresponding elastic modulus in this study. Meanwhile, the temperature effect on the elastic modulus of the CA mortar was taken into account in the CA void identification of the slab track. The detailed procedures of void identification considering temperature effect are summarized in the flowchart of Figure 3.

Firstly, impact hammer tests were carried out on the slab track structure under different ambient temperatures. During each test, the excitation and temperature data as well as the accelerations at interested DOFs were collected. Subsequently, the measured excitation and corresponding temperature data were put into the FE model to calculate the model predicted responses. With the measured and calculated accelerations, the marginal posterior PDFs of the uncertain model parameters can be calculated by employing the proposed MCMC-based Bayesian model updating methodology. The most probable values (MPVs) of the uncertain model parameters as well as the associated uncertainties can be ultimately estimated and then the CA void can be identified. In this way, the temperature effect was considered in the CA mortar void detection in a simple and proper manner.

Numerical case studies

Description of temperature setting and damage scenarios

According to the monitoring temperature records of the slab track model located in Wu Han (refer to Figure 9(a)), the temperature range of the whole CA mortar layer in the computer simulated FE model was set to vary from 0°C to 40°C. To simplify the analysis, the three-dimensional temperature field in the entire CA mortar layer was simulated to be constant under a certain temperature condition, which was only adjusted by the predefined load module in the FE model. Five different temperature conditions were considered in this study and they are uniform temperature fields of 0°C, 10°C, 20°C, 30°C and 40°C, respectively.

In Bayesian model updating methods for void detection, model class selection is needed to determine the model divisions. More details can be found in the references (Hu et al. (2021); Hu and Shen (2023)). For given model class, two damaged cases were considered to demonstrate the feasibility and practicability of the proposed method. According to the service state of the CA mortar layer in practice, the CA void was usually arisen at the edge of the track (Xin and Ren, 2022), therefore two different damaged cases were set with the damage at edge of the CA mortar layer, and the corresponding schematic is exhibited in Figure 4. In damaged case 1, the CA mortar layer was equally divided into $2 \times 2$ regions, and only the region ① was damaged with the damage extent to be 30%. Damaged case 2 is a multiple damaged case, and the CA mortar layer was equally divided into $2 \times 4$ regions. The regions ①, ⑦ were damaged, and the damage extents of these two regions are 30% and 50%, respectively. The 30% damage extent is determined based on the laboratory model with an approximate damage severity of 30%. Considering the 2 × 4 division of the model, where the damage region is relatively small, a 50% damage extent was set to evaluate the feasibility of the proposed method in detecting damage within small damaged regions.

Void detection results

Damaged case 1

Firstly, the damaged case 1 was utilized to validate the feasibility of the proposed Bayesian CA void detection method for the slab track considering the temperature effect. Structural damage generally leads to a reduction in stiffness, and in this study, the stiffness reduction induced by the CA void is simulated by decreasing the elastic modulus of CA mortar layer. For a given region, the stiffness was treated as a constant, whereas the identified stiffness reduction indicated the potential presence of void in the corresponding region of the CA mortar layer. Thus, the dimensionless scaling factor of the CA mortar’s elastic modulus in a region was considered as an uncertain parameter. Additionally, this study focuses on considering the temperature effect in CA void detection. When there are many uncertain parameters, calculating high-dimensional posterior integrals becomes highly complicated. To improve the computational efficiency, the elastic modulus of prefabricated slab, concrete base and elastic foundation has not been updated. In damaged case 1, the CA mortar layer was divided into four regions, therefore, the model parameter vector can be represented by $θ = {θ_{1}, θ_{2}, θ_{3}, θ_{4}}$ , for example, $θ_{1}$ is utilized to scale the stiffness of region ① of the CA mortar, and in the i-th region, the stiffness of the CA mortar layer can be obtained as $E_{i} = θ_{i} E_{T} (i = 1, . ., 4)$ . Under the temperature conditions of 0°C, 10°C, 20°C, 30°C and 40°C, the elastic modulus $E_{T}$ of the CA mortar layer can be calculated by using equation (1), which are 1033 MPa, 967 MPa, 900 MPa, 833 MPa and 767 MPa, respectively. Consequently, the damage extent of the i-th region of the CA mortar layer can be easily calculated by using $1 - θ_{i}$ . When the damage extent is set to 30%, the theoretically true value of $θ_{1}$ is 0.70.

