Quantifying the Impact of Subgrade Stiffness on Track Quality and the Development of Geometry Defects

The VTD measurements were recorded using an MRail rolling deflection measurement system that was developed at the University of Nebraska at Lincoln in collaboration with the Federal Railroad Administration (FRA) (Norman 2004; Norman et al. 2004; McVey et al. 2005; Farritor 2006; McVey 2006; Arnold et al. 2006, Lu 2008; Greisen 2010; Farritor and Fateh 2013). The system consists of a laser and camera that measure the deflection of track at a distance of 1.22 m toward the center of the car from an inboard wheel, relative to a datum at the base of that wheel (Fig. 1). VTD measurements were taken every 0.305 m (1 ft) along the track on both rails. This VTD system was used over the two subdivisions between May and July of 2015. The collected VTD measurements were processed to calculate ${\mathrm{VTD}}_{\text{sub}}$ using the methodology presented in Roghani and Hendry (2016). In both measurement runs, the MRail system was installed on a ballast car loaded to 117.7 t and axle load of $29\pm 1.6\mathrm{t}$. The measurement system was operated within revenue service and, as a result, the weight of the axles of the adjacent car or locomotive could not be specified, only measured. The variations in loading of the car or locomotive adjacent to the measurement system may change VTD measurements and the resulting ${\mathrm{VTD}}_{\text{sub}}$ (Roghani and Hendry 2016). For the Prairie subdivision, the instrumented car was adjacent to a six-axle locomotive with axle loads of $31.3\pm 0.7\mathrm{t}$, and for the Shield subdivision, it was adjacent to a freight car with axle loads of $27.7\pm 1.4\mathrm{t}$.

Track geometry measurements are used by the railway industry to ensure that the shape of the rail allows for the safe passage of trains at the designated maximum speed of the track (AREMA 2012). The ability to maintain operable track geometry is the primary function of the infrastructure beneath the track; thus, the ability of these structures to maintain this geometry is the metric by which the subsequent analyses defines performance.

The published works and research regarding soft subgrades have quantified the condition of track using a track modulus ($u$), not VTD, where $u$ is defined as the ratio of VTD and the pressure between base of the rail and the underlying ties and foundation and is a measure of the stiffness of the structure (Cai et al. 1994). Parametric studies have shown $u$ to be primarily influenced by the subgrade conditions (Stewart and Selig 1982; Stewart 1985; Selig and Li 1994; Shahu et al. 1999; Shahin and Indraratna 2006; Rose and Konduri 2006). The numerical analysis conducted by Selig and Li (1994) concluded that changes in track modulus, where the modulus is $<\hspace{0.17em}28\mathrm{MPa}$ (4,000 psi), results in substantial increase in track deflection, rail and tie bending stress, and subgrade stresses. Hay (1982) and the American Railway Engineering and Maintenance of Way Association (AREMA 2012) manual suggested that 14 MPa (2,000 psi) is the minimum value of $u$ required for satisfactory performance of the track. Similarly, Ahlf (1975) found through field observations that a track modulus $<14\mathrm{MPa}$ resulted in track that required an exceptional amount of maintenance to maintain, a track modulus between 14 and 28 MPa was average, and a track modulus greater than 28 MPa was good. The terms satisfactory, poor, average, and good used by AREMA (2012) and Ahlf (1975) are qualitative and describe the amount of maintenance that is required to maintain the track in an operational condition. Poorer performance requires more maintenance and monitoring to ensure operational conditions.

