U.S. Department of 

Transportation

Federal Railroad 

Administration

Measurement of Vertical Track Deflection from a 

Moving Rail Car

Office of Research

and Development

Washington, DC 20590

DOT/FRA/ORD-13/08 Final Report

February 2013
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

February 2013

3. REPORT TYPE AND DATES COVERED

Technical Report

4. TITLE AND SUBTITLE

Measurement of Vertical Track Deflection from a Moving Rail Car

5. FUNDING NUMBERS

6. AUTHOR(S) and FRA Technical Representative

Shane Farritor and Mahmood Fateh

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

University of Nebraska-Lincoln

Department of Mechanical Engineering

Lincoln, NE 68588

8. PERFORMING ORGANIZATION 

REPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

U.S. Department of Transportation 

Federal Railroad Administration

Office of Research and Development

Washington, DC 20590

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

DOT/FRA/ORD-13/08

11. SUPPLEMENTARY NOTES

Program Manager: Mahmood Fateh

12a. DISTRIBUTION/AVAILABILITY STATEMENT

This document is available to the public through the FRA Web site at http://www.fra.dot.gov.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The University of Nebraska has been conducting research sponsored by the Federal Railroad Administration’s Office of Research 

and Development to develop a system that measures vertical track deflection/modulus from a moving rail car. Previous work has 

suggested the system can find critical maintenance problems not found by other inspection methods including standard track 

geometry.

The University of Nebraska system uses cameras and lasers to measure a portion of the track deflection basin, and this 

measurement is used to estimate vertical rail deflection and/or track modulus. This report presents the measurement system along 

with development of the theory behind the measurement to support diagnosis of track conditions. The results of significant field 

testing are presented, including the results of a test designed to verify the accuracy of the measurement. A Finite Element 

Analysis of the measurement is presented along with several variations on closed form solution analysis, based on beam on elastic 

foundation as well as a discrete support model for rail.

The University of Nebraska system has identified several critical maintenance problems not found by other inspection methods,

and all previous studies indicate that measuring track deflection provides unique and valuable insight to track conditions that can 

improve inspection, maintenance, and safety. Finally, some guidelines are suggested for development of thresholds to guide data 

interpretation and track condition assessment, but more study and tests are needed to better define threshold criteria.

14. SUBJECT TERMS

Vertical track deflection, track deflection, rail deflection, track measurement system, vertical 

track modulus, track modulus, track stiffness, track geometry measurement

15. NUMBER OF PAGES

136

16. PRICE CODE

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19. SECURITY CLASSIFICATION

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20. LIMITATION OF ABSTRACT

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. 239-18

298-102
ii

METRIC/ENGLISH CONVERSION FACTORS

ENGLISH TO METRIC METRIC TO ENGLISH

LENGTH (APPROXIMATE) LENGTH (APPROXIMATE)

1 inch (in) = 2.5 centimeters (cm) 1 millimeter (mm) = 0.04 inch (in)

1 foot (ft) = 30 centimeters (cm) 1 centimeter (cm) = 0.4 inch (in)

1 yard (yd) = 0.9 meter (m) 1 meter (m) = 3.3 feet (ft)

1 mile (mi) = 1.6 kilometers (km) 1 meter (m) = 1.1 yards (yd)

1 kilometer (km) = 0.6 mile (mi)

AREA (APPROXIMATE) AREA (APPROXIMATE)

1 square inch (sq in, in2) = 6.5 square centimeters (cm2) 1 square centimeter (cm2) = 0.16 square inch (sq in, in2)

1 square foot (sq ft, ft2) = 0.09 square meter (m2) 1 square meter (m2) = 1.2 square yards (sq yd, yd2)

1 square yard (sq yd, yd2) = 0.8 square meter (m2) 1 square kilometer (km2) = 0.4 square mile (sq mi, mi2)

1 square mile (sq mi, mi2) = 2.6 square kilometers (km2) 10,000 square meters (m2) = 1 hectare (ha) = 2.5 acres

1 acre = 0.4 hectare (he) = 4,000 square meters (m2)

MASS - WEIGHT (APPROXIMATE) MASS - WEIGHT (APPROXIMATE)

1 ounce (oz) = 28 grams (gm) 1 gram (gm) = 0.036 ounce (oz)

1 pound (lb) = 0.45 kilogram (kg) 1 kilogram (kg) = 2.2 pounds (lb)

1 short ton = 2,000 pounds 

(lb)

= 0.9 tonne (t) 1 tonne (t) =

=

1,000 kilograms (kg)

1.1 short tons

VOLUME (APPROXIMATE) VOLUME (APPROXIMATE)

1 teaspoon (tsp) = 5 milliliters (ml) 1 milliliter (ml) = 0.03 fluid ounce (fl oz)