The results of model updating for the damaged case 1 are shown in Table 1. It can be seen that the MPVs of the uncertain model parameters of region ① were all around 0.70 under all considered temperature conditions, and the average of the MPVs under the five temperature conditions were 0.6990, which is very close to the true value (0.70). The closeness indicates that the damage location and extent of the CA void have been successfully detected through the proposed method. The corresponding damage extent is estimated as 30.10%, which are consistent with the simulated damage situation. Then the average coefficients of variation (COVs) of all uncertain model parameters under different temperature conditions are calculated and they are 15.26%, 14.56%, 13.15%, 13.34% and 13.99%, respectively, which are all within an acceptable level. And the average relative errors (REs) are 0.95%, 0.50%, 0.49%, 1.58% and 0.76%, and the maximum REs is only 2.99%, which are small and acceptable. And it is easy to find that the average COV and RE at temperature of 20°C is the smallest one.

Table 1.

Results of model updating for the damaged case 1 with considering temperature effect.

Temperature	Parameters	$θ_{1}$	$θ_{2}$	$θ_{3}$	$θ_{4}$	Average
/	Theoretical value	0.7000	1.0000	1.0000	1.0000	/
0°C	MPVs	0.6884	1.0032	1.0000	1.0183	/
	REs (%)	1.66	0.32	0.00	1.83	0.95
	COVs (%)	20.99	13.89	13.01	13.14	15.26
10°C	MPVs	0.7012	0.9864	1.0009	0.9961	/
	REs (%)	0.17	1.36	0.09	0.39	0.50
	COVs (%)	20.66	12.05	13.04	12.49	14.56
20°C	MPVs	0.7020	0.9965	0.9950	1.0084	/
	REs (%)	0.29	0.35	0.50	0.84	0.49
	COVs (%)	18.46	11.04	12.99	10.09	13.15
30°C	MPVs	0.7197	1.0299	0.9955	0.9993	/
	REs (%)	2.81	2.99	0.45	0.07	1.58
	COVs (%)	16.26	12.40	11.42	13.31	13.34
40°C	MPVs	0.6838	1.0011	0.9978	0.9959	/
	REs (%)	2.31	0.11	0.22	0.41	0.76
	COVs (%)	17.69	10.80	14.61	12.85	13.99

In order to evaluate the effect of temperature variation on the CA void detection results, the CA mortar void detection without considering temperature effect was investigated, and the elastic modulus of CA mortar layer in the FE model remains 900 MPa under the aforementioned temperature conditions. The results of model updating for the damaged case 1 without considering temperature effect are shown in Table 2. It can be easily noted that the region ① was damaged and the average MPVs of the model parameter

θ_{1}

was 0.7304, and the corresponding damage extent is 26.96%, which indicates a decreasing accuracy compared to the CA void detection results with considering the temperature effect. Compared with the COVs of the model parameters considering temperature effect, the average COVs of all model parameters increased, especially for the model parameter of the damaged region ①.

Table 2.

Results of model updating for the damaged case 1 without considering temperature effect.

Temperature	Parameters	$θ_{1}$	$θ_{2}$	$θ_{3}$	$θ_{4}$	Average
/	Theoretical value	0.7000	1.0000	1.0000	1.0000	/
0°C	MPVs	0.7328	0.9971	0.9995	0.9874	/
	REs (%)	4.69	0.29	0.05	1.26	1.57
	COVs (%)	26.74	13.20	13.42	13.58	16.74
10°C	MPVs	0.7835	1.0085	1.0033	0.9891	/
	REs (%)	11.93	0.85	0.33	1.09	3.55
	COVs (%)	21.22	14.14	10.76	12.95	14.77
30°C	MPVs	0.7495	1.0012	0.9766	0.9939	/
	REs (%)	7.07	0.12	2.34	0.61	2.54
	COVs (%)	31.00	14.10	13.68	13.68	18.12
40°C	MPVs	0.6841	0.9797	0.9983	1.0172	/
	REs (%)	2.27	2.03	0.17	1.72	1.55
	COVs (%)	45.23	17.48	18.89	18.52	25.03

The comparison of the COVs and REs of model parameters with and without the temperature effects can be seen in Figure 5. For COVs, it is obvious that significant difference can be found at the temperature of 40°C. Similarly, the maximum difference in RE with and without considering temperature is 11.76%, and the average RE of $θ_{1}$ increases by 64% compared to that considering the temperature effect in the void detection process. It can be concluded that the accuracy and reliability of the void detection results of the CA mortar of the slab track by utilizing the presented method will be significantly affected if the temperature effect was not considered. In real-world scenarios, the service environment of the slab track is subject to more complex temperature variations, which leads to increased uncertainties and low robustness of CA void detection results. Therefore, it is quite necessary to consider the temperature effect in the void detection of the CA mortar layer of the slab track.