The analyses presented within this paper are conducted with VTD measurements, and the thresholds for $u$ are converted to ${\mathrm{VTD}}_{\text{sub}}$ because ${\mathrm{VTD}}_{\text{sub}}$ can be measured and evaluated. This conversion was based on Fallah et al. (2016), which showed that the average $u$ over a 20-m section of track with continuously welded rail (CWR) could be determined with the Winkler model and the ${\mathrm{VTD}}_{\text{sub}}$ measurement from the MRail system and axle loads. Thus, conversions for $u$ to ${\mathrm{VTD}}_{\text{sub}}$ were developed for both loading conditions (adjacent car or locomotive) using the Winkler model; these modeled relationships between ${\mathrm{VTD}}_{\text{sub}}$ and $u$ are compared in Fig. 2. It is evident from Fig. 2 that there is very little difference ($<2\%$) between the ${\mathrm{VTD}}_{\text{sub}}$ values obtained when a locomotive is adjacent to the instrumented car as opposed to when a loaded rail car is adjacent. Thus, a single equation was developed to provide a conversion between $u$ and ${\mathrm{VTD}}_{\text{sub}}$ for both of these loading conditions [Eq. (2)]where $u$ = track modulus (MPa) and ${\mathrm{VTD}}_{\text{sub}}$ (mm). From Fig. 2 and Eq. (2), ${\mathrm{VTD}}_{\text{sub}}>4.4\mathrm{mm}$ is equivalent to the lower threshold of $u<$ 14 MPa, and ${\mathrm{VTD}}_{\text{sub}}<3.1\mathrm{mm}$ is equivalent to upper threshold of $u>28\mathrm{MPa}$.

The distribution of ${\mathrm{VTD}}_{\text{sub}}$ from both study subdivisions is presented in Fig. 3(a), along with the threshold values for ${\mathrm{VTD}}_{\text{sub}}$ derived from the AREMA thresholds for $u$. Overall, ${\mathrm{VTD}}_{\text{sub}}$ has a normal distribution, where the mean (and median) of ${\mathrm{VTD}}_{\text{sub}}$ is 3.7 mm, and the standard deviation ($\sigma$) is 0.5 mm. From Fig. 3(a), the AREMA thresholds provide a reasonable agreement with the measured distributions, with 12% of the track classified as good and the threshold $1.2\sigma$ below the mean, 78% as average, and 10% as poor, with the threshold $1.4\sigma$ above the mean. The Prairie subdivision shows a higher prevalence of poor track than the Canadian Shield subdivision, and thus softer subgrade conditions. The average range was subdivided at the mean to result in four categories and increased resolution.

There is no precedent for the quantifying the change in modulus or the corresponding change in VTD; thus, a metric was devised for this study. The slope of ${\mathrm{VTD}}_{\text{sub}}$ versus distance plot was adopted as a simple and transparent metric to quantify change of track deflection ($\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$) over a distance, where $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ is calculated as the absolute value of secant slope of ${\mathrm{VTD}}_{\text{sub}}$, and distance ($d$) is the length of track over which this slope is evaluated [Eq. (3)]. For this analysis, $d$ was set equal to 20 m to be consistent with the other metrics and filtering used for the analysis of VTD measurements

The meaning of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ is demonstrated in Fig. 4, which shows the stratigraphy of an 800-m section of track. Fig. 4(b) plots the ${\mathrm{VTD}}_{\text{sub}}$ measured over this track, and Fig. 4(c) plots the $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ calculated from the ${\mathrm{VTD}}_{\text{sub}}$. An increase in $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ is the result of an increase in slope in the ${\mathrm{VTD}}_{\text{sub}}$ plot and corresponds to a higher spatial change of ${\mathrm{VTD}}_{\text{sub}}$ and thus $u$.

The distribution of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ from both study subdivisions is presented in Fig. 3(b). Fig. 3(b) shows that the 99% of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ values vary between 0 and $0.05\mathrm{mm}/\mathrm{m}$ within the two subdivisions evaluated. The folded-normal distribution of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ is a result of the use of an absolute value within Eq. (3), and results in a mode of 0 and a mean value of $0.009\mathrm{mm}/\mathrm{m}$. The Prairie subdivision shows a higher prevalence of high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. There are no thresholds that can be adopted to quantify the track conditions based on $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$; thus, arbitrary values are imposed that divide the track into four sections that comprise equal lengths of track (quartiles). This division is based on the cumulative distribution of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ presented in Fig. 3(c), where the 25, 50, and 75% quartile thresholds correspond to $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ values of 0.003, 0.008, and $0.013\mathrm{mm}/\mathrm{m}$.