1 tablespoon (tbsp) = 15 milliliters (ml) 1 liter (l) = 2.1 pints (pt)

1 fluid ounce (fl oz) = 30 milliliters (ml) 1 liter (l) = 1.06 quarts (qt)

1 cup (c) = 0.24 liter (l) 1 liter (l) = 0.26 gallon (gal)

1 pint (pt) = 0.47 liter (l)

1 quart (qt) = 0.96 liter (l)

1 gallon (gal) = 3.8 liters (l)

1 cubic foot (cu ft, ft3) = 0.03 cubic meter (m3) 1 cubic meter (m3) = 36 cubic feet (cu ft, ft3)

1 cubic yard (cu yd, yd3) = 0.76 cubic meter (m3) 1 cubic meter (m3) = 1.3 cubic yards (cu yd, yd3)

TEMPERATURE (EXACT) TEMPERATURE (EXACT)

[(x-32)(5/9)] °F = y °C [(9/5) y + 32] °C = x °F

QUICK INCH - CENTIMETER LENGTH CONVERSION

0 1 2 3 4 5

Inches

Centimeters 0 1 2 3 4 5 6 7 8 9 10 11 12 13

QUICK FAHRENHEIT - CELSIUS TEMPERATURE CONVERSIO

 -40° -22° -4° 14° 32° 50° 68° 86° 104° 122° 140° 158° 176° 194° 212°

 

°F

 °C -40° -30° -20° -10° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°

For more exact and or other conversion factors, see NIST Miscellaneous Publication 286, Units of Weights and 

Measures. Price $2.50 SD Catalog No. C13 10286 Updated 6/17/98
iii

Contents

Executive Summary........................................................................................................................ 1

1. Introduction ................................................................................................................. 3

1.1 Statement of Problem and Task Goals........................................................................ 3

1.2 Previous United States Track Modulus Measurement Systems.................................. 4

1.3 Related Track Measurements...................................................................................... 4

2. Analytical Models ....................................................................................................... 6

3. Overall approach ....................................................................................................... 19

3.1 The UNL Measurement............................................................................................. 19

3.2 Relationship to Existing Profile Variations............................................................... 25

3.3 End Chord Offset (ECO)........................................................................................... 28

3.4 Relative Track Deflection ......................................................................................... 29

4. The UNL Measurement Car...................................................................................... 32

4.1 Early Deployment...................................................................................................... 32

4.2 Caboose ..................................................................................................................... 32

4.3 Tank Car.................................................................................................................... 33

4.4 UNLX002 Autonomous Measurement System......................................................... 33

5. Validation of the UNL System.................................................................................. 54

5.1 Measurement of Deflection....................................................................................... 54

5.2 Measurement of Strain .............................................................................................. 61

6. Testing on the Union Pacific South Morrill Subdivision .......................................... 69

6.1 Derailments Following the 2007 Test ....................................................................... 72

6.2 System Measurement Under Various Conditions..................................................... 73

6.3 Trending Analysis ..................................................................................................... 77

6.4 Comparison between Different Measurement Systems ............................................ 82

6.5 Field Investigations ................................................................................................... 87

6.6 Analysis of Bridge Approaches................................................................................. 94

6.7 Identifying Thresholds for Maintenance ................................................................... 97

7. Finite Element Analysis (FEA) from System Perspective ...................................... 100

7.1 FEA Model Development ....................................................................................... 100

7.2 Verification of FEA Model ..................................................................................... 109

7.3 FEA Analysis and Results....................................................................................... 111

7.4 Suggestions for Further Development..................................................................... 119

8. Summary & Conclusions......................................................................................... 122

9. References ............................................................................................................... 123

Appendix A. Abbreviations and Acronyms............................................................................... 125
iv

Illustrations

Figure 1-1: Relative Rail Displacement under a Railcar ............................................................... 7

Figure 1-2: Discrete Model and Free Body Diagram .................................................................... 8

Figure 1-3: Middle Segment of Discrete Tie Model...................................................................... 9

Figure 1-4: Comparison of Winkler and Discrete Models........................................................... 12

Figure 1-5: Deflection of Track under Three Loads.................................................................... 14

Figure 1-6: Experimental Data and Curve Fitting (Zarembski and Choros, 1980) ..................... 15

Figure 1-7: Comparison of Cubic and Winkler Models .............................................................. 17

Figure 1-8: Modulus Calculations in Winkler and Cubic Model ................................................ 18

Figure 2-1: Diagram of Measurement Principle .......................................................................... 19

Figure 2-2: Camera/Laser System ............................................................................................... 20

Figure 2-3: Sensor Geometry....................................................................................................... 21

Figure 2-4: Superposition of the Deflections from Two Loads................................................... 22

Figure 2-5: Relation between Yrel and Modulus (Winkler model)............................................. 24