Damaged case 2

To further validate the applicability of the proposed method under more complicated damaged conditions, multiple damaged case 2 with smaller region division was considered. In this case, the dimensions of the two damaged regions ① and ⑦ of the CA mortar layer are

0.4 m \times 0.7 m

, and the corresponding damage extents are 30% and 50%, respectively. Similarly, the scaling factors of the CA mortar’s elastic modulus of the eight equally divided regions were treated as the uncertain parameters, which is represented as

θ = {θ_{1}, θ_{2}, θ_{3}, \dots, θ_{8}}

. The results of model updating for the damaged case 2 are shown in Table 3.

Table 3.

Results of model updating for the damaged case 2.

Temperature	Parameters	$θ_{1}$	$θ_{2}$	$θ_{3}$	$θ_{4}$	$θ_{5}$	$θ_{6}$	$θ_{7}$	$θ_{8}$	Average
/	Theoretical value	0.7000	1.0000	1.0000	1.0000	1.0000	1.0000	0.5000	1.0000	/
0°C	MPVs	0.6992	1.0099	1.0046	1.0068	0.9900	1.0161	0.5010	0.9866	/
	REs (%)	0.11	0.99	0.46	0.68	1.00	1.61	0.20	1.34	0.80
	COVs (%)	21.30	18.13	16.13	15.93	16.60	15.94	30.26	18.96	19.16
10°C	MPVs	0.6762	0.9975	1.0023	1.0088	0.9984	0.9990	0.5032	0.9958	/
	REs (%)	3.40	0.25	0.23	0.88	0.16	0.10	0.64	0.42	0.76
	COVs (%)	24.86	16.69	16.62	16.02	15.74	15.49	34.06	15.34	19.35
20°C	MPVs	0.7020	1.0049	1.0130	1.0057	0.9917	0.9988	0.4970	1.0001	/
	REs (%)	0.29	0.49	1.30	0.57	0.83	0.12	0.60	0.01	0.53
	COVs (%)	24.44	16.07	13.96	15.04	14.63	17.52	30.50	15.45	18.45
30°C	MPVs	0.6936	0.9873	0.9747	0.9884	0.9881	0.9989	0.5134	1.0000	/
	REs (%)	0.91	1.27	2.53	1.16	1.19	0.11	2.68	0.00	1.23
	COVs (%)	18.62	18.59	16.98	17.12	16.42	16.24	31.91	15.99	18.98
40°C	MPVs	0.6954	0.9878	0.9897	0.9858	1.0087	0.9947	0.4994	1.0149	/
	REs (%)	0.66	1.22	1.03	1.42	0.87	0.53	0.12	1.49	0.92
	COVs (%)	26.95	16.46	13.99	16.50	16.65	18.00	32.57	18.00	19.89

It can be easily noted that the average MPVs of the uncertain parameters of regions ① and ⑦ were 0.6933 and 0.5028, respectively. Therefore, the damage extents of these two regions can be calculated as 30.67% and 49.72%, respectively, which implies that the two regions ① and ⑦ are damaged. And these identified results are consistent with the simulated damage scenario. The average COVs and REs of all model parameters are not large under all considered temperature conditions, and the average COV and RE at temperature of 20°C is the smallest one.

The MCMC samples in the final sampling level at several representative temperatures (e.g. 0°C and 20°C) of damaged case 2 are depicted in Figure 6. It can be noted that the generated samples of all uncertain model parameters had eventually converged to the important regions at the considered temperatures. Based on the MCMC samples in the final sampling level at the temperature of 0°C in the damaged case 2, the marginal posterior PDFs of the uncertain model parameters are plotted in Figure 7. It is not difficult to find that the marginal posterior PDF curves almost overlap with the fitted Gaussian distributions. The mean of the Gaussian corresponds to the MPV, while the degree of flatness characterizes the uncertainty level of void detection results. The sharp distributions of the model parameters indicated relatively low associated uncertainties.

