Results and Discussion
The regularly reported strain, rail temperature, and RNT data
reveal that the rail axial stress state is continuously changing
over time. As demonstrated in Figure 6, the axial strain
behavior follows that of rail temperature throughout a single
day, but the RNT varies in a different way, especially during the
daytime. The varying RNT response implies that factors other
than temperature also contribute to changes in strain. The
phenomenon is discussed in detail by Kish et al. (1987). The
long-term RNT variation is revealed by the nighttime averaged
RNT shown in Figure 7, where the RNT values recorded before
6:00 a.m. and after 7:00 p.m. are averaged in order to elim-
inate the effects of daytime variation. The RNT plummeted
following the rail-cutting procedure and then fluctuates fol-
lowing a seasonal pattern: the RNT typically starts to decrease
during summer months and then increases in springtime.
Figure 7 also demonstrates the RNTs at the two different loca-
tions behave distinctly. This difference in behavior could be
caused by different rail support conditions at the two loca-
tions, because changes in rail support conditions have been
identified as one factor that affects global rail stress state and
RNT (Kish et al. 1987). Compared to the East location, the
RNT on the West location is lower and fluctuates more signifi-
cantly before June 2020 after this time, the fluctuation at the
two locations becomes similar and synchronized. A possible
explanation for this observed behavior is that at the time of
the rail-cutting procedure the track on either side of the cut
moved (contracted) differently in response to that stress relief
because of distinct support conditions on either side. The rail
support conditions at the two locations have been modified
over time by the train traffic, track maintenance procedures,
and mechanical relaxation of the system such that boundary
conditions and the support structure become more similar at
the two locations as the track structure relaxes after the rail cut
destressing procedure.
The frequency values of the four selected resonances are
correlated with strain at the corresponding measurement
time for each testing location. The daily frequency-strain rela-
tions for three of the modes at both locations are compared
in Figure 8. The results show that the resonances around
37, 39, and 76 kHz do not exhibit a unique correlation with
strain throughout all testing days in other words, frequency
value alone cannot be used to uniquely define in-place strain
with a reasonable degree of accuracy. We surmise that other
unknown effects disrupt the strain-frequency relation for these
modes. It is also evident that the resonance frequency behavior
of these modes is different at the two locations. In contrast, the
resonance around 31 kHz demonstrates a much more consis-
tent linear correlation with strain, which is maintained over
all testing days regardless of temperature however, the unique
correlation is still restricted to an individual location, as shown
in Figure 9a. To illustrate the potential of using the frequency of
the 31 kHz resonance for predicting strain/RNT, a linear model
is fit to the frequency-strain data from each location, and the
resulting fits are used to predict strain at known rail tempera-
ture and frequency. The predicted strain is then converted to
RNT, knowing the rail temperature and using Equation 2 the
predicted RNT value is then compared to the system-reported
RNT, and the difference between them defines RNT prediction
error. The RNT prediction error distributions for East and West
testing locations are shown in Figures 9b and 9c, respectively.
The RNT errors range between ±3.33 °C (±6 °F) overall and
mostly between ±1.11 °C (±2 °F). The standard deviation and
mean of the errors for the East location are ±1.16 °C (±2.1 °F)
and –1.61 × 10–9 °C (–2.87 × 10–9 °F), respectively, compared to
ME
|
RAILROADS
Time of day
Time of day
00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00
Time of day
00:00
100
100
80
60
92
91
90
0
–100
–200
03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00
00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00
Figure 6. Typical daily records measured from midnight to midnight
on 15 August 2019 of (a) rail temperature (b) axial strain reported
from full-bridge strain gauge readings and (c) the corresponding RNT
calculated based on the reported temperature and strain from the data
logging system.
Date
2019- 09 2019- 11 2020- 01 2020- 03 2020- 05 2020- 07 2020- 09 2020- 11 2021- 01 2021- 03 2021- 05
65
70
75
80
85
90
East
West
Test days
Figure 7. Average nighttime RNT variation from the East and West test
locations during the period from 31 July 2019 through 20 June 2021.
Vibration testing days are indicated with black dots.