Figure 2-6: Relation between the Total Deflection and Yrel (Winkler model)........................... 25

Figure 2-7: An Example Site with both Significant Unloaded Geometry and Low Track Stiffness

............................................................................................................................................... 26

Figure 2-8: Simulation on the Effects of Track Geometry .......................................................... 27

Figure 2-9: Effects of Unloaded Geometry of Various Length (L) and Depth (d)...................... 28

Figure 2-10: 10’ ECO Calculation from Rail Profile................................................................... 29

Figure 2-11: Deflection Calculation ............................................................................................ 30

Figure 3-1: Experiments in Early Deployment............................................................................ 32

Figure 3-2: The Hopper Car Used in Early Deployment............................................................. 32

Figure 3-3: Exterior and Interior of the Caboose......................................................................... 33

Figure 3-4: The Tank Car Used in Early Stage of the Project..................................................... 33

Figure 3-5: System Instrumentation............................................................................................. 34

Figure 3-6: The Rigid Beam on the Side Frame .......................................................................... 35

Figure 3-7: Sensor Head Assembly ............................................................................................. 35

Figure 3-8: Typical Test Image.................................................................................................... 36

Figure 3-9: Enclosed Box for Computers.................................................................................... 37

Figure 3-10: Power Supply System ............................................................................................. 38

Figure 3-11: Power Supply System Monitoring Information in the April 2008 Test.................. 39
v

Figure 3-12: Database Website Screenshot ................................................................................. 40

Figure 3-13: Exception Locations List from the Website............................................................ 41

Figure 3-14: Reproduced Laser Curves....................................................................................... 41

Figure 3-15: An Imperfect Image Example ................................................................................. 42

Figure 3-16: Converting Number of Pixels into Distance in Inches............................................ 43

Figure 3-17: Calibration Plate on the Top of the Rail ................................................................. 44

Figure 3-18: Calibration Plate on Top of the Rail (Side View)................................................... 44

Figure 3-19: Captured Image of the Calibration Plate................................................................. 44

Figure 3-20: Calibration Results.................................................................................................. 45

Figure 3-21: Capturing the Rail Deflection with a Video Camera .............................................. 46

Figure 3-22: Captured Video Showing the Rail Deflection......................................................... 47

Figure 3-23: The Deflection Curve of the Rail from Calibration ................................................ 48

Figure 3-24: Captured Image when Sensor Head Passing by the Marker................................... 49

Figure 3-25: Yrel Data from the Mechanical Shop ..................................................................... 50

Figure 3-26: Limited Sampling Rate Causing Measurement Errors ........................................... 51

Figure 3-27: Laser Line Width .................................................................................................... 52

Figure 3-28: Laser Beam Drifting................................................................................................ 53

Figure 4-1: String Measurement Diagram................................................................................... 54

Figure 4-2: Field String Measurement......................................................................................... 55

Figure 4-3: Instruments Used in Survey Measurements.............................................................. 56

Figure 4-4: Measurement of Vertical Rail Position by Surveying .............................................. 56

Figure 4-5: Survey Measurements............................................................................................... 57

Figure 4-6: Wayside Camera Measurements Setup..................................................................... 58

Figure 4-7: Sample Data of Absolute Deflection from Wayside Cameras.................................. 59

Figure 4-8: Deflection Data from Camera Measurement ............................................................ 60

Figure 4-9: Rail Stresses and Fatigue .......................................................................................... 61

Figure 4-10: Winker Shape of Rail with Yrel Value ................................................................... 62

Figure 4-11: Relationship between Yrel and Rail Strain............................................................. 63

Figure 4-12: Two Gage Wheatstone Bridge Configuration......................................................... 64

Figure 4-13: Strain as Test Car is Spotted ................................................................................... 65

Figure 4-14: Measurements from Tangent Track (MP 231.6)..................................................... 67

Figure 4-15: Measurements from Pumping Track (MP 228.6) ................................................... 67

Figure 5-1: System in Revenue Service Testing.......................................................................... 69
vi

Figure 5-2: Yrel Data Overlaid on a Satellite Map...................................................................... 70

Figure 5-3: A Rough High-Speed Crossover............................................................................... 71

Figure 5-4: Track with Consistent Modulus................................................................................ 71

Figure 5-5: Site of Broken Field Weld 14 Days after Test.......................................................... 72

Figure 5-6: Failed Non-Insulated Joint 30 Days Post-Test.......................................................... 73

Figure 5-7: Measurements from Different Speeds....................................................................... 74

Figure 5-8: Dynamic Loads Affecting Measurements................................................................. 75

Figure 5-9: Variations of the Measurements ............................................................................... 76

Figure 5-10: Variations in Some Sections of Track..................................................................... 77