Effect of the measurement noise

To ascertain the robustness of the proposed Bayesian CA void detection method, different levels of Gaussian white noise were added to the simulated vibration responses under the reference temperature (20°C) in damaged case 1. Given the environment noise at a distance of 25 m from the track is regulated within the range of 70 to 75 dBA, corresponding to an approximate noise level of 3% to 5% (Miskinyte et al., 2016), three noise levels were set, denoted as

λ

= 2%, 5%, and 10% respectively to comprehensively consider factors such as environment, measurement techniques, and equipment aging. The results of model updating under different noise levels at the temperature of 20°C for the damaged case 1 are summarized in Table 4. It can be easily seen that when the noise level increased from 2% to 10%, the REs of the model parameter of the damaged region ① increased from 0.56% to 4.89%. Additionally, the average COVs of all model parameters increased from 18.26% to 26.36%, which implies that the uncertainties of the model parameters increased with the increase of the noise level. However, even at the 10% noise level, the maximum RE remained at 4.89%, which is within an acceptable range. This demonstrates the robustness and reliability of the proposed method.

Table 4.

Results of model updating under different noise levels at the temperature of 20°C for damaged case 1.

Temperature	Parameters	$θ_{1}$	$θ_{2}$	$θ_{3}$	$θ_{4}$	Average
/	Theoretical value	0.7000	1.0000	1.0000	1.0000	/
$λ = 2 %$	MPVs	0.6961	1.0144	0.9917	0.9964	/
	REs (%)	0.56	1.44	0.83	0.36	0.80
	COVs (%)	19.87	15.57	17.82	19.78	18.26
$λ = 5 %$	MPVs	0.6829	1.0129	0.9876	1.0488	/
	REs (%)	2.44	1.29	1.24	4.88	2.46
	COVs (%)	27.94	20.49	17.76	17.67	20.96
$λ = 10 %$	MPVs	0.7342	0.9852	0.9901	1.0022	/
	REs (%)	4.89	1.48	0.99	0.22	1.90
	COVs (%)	28.11	26.79	27.00	23.53	26.36

The MCMC samples at the final sampling level for the damaged case 1 under the three noise levels are exhibited in Figure 8. It can be observed that the discrete samples under three noise levels all converged to the important region, which indicates the proposed method can effectively handle the associated uncertainties induced from the measurement noise.

Experimental case studies

Impact hammer test

To investigate the feasibility and practicability of the proposed CA void detection method, a scaled slab track model with CA void was constructed in the laboratory of Huazhong University of Science and Technology, located in Wuhan, Hubei Province, as shown in Figure 9(a). This study does not focus on investigating the mechanical properties of the structure. Since the vertical response of the prefabricated slab is measured to detect the CA void, therefore, the vertical dimensions of each substructure are kept consistent with the standard CRTS Ⅱ slab track system. In order to conveniently simulate the void in the CA mortar layer and enhance interfacial bonding strength, a T-shaped section was set in the CA mortar layer along its x-axis. The detailed dimensions of the cross section of the CA mortar layer can refer to Figure 1.

With the scaled slab track model, impact hammer tests have been conducted monthly since September 2021. During these vibration tests, accelerometers were installed on the surface of the prefabricated slab to measure the vertical vibrations of the slab track model, and the impact hammer with green color tip was utilized to hit the structure. The equipment involved in the impact hammer tests is showed in Figure 9(b). The data acquisition system was connected to a computer equipped with the DHDAS dynamic signal acquisition and analysis system. The sampling frequency and measurement duration were set to be 5000 Hz and 15 sec, respectively, and approximately 5 impulses were recorded in each set of measurement.

Temperature monitoring

Since the slab track model is located in Wuhan, Hubei Province, which belongs to the humid subtropical climate. Consequently, there exists an obvious change in ambient temperature with the seasons, which can significantly affect the dynamic characteristics of the structure. Therefore, the ambient temperature of the CA mortar layer was monitored nearly for 6 months from mid-August 2021 to the end of January 2022, then the feasibility of the proposed method for void detection in the CA mortar layer considering the temperature effect can be systematically investigated.