64
M A T E R I A L S E V A L U A T I O N • J A N U A R Y 2 0 2 4
2401 ME January.indd 64 12/20/23 8:01 AM
RNT
(°F)
Microstrain
T
(°F)
RNT
(°F)

1.37 °C (2.46 °F) and –2.778 × 10–10 °C (4.68 × 10–10 °F) for the
West location. Considering the practical RNT accuracy range
established by the Transportation Technology Center Inc.
(TTCI, now MxV Rail) is ±5.6 °C (±10 °F) (Read 2005) and the
optimal RNT accuracy range established by Kish et al. (2013)
is ±2.8 °C (±5 °F), the 31 kHz resonance frequency prediction
shows the ability to predict strain/RNT around these accuracy
levels at a single fixed location. However, the results from one
location cannot be applied at another. In the case where the fit
to data from the East location is applied to predict the strain
and thus RNT on the West location, the RNT error ranges from
17 to 24 °C (32 to 44 °F) and the RNT error ranges from –21 to
–12 °C (–38 to –22 °F) for the converse case.
Despite the fact that selected resonance frequencies exhibit
high correlation with strain at one particular location, it is
important to understand that other factors may influence the
resonance frequency values even at the high-frequency range
for example, the rail damage condition, profile geometry, and
other material properties of the rail.
Conclusions
This paper presents an approach for nondestructive estima-
tion of axial rail stress state (strain) in situ. The approach is
based on contactless sensing of impulse-driven rail vibration
resonances. Instead of using low-frequency vibrations, where
corrupting influences from rail support structures are known,
we investigated relatively high-frequency resonances above
20 kHz, four of which are prominent and consistently excited.
The approach using data collected from a rail structure in
active service is demonstrated, where the rail temperature,
axial strain, and RNT have been continuously recorded over
two years. We studied correlations between the resonance fre-
quency of the four modes and axial rail strain across a range
of temperatures and RNT conditions at two distinct test loca-
tions. All modes show a coupled influence of temperature and
stress state on the resonance frequency, and, for most of the
vibration modes, this coupled behavior and other unconfirmed
influences disrupt the correlation with strain across varying
temperature. However, the resonance around 31 kHz does
exhibit consistent and strong correlation with strain and can
be used to predict in-place RNT considering the full two-year
dataset within an accuracy of ±3.33 °C (±6 °F) at one test
location. On the other hand, the correlation only applies to a
specific individual test location, and the correlation at another
test location must be determined separately. Nevertheless,
RNT prediction using a linear fit to frequency-strain data for
this one resonance mode at a specific test location for which
the relation has been established and the rail temperature is
known shows acceptable accuracy. The result thereby demon-
strates the potential of resonance frequency prediction of
strain/RNT from in-service rail structures assuming an appro-
priate rail vibration mode is identified.
Frequency (Hz)
200
–200
37 000 37 100 37 200
0
Frequency (Hz)
200
–200
39 300 39 400 39 500
0
Frequency (Hz)
200
–200
76 400 76 600 76 800
0
East
West
East
West
East
West
Figure 8. Axial strain
as a function of
frequency of the mode
at (a) 37 kHz (b) 39 kHz
and (c) 76 kHz at the East
and West test locations.
Each line represents
the measurements
during one day. Positive-
valued strain represents
compression.
East
y =–3.017 x +93932.330
R2 =0.98
y =–2.518 x +78191.829
R2 =0.94
West
Frequency (Hz)
RNT error (°F)
400
300
200
30 950 31 000 31 050 31 100 31 150 31 200 31 250
–8 –6 –4 –2 0 2
25
20
15
10
5
0 4 6 8
100
0
–100
–200
–300
–400
RNT error (°F)
–8 –6 –4 –2 0 2
10
8
6
2
4
0 4 6 8
Figure 9. (a) Axial strain as a
function of frequency for the
31 kHz mode at both East and
West test locations. Each line
represents the measurements
during one day of testing.
Positive-valued strain
represents compression. The
error between the frequency-
predicted RNT and system-
reported RNT: (b) the East
location and (c) West location.
J A N U A R Y 2 0 2 4 • M A T E R I A L S E V A L U A T I O N 65
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Microstrain Microstrain Microstrain
Microstrain
Counts
Counts