ABSTR ACT
This paper presents the latest findings of a
nondestructive evaluation technique currently
under development at the University of Pittsburgh
to determine the rail neutral temperature (RNT) in
continuous welded rails. The technique is based
on the extraction of relevant features from rail
vibrations and the use of machine learning (ML) to
associate these features to the longitudinal stress of
the rail of interest. The features contain the spectral
information of the vibrations and are pooled together
by frequency domain decomposition for input to
ML algorithms. Minimum redundancy–maximum
relevance and neighboring component analysis are
used to identify relevant features to reduce the size of
the input vector. In addition, seven algorithms were
considered to identify the most accurate model for
neutral temperature with respect to the ground truth
RNT measured with a strain-gage rosette. The data
used in this study were collected from a 5° curved rail
on concrete ties. The vibrations were triggered with a
hammer and recorded with a few wireless and wired
accelerometers attached on the railhead. The results
showed that the Gaussian process regressor performs
best, and as few as 20 frequencies can be used to
predict the RNT with sufficient accuracy.
KEYWORDS: machine learning, nondestructive evaluation,
continuous welded rails, feature extraction, rail neutral
temperature
Introduction
Continuous welded rail (CWR) refers to an uninterrupted
rail that is formed by many rails welded together to form one
continuous rail that may be several miles long. Structurally
speaking, CWR can be simplified as very long beams that
are in tension when the temperature of the rail is below the
so-called rail neutral temperature (RNT), and in compression
otherwise. On or above the RNT, the metal dilation induced by
a rise in temperature is counteracted by a compressive force
that may be significantly increased by an approaching train.
This scenario may lead to a buckle (called a sun kink in the
US), one of the most expensive causes of damage in the North
American rail network. When considered equivalent to an ideal
column, CWR buckles when the temperature TR of the rail
reaches the critical temperature Tcr, calculated as: ​ ​
(1)​ ​ ​ σ​cr​​
Eα ​
+​ TN​​ ​ =​ Tcr​​​​
where ​ ​
T​N​​​ is the RNT, ​ ​
σ​cr​​​ is the critical (Euler) stress of the rail, and
E and ​ ​ are the Young’s modulus and the coefficient of
thermal expansion of the steel, respectively.
As ​​ ​ cr​​​ E, and ​ ​ are known by design, the determination of
TN becomes relevant and can be obtained at any temperature
TR from the formula: ​ ​
(2)​ ​ TN​​ ​ =​ TR​​ ​ − ​ ​ σ​R​​
Eα ​ ​
where​​
σ​R​​​ is the longitudinal stress and is considered positive when
the rail is under compression.
In common practice, pretension is applied before welding
to compensate thermal expansion when a new rail segment
is laid. The level of prestress, and therefore the value of the
RNT, depends upon the geographic location of the site (i.e., by
the local climate history). However, operational maintenance
and stress relaxation decrease the designed RNT to unknown
values over time, increasing the risk of buckling. This calls for
nondestructive methods to measure ​​ ​ R​​​ and thus TN such that
slow orders or track closures can be issued properly. In current
practice, the desired accuracy for the estimation of the RNT is
±2.78 °C (or ±5 °F) (Zhu and Lanza di Scalea 2017).
NONDESTRUCTIVE ESTIMATION OF NEUTRAL
TEMPERATURE IN RAILS: A COMPARATIVE
STUDY OF MACHINE LEARNING STRATEGIES
MATTHEW BELDING*, ALIREZA ENSHAEIAN†, AND PIERVINCENZO RIZZO†‡
*Department of Electrical and Computer Engineering, University of Pittsburgh,
Pittsburgh, PA 15261
† Department of Civil and Environmental Engineering, University of Pittsburgh,
Pittsburgh, PA 15261
‡ pir3@pitt.edu 1-412-624-9575
Materials Evaluation 82 (1): 67–78
https://doi.org/10.32548/2024.me-04384
©2024 American Society for Nondestructive Testing
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Current methods to estimate RNT are based on strain
gages or the lift method. The first approach requires rail
cutting, prestressing, and re-welding (Wang et al. 2016). The lift
method requires unfastening 30 m of rail (Pandrol 2019). Pure
nondestructive testing (NDT) methods target the measurement
of the axial stress or the estimation of the RNT using physical
principles such as ultrasounds (Nucera et al. 2013 Nucera and
Lanza di Scalea 2014a, 2014b Lanza di Scalea and Nucera 2014
Szelaz˙ek ˛ 1992 Niu et al. 2023), digital image correlation (Knopf
et al. 2021), or acoustics (Bagheri et al. 2016 Nasrollahi and
Rizzo 2018, 2019), just to name a few. As the cross comparison
of these methodologies is beyond the scope of this paper, inter-
ested readers are referred to the review articles by Enshaeian
and Rizzo (2021) and Huang et al. (2023).