The wireless temperature measurement system included two BM100 wireless temperature sensors (Figure 10(a)) and an NT1000 temperature indicator (Figure 10(b)), which can allow for conveniently and accurately monitoring of the temperature within the CA mortar layer of the slab track model. The BM100 temperature sensor was designed to automatically measure the temperature of the structure at pre-set intervals. Two BM100 wireless temperature sensors, denoted as T1183 and T1184, respectively, were strategically arranged at the outer edge and inner space of the CA mortar layer by using a PVC plastic tube, as shown in Figure 10(c), which was inserted into the reserved cavity of the CA mortar layer. The placement of the temperature sensors is illustrated in Figure 10(d), the sensor A was employed to measure the temperature at the inner space of the CA mortar layer, and the sensor B was utilized to measure the temperature at the edge of the CA mortar. The sampling rate of the temperature sensors is 10 min per reading.

Since the temperature sensor located at the outer edge of the CA mortar layer was more susceptible to the influence of the external environment temperature, resulting in more significant temperature variations compared to the sensor at the inner space, therefore, only the average daily temperature of the temperature sensor A (located at the inner space of the CA mortar layer) from mid-August 2021 to the end of January 2022 are displayed in Figure 11(a). It is observed that the temperature varied from 0°C to 30°C in the monitoring period and the temperature in summer (fluctuated around 30°C) was much higher than that in winter, and the temperature had a sudden decrease in mid-October 2021, therefore, it is necessary to consider the temperature effect in the void detection of the CA mortar of the slab track, especially in the area with humid subtropical climate, e.g., Wu Han.

The daily temperature variations of a representative month (e.g., December 2021) of the two temperature sensors are compared and only the temperatures at 9:00, 12:00, 15:00 and 18:00 per day are plotted, which is shown in Figure 11(b) and (c). It can be easily noted that the daily temperature variation of sensor A was considerably smaller than that of sensor B. The daily temperature change of sensor A was about 0.1∼0.7°C, while the daily temperature change of sensor B was 0.1∼2.2°C, which emphasizes the aforementioned statement that the temperature of sensor B was more susceptible to the influence of the external environment temperature, therefore, the temperature of sensor A located at the inner space of CA mortar layer was implemented to represent the average temperature of CA mortar in the void detection process.

Void detection by the proposed method

Model updating and void detection

Three different temperature conditions (e.g. 26°C, 12°C and 5.6°C) were considered in the experimental verification, and the process of model updating was similar to that of the above numerical case studies, similarly, only the CA mortar’s elastic modulus was considered as uncertain parameter. The void is located at one quarter of the CA mortar layer (region ①), and the other three regions are undamaged. Then the uncertain model parameter vector can be written as $θ = {θ_{1}, θ_{2}, θ_{3}, θ_{4}}$ , therefore, the updated elastic modulus of the corresponding region can be obtained as $E_{i} = θ_{i} E_{T} (i = 1, . ., 4)$ . Under the temperature conditions of 26°C, 12°C and 5.6°C, the elastic modulus $E_{T}$ of the CA mortar layer can be calculated by using equation (1), which are 860 MPa, 953 MPa and 996 MPa, respectively. The damage extent of each region is defined as $1 - θ_{i}$ .

The MPVs and corresponding COVs of the model parameters under different temperature conditions are presented in Table 5. It is noted that the dimensionless scaling factor of the elastic modulus of region ① is approximately 0.7499, which indicates that the region ① was damaged, and the damage extent was estimated to be around 25.01%. Meanwhile, the model parameter values of the other three regions all remained around 1.0000, which implies that these regions are nearly undamaged. This result is consistent with the simulated damage situation. And the average COVs of the model parameters under the three temperature conditions were 25.88%, 23.18% and 23.21%, respectively, which indicates that the associated uncertainties of the three damage situations were relatively comparable. Therefore, it can be concluded that the proposed Bayesian void detection method with considering temperature effect can successfully detect the location and extent of the void in the CA mortar layer of the slab track system. Additionally, it is worth noting that the MPVs of all uncertain model parameters under different temperature conditions were quite close, which demonstrates the proposed method for CA void detection can effectively consider the temperature effect and handle the associated uncertainties caused by temperature variations.

Table 5.

MPVs and COVs of the model parameters under different temperature conditions.