The defect data are composed of discrete locations at which priority and urgent threshold values were exceeded. The comparison of the location of defects and the VTD measurements was based on the coordinates from the global positioning systems (GPS) used for the measurements. Both the GPS used for the track geometry and the VTD measurements had a specified R95 (the radius of a circle centered at the true position, containing the position estimate with probability of 95%) of 3.7 m. Thus, the ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ evaluated at the location of each identified defect were the average of the values measured within 7.4 m of the defect. These results were found to be insensitive to the variation of this 7.4-m window from 1 to 10 m; this insensitivity is attributed to the 20-m filtering applied to generate the ${\mathrm{VTD}}_{\text{sub}}$.

This defect data were analyzed to determine the prevalence of defects per kilometer of track within the different categories defined by the divisions of ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. Thus, the combined defects from both subdivisions were sorted into one of 16 categories based on the four divisions of ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. The number of defects within each of these categories was divided by the number of kilometers of track within each category to allow for a comparison of the intensity of the occurrence of defects generated by each classification of track. The results of this categorization are presented in Fig. 5 for Class 4 track and Fig. 6 for Class 3 track. There were too few alignment defects to include in this analysis, with no Class 4 alignment defects and only 50 Class 3 alignment defects (or 3% of total number of defects).

Surface defects and gauge defects make up 30 and 70% of the total number of Class 4 defects, respectively. Figs. 5(a) and 6(a) show a strong relationship between gauge defects and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$, but not with ${\mathrm{VTD}}_{\text{sub}}$, with the highest number of defects occurring with high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ and over the stiffest track (good ${\mathrm{VTD}}_{\text{sub}}$). The authors hypothesize this is the result of higher dynamic loads that occur at transitions between differing track moduli, a mechanism suggested in Li et al. (2015), with the stiffest track providing less attenuation for impacts. The number of defects over the track with the most consistent ${\mathrm{VTD}}_{\text{sub}}$ (low $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$) suggests that there is only a very small contribution by factors not represented within this comparison. Figs. 5(b) and 6(b) show a strong relationship between surface defects and both ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$, with the highest number of defects occurring with high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ and poor ${\mathrm{VTD}}_{\text{sub}}$. These surface defects appear to be the result of both high deflections (low stiffness) and dynamic loads resulting from the changes in stiffness. The very small number of defects occurring where there are good ${\mathrm{VTD}}_{\text{sub}}$ and low $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ suggests that these two metrics describe the primary conditions that result in surface defects. These trends between gauge, surface defects, ${\mathrm{VTD}}_{\text{sub}}$, and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ were found to be consistent between the two subdivisions, with only slight variations in the magnitudes.

The surface defects [Fig. 5(b)] can be further divided into warp [Fig. 7(a)], crosslevel [Fig. 7(b)], and profile [Fig. 7(c)] defects, each with a distribution within the ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ classifications. These plots show that the highest frequency of warp, crosslevel, and profile defects occur with poor ${\mathrm{VTD}}_{\text{sub}}$ and high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. Both ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ contribute to the generation of defects, though warp and profile defects show a greater impact of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ [Figs. 7(a and c)]. Similar correlations were found between Class 3 warp, crosslevel, profile defects, and ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ from both subdivisions (Fig. S1).

The significance of poor ${\mathrm{VTD}}_{\text{sub}}$ and high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ on the development of surface defects is evident from all the plots presented in Fig. 7. The number of Class 4 warp, crosslevel, and profile defects per km generated at locations with $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ from the highest 25% was 2.6, 1.2, and 2.8 times that from the remaining 75% of the track, respectively [Figs. 7(a–c)]. Similarly, the number of Class 4 warp, crosslevel, and profile defects per km generated at locations with poor ${\mathrm{VTD}}_{\text{sub}}$ was 1.0, 1.7, and 1.7 times that from the remaining 82% of the track [Figs. 7(a–c)].