Over the past few years, the authors have developed an
NDT approach based on low-frequency vibrations, finite
element modeling, and machine learning (ML) (Enshaeian et
al. 2021 Belding et al. 2022, 2023a, 2023b, 2023c). The overar-
ching idea consists of triggering low-frequency (below 1 kHz)
vibrations of the rail of interest with a hammer and recording
them with a few accelerometers. The power spectral densities
(PSD) of the vibrations are calculated from the time domain to
become part of the input vector of a ML algorithm developed
to associate the spectral densities with the longitudinal stress
and then, using Equation 2, the RNT. The method was tested
in the field on a tangent track on wood crossties and a curved
track on concrete ties using an instrumented hammer and
wired and wireless accelerometers attached to the gage side of
the head. For both the tangent (Belding et al. 2023b) and the
curved rail (Belding et al. 2023c), it was found that an artificial
neural network (ANN) trained with experimental data was able
to predict the neutral temperature within the desired margin
of error of 2.78 °C. However, a few unanswered research ques-
tions remained, and the study presented in this paper answer
two of them, by analyzing the data collected from the curved
track on concrete ties. The first question pertained to the
performance of different ML algorithms to find the one that
outperforms the others in other words, the one that provides
the most accurate results in terms of neutral temperature pre-
dictions. The second question aimed to identify the number
of features, namely narrow bandwidths within the calculated
PSD, which are the most sensitive to the change of vibration
characteristics and neutral temperature. This second question
addresses the need to reduce the computational effort required
to train a “black box” algorithm with high dimensionality/
redundancy. This is because the extraction of exact features/
frequencies that are sensitive to stress is not trivial, especially
without the support of adequate modeling. To achieve the
first scope of the study, the following popular ML algorithms
were considered: linear regression (LR), decision trees, support
vector machine (SVM), ensembles, Gaussian process regres-
sion (GPR), and kernel approximation. These algorithms were
all trained and tested with the same set of experimental data.
To extract the relevant information for the PSDs, the minimum
redundancy–maximum relevance (mRMR) (Ding and Peng
2005) algorithm and the neighboring component analysis
(NCA) algorithm (Yang et al. 2012) were applied. mRMR seeks
to find the optimal set of features that maximize the relevance
and minimize the redundancy of a set of data to represent the
response variable effectively. Relevance is related to mutual
information between a feature and the output (RNT) and is
measured by using equations that will be presented later. NCA
is a nonparametric method that seeks to obtain features that
maximize the prediction accuracy of a regression problem and
acts as an alternative method to determining the most preva-
lent features.
This paper is organized as follows. The next section sum-
marizes the experimental setup and discusses the challenges
associated with the nondestructive estimation of the neutral
temperature. For the sake of completeness, this section also
describes the post-processing analysis of the experimental
vibrations. For more details, the reader is referred to Belding
et al. (2023a, 2023b). The Data Prepping section presents the
procedure to prep the input vector in support of the ML algo-
rithms. The Feature Extraction and RNT Prediction section
presents the results relative to the determination of the band-
width that should be considered to optimize the computational
efforts of the proposed vibration-based NDT. The section titled
ML Algorithms Comparison describes the results associated
with the determination of the algorithm that minimizes the
error between the predicted RNT and the ground RNT deter-
mined with a strain gage system. In addition, this section
includes an ablation study to measure the performance impact
of removing features from the models. Finally, the paper ends
with some concluding remarks.
Experiment
This section summarizes the experimental setup, discusses the
challenges associated with the nondestructive estimation of the
RNT, and describes the post-processing analysis of the experi-
mental vibrations.
Setup
Two field tests were performed at the Transportation
Technology Center Inc. (TTCI) in Pueblo, Colorado, in May
2021 and May 2022. The center is a facility owned by the
Federal Railroad Administration, managed by MxV Rail (here-
inafter referred to as the host) at the time of the experiments.
A tangent 136RE rail on wood crossties and a 5° curved 141RE
rail on concrete crossties were tested, and the data from the
latter were considered in this study. Vibrations on the rails
were induced with a hammer impacting the field side of the
railhead, in alternation above one tie and at the mid-span. The
vibrations were recorded with a few accelerometers. In May
2021, two wired accelerometers were bonded to the rail. One
year later, two wireless accelerometers were added and all four
were attached to the rail using magnets instead of epoxy glue.
This latter setup is schematized in Figure 1. The performance
of the wireless accelerometers with respect to their wired coun-
terparts and the advantages of this modified setup are amply
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