Temperature	--	$θ_{1}$	$θ_{2}$	$θ_{3}$	$θ_{4}$	Average
26°C	MPVs	0.7740	1.0447	1.0101	0.9677	/
26°C	COVs (%)	15.34	25.25	32.95	29.99	25.88
12°C	MPVs	0.7434	1.0394	1.0368	0.9861	/
12°C	COVs (%)	9.00	28.17	25.00	30.56	23.18
5.6°C	MPVs	0.7322	1.0034	0.9780	0.9675	/
5.6°C	COVs (%)	17.65	26.22	24.17	24.82	23.21

Uncertainty quantification of the model parameters

There usually exists deviations between the updated model parameters and the true values based on the model updating techniques due to the uncertainties induced from the inevitable measurement noise and modeling error. Therefore, the associated uncertainties of the void identifications were investigated in this section. The MCMC samples of the model parameters at the final sampling level at different temperatures are shown in Figure 12. It is evident that the discrete samples of all the dimensionless model parameters converged to the important regions under the three temperature conditions.

The marginal posterior PDFs and the comparison of the COVs of the model parameters at different temperatures are given in Figure 13. The consistent trends in the posterior PDFs of model parameters under three temperature conditions can be observed from the Figure 13(a)–(c), which denotes the robustness of the proposed methodology with respect to the varying temperature. Furthermore, it can be noted that the posterior PDF of the dimensionless model parameter $θ_{1}$ in the region ① of the CA mortar layer at different temperatures was relatively steeper compared to that of other regions, which indicates relatively low uncertainties and high reliabilities associated with the void detection results in this region.

Conclusions and discussions

This study proposes a CA mortar void detection method by utilizing the time-domain MCMC-based Bayesian model updating technique considering the temperature effect. In order to reduce the temperature effect on the CA void detection results, the relationship between the elastic modulus of the CA mortar layer and the temperature has been established, which was utilized in the FE model of the slab track structure during model updating process. Comprehensive numerical and experimental case studies were conducted to validate the proposed methodology, and nearly a half year temperature monitoring was conducted on a scaled slab track model in the laboratory. The main conclusions are summarized as follows:

(1) The comprehensive numerical case studies reveal that compared to CA void detection without considering temperature effect, the proposed method can improve detection accuracy and robustness while handling uncertainties introduced by measurement noise and modeling error. Especially for damaged region ①, the maximum difference in RE with and without considering temperature is 11.76%, and the average RE decreases by 64% when temperature effects are considered.

(2) The proposed method can effectively handle the impact of uncertainties caused by measurement noise on CA void detection. At $λ$ = 10%, the maximum RE is 4.89%, which remains within an acceptable range. Additionally, the associated uncertainties of the uncertain model parameters can be quantitatively described simultaneously.

(3) The temperature at the outer edge of the CA mortar layer was more susceptible to the influence of the external environment temperature change, resulting in more significant temperature variations compared to that at the inner space of the CA mortar.

(4) In the experimental case studies, the void detection results remain quite close across different temperature conditions, with an estimated average damage extent of 25.01%, and low uncertainties of the results for the damaged region can be investigated, which strengthens the robustness of the proposed method under temperature variations.

In real-world railway slab tracks maintenance, it is essential to identify the location and extent of CA void immediately. The proposed method enables real-time detection of CA void via a continuous monitoring of structural dynamic responses. Based on predefined thresholds, if the system detects a CA void that may pose a risk to railway safety, timely and accurately maintenance can be implemented. However, there are several difficulties to be overcome before the proposed methodology can be put into industrial applications:

(1) The study focuses on a scaled slab track model, and the size effects may impact the accuracy of CA void detection. It is proposed to further verify the proposed methodology through field tests.

(2) In this study, impact hammer tests were employed to collect accelerations of the slab track. Future research should involve track-induced vibrations to verify the practicality of the proposed method.

(3) The void is assumed to fully occupy a uniformly divided region in this study. Future research should investigate cases involving irregularly shaped void, multiple voids, or smaller void sizes.

(4) The laboratory model is situated in a humid subtropical zone with mild temperature fluctuations, and the method employed to account for temperature effects in this study is relatively simple. Further research is required to assess its applicability in more complex temperature conditions.

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 National Natural Science Foundation of China (Grant No. 52178287).

ORCID iDs

Qin Hu

Yunhao Guan

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