An examination of the VTD and TQI data was conducted to observe whether locations with higher ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ correspond to higher local TQI values obtained from one measurement of track geometry. This examination showed that elevated TQI coincided with two cases of VTD: the first are locations with elevated ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$, and the second are locations with elevated $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ but lower ${\mathrm{VTD}}_{\text{sub}}$. An example of the first condition is presented in Fig. 8, which shows a 2-km section of track of poor track conditions (${\mathrm{VTD}}_{\text{sub}}\ge 4.4\mathrm{mm}$) with more competent track on both ends [Fig. 8(a)], with high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ at transitions and variations of stiffness [Fig. 8(b)]. TQI measures, excluding ${\mathrm{TQI}}_{GA}$, increase within the section of poor ${\mathrm{VTD}}_{\text{sub}}$ [Figs. 8(c and d)]. Local peaks in TQI, including ${\mathrm{TQI}}_{GA}$, are coincident with peaks in $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ [Figs. 8(c and d)]. An example of the second condition is presented in Fig. 9, which shows an 800-m section of track of average to good track conditions based on ${\mathrm{VTD}}_{\text{sub}}$ [Fig. 9(a)], with a 270-m section of track with high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ [Fig. 9(b)]. TQI measures increase within the section of high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ [Figs. 9(c and d)], though the ${\mathrm{VTD}}_{\text{sub}}$ alone would suggest this to be a relatively competent structure.

Plots of ${\mathrm{TQI}}_{CR}$ versus ${\mathrm{VTD}}_{\text{sub}}$ are presented in Fig. 10(a) and ${\mathrm{TQI}}_{CR}$ versus $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ in Fig. 10(b). The plots in Fig. 10 show that at any given ${\mathrm{VTD}}_{\text{sub}}$ or $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$, there is a spectrum of ${\mathrm{TQI}}_{CR}$ values that result in a poor correlation, with a coefficient of determination ($R^2$) of 0.38 and 0.40 from the linear regression. Plots for ${\mathrm{TQI}}_{PR}$, ${\mathrm{TQI}}_{GA}$, and ${\mathrm{TQI}}_{\mathrm{AL}}$ show very similar plots, with $R^2$ values ranging from 0.35 to 0.56. Because these regressions are developed from 2,235,328 data points, there is a $>99.9\%$ probability that an underlying correlation between increasing TQI, ${\mathrm{VTD}}_{\text{sub}}$, and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ exists (Smith 2015). Nonlinear correlations and further data processing to match the peaks of TQI and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ that are offset from one another (Fig. 8) to account for the accuracy of the GPS locations and selection of TQI from differing track geometry measurements did not result in a stronger relationship between TQI, ${\mathrm{VTD}}_{\text{sub}}$, and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. This relationship is obscured by the impact of maintenance. The authors have not included the equations so they are not used to predict track conditions.

A simpler comparison between TQI, ${\mathrm{VTD}}_{\text{sub}}$, and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ can be shown with the distribution of TQI values within the differing catagories of ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ (Fig. 11). Fig. 11(a) presents the skewed, but similar, distributions of ${\mathrm{TQI}}_{PR}$ values from both the Prairies and Canadian Shield subdivisions from a single measurement of track geometry; similar plots of the distributions of ${\mathrm{TQI}}_{CR}$, ${\mathrm{TQI}}_{GA}$, and ${\mathrm{TQI}}_{\mathrm{AL}}$ are presented in Fig. S2. The range of values for the TQI is between 0.5 and 5.0 mm, with the exception of ${\mathrm{TQI}}_{AL}$, which has a narrower range of 0.5 and 3.0 mm. Fig. 11(b) presents the distributions of the ${\mathrm{TQI}}_{PR}$ for track divided into subsets of good and poor ${\mathrm{VTD}}_{\text{sub}}$, and Fig. 11(c) presents the distributions of the ${\mathrm{TQI}}_{PR}$ for track divided into subsets of high and low $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. It is clear from the distributions presented in Figs. 11(b and c) that the trend of higher VTD metrics resulting in higher ${\mathrm{TQI}}_{PR}$ values does exist within the data. There is a significant amount of overlap in the distributions of ${\mathrm{TQI}}_{PR}$ for the different catagories of ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$; this was also evident in this data as plotted in Fig. 8, and again shows the inability to predict ${\mathrm{TQI}}_{PR}$ from ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. The poor and high categories show a wider distribution [Figs. 11(b and c)], whereas the good and low categories show a more concentrated distribution. From Figs. 11(b and c), the categories of ${\mathrm{VTD}}_{\text{sub}}$ are a slightly better discriminator of expected ${\mathrm{TQI}}_{PR}$ than the categories of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ because the difference between the peaks (modes) in Fig. 11(b) is greater than those in Fig. 11(c). However, the categories of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ appear to be better indicators of high ${\mathrm{TQI}}_{PR}$ because they show very similar densities for the differing categories of ${\mathrm{VTD}}_{\text{sub}}$ for ${\mathrm{TQI}}_{PR}>4\mathrm{mm}$ [Fig. 11(b)], in contrast to the large difference between the densities between the differing $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ categories [Fig. 11(c)]. Similar observations can be made for the distributions of ${\mathrm{TQI}}_{CR}$, ${\mathrm{TQI}}_{GA}$, and ${\mathrm{TQI}}_{AL}$, which are presented in Fig. S3. The exceptions are the relationship between the alignment and crosslevel with the ${\mathrm{VTD}}_{\text{sub}}$, for which the distributions for good and poor ${\mathrm{VTD}}_{\text{sub}}$ were nearly identical, suggesting that ${\mathrm{VTD}}_{\text{sub}}$ does not impact the roughness of the alignment and crosslevel.

The contrast between the strength of the relationship between defects and VTD but not between TQI and VTD led to further examination of the TQI distributions to determine why this difference exists. The authors suggest that this difference is a result of the use of threshold values for geometry in the evaluation of defects versus analyses of the full spectrum of possible TQI values. Additionally, poor ${\mathrm{VTD}}_{\text{sub}}$ and high $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ have a much higher representation at higher values of TQI (Fig. 11). Thus, an arbitrary threshold of 3.0 mm was applied to the TQI, and the locations that exceeded this value were given the same treatment as the geometry defects. The sections of track from both subdivisions were divided into 16 categories based on ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$. Figs. 12(a and b) show the percentage of total amount of track with ${\mathrm{TQI}}_{GA}$ and surface-related TQI (${\mathrm{TQI}}_{PR}$ and ${\mathrm{TQI}}_{CR}$) that exceed the 3.0-mm threshold. The correlation between ${\mathrm{TQI}}_{GA}$ and surface-related TQI was strong with $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ and showed no correlation with ${\mathrm{VTD}}_{\text{sub}}$. This shows the significance of $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ in development of poor track roughness, with the average percentage of track with ${\mathrm{TQI}}_{GA}>3.0\mathrm{mm}$ increasing from 9.4 to 19.6% when $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ changes from low ($<25\%$) to high ($>75\%$). This impact is even more pronounced for the surface-related TQIs, where a change in $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ from low to high increases the average percentage of track with $\mathrm{TQI}>3.0\mathrm{mm}$ from 21.0 to 47.2%.

The surface-related TQIs [Fig. 12(b)] could be further divided into ${\mathrm{TQI}}_{PR}$ [Fig. 13(a)] and ${\mathrm{TQI}}_{CR}$ [Fig. 13(b)]. ${\mathrm{TQI}}_{CR}$ shows a strong correlation with $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$, whereas for ${\mathrm{TQI}}_{PR}$, both ${\mathrm{VTD}}_{\text{sub}}$ and $\mathrm{\Delta }{\mathrm{VTD}}_{\text{sub}}$ contribute. A sensitivity analysis was also conducted on the threshold value for TQI. It was found that these trends become obscure if the threshold is reduced below 2.5 mm, and remain strong with increasing thresholds until the data set above the threshold becomes too small to trend.