Figure 5-11: The Original data from Two Tests.......................................................................... 78

Figure 5-12: Cross Correlation .................................................................................................... 79

Figure 5-13: The Shifted Data from Two Tests........................................................................... 79

Figure 5-14: Data from Three Tests at MP A.74......................................................................... 80

Figure 5-15: Trending at MP A.74 and A.76............................................................................... 81

Figure 5-16: Test Data at MP A.74 as a Function of Time.......................................................... 81

Figure 5-17: Trending from MP A.70 to A.74............................................................................. 82

Figure 5-18: Data at the Crushed Rail Head Site......................................................................... 87

Figure 5-19: A Crushed Rail Head .............................................................................................. 88

Figure 5-20: Data at the Muddy Crossing Site ............................................................................ 89

Figure 5-21: The Muddy Crossing............................................................................................... 89

Figure 5-22: Data at the Failing Joint Site................................................................................... 90

Figure 5-23: The Failing Insulated Joint...................................................................................... 91

Figure 5-24: Data at the Broken Ties Site ................................................................................... 92

Figure 5-25: Six Broken Ties in a Row ....................................................................................... 93

Figure 5-26: Track Taken Out of Service .................................................................................... 94

Figure 5-27: Deflection Change Between Wood and Concrete Ties........................................... 95

Figure 5-28: Concrete Boxed Culvert.......................................................................................... 96

Figure 5-29: Three Span Bridge .................................................................................................. 97

Figure 5-30: Summary of Maintenance Priority of Sites............................................................. 99

Figure 6-1: Different Shapes in Yrel and ECO Data ................................................................. 101

Figure 6-2: Hermite Cubic Shape Functions ............................................................................. 104

Figure 6-3: Flowchart of Custom FEA Computer Program ...................................................... 106

Figure 6-4: Visual Diagram of FEA Model............................................................................... 107
vii

Figure 6-5: 132 RE Rail Section Properties............................................................................... 108

Figure 6-6: Schematic Representation of Model Input Variables ............................................. 109

Figure 6-7: Single-load Simulation with FEA Program Compared to Winkler Model............. 110

Figure 6-8: Two-load Simulation with FEA Program Compared to Winkler Model................ 110

Figure 6-9: Gap Element Simulation in ALGOR® Compared to Winkler Model.................... 111

Figure 6-10: Schematic of FEA Model with Pin Joint .............................................................. 112

Figure 6-11: Movie Frame from FEA Simulation with Pin Joint.............................................. 113

Figure 6-12: Yrel and ECO Results from FEA Simulation with Pin Joint................................ 113

Figure 6-13: Schematic of FEA Model with Pin Joint and Two Bad Ties................................ 114

Figure 6-14: Movie Frame from FEA Simulation with Pin Joint and Two Bad Ties ............... 114

Figure 6-15: Yrel and ECO Results from FEA Simulation with Pin Joint and Two Bad Ties . 115

Figure 6-16: Schematic of FEA Model with Pin Joint, Bad Ties, and Voids............................ 116

Figure 6-17: Nonlinear Deflection Curve for FEA Model with Voids...................................... 116

Figure 6-18: Yrel and ECO Results from FEA Simulation with Pin Joint, Bad Ties, and Voids

............................................................................................................................................. 117

Figure 6-19: Schematic of FEA Model with Pre-Existing Geometry........................................ 117

Figure 6-20: Yrel and ECO Results from FEA Simulation with Pre-Existing Geometry......... 118

Figure 6-21: Schematic of FEA Model with 10 Bad Ties......................................................... 118

Figure 6-22: Yrel and ECO Results from FEA Simulation with 10 Bad Ties........................... 119
viii

Tables

Table 2-1: Modulus and Yrel for Typical Track Conditions....................................................... 24

Table 4-1: String Measurement and Yrel Measurement.............................................................. 55

Table 4-2: Comparison between System Measurement and Survey Measurement..................... 57

Table 4-3: Comparison between Measurements from Surveying and Camera ........................... 60

Table 4-4: Comparison for Strain Measurement ......................................................................... 67

Table 5-1: Prioritized Exceptions of VTI Data............................................................................ 83

Table 5-2: Comparison between Vertical Track Deflection and ECO ........................................ 85

Table 5-3: Comparison between Vertical Track Deflection and ECO (Ranked by ECO) .......... 86

Table 5-4: Modulus Values from Kerr (2003)............................................................................. 94

Table 5-5: Average Deflection Over and Around Culvert........................................................... 96

Table 5-6: Average Deflection Over and Around Three-Span Bridge ........................................ 97

Table 5-7: Maintenance Summary............................................................................................... 98

Table 6-1: Results of FEA Simulations..................................................................................... 119
1

Executive Summary

This report documents the research and development work, “Track Modulus Measurement from 

a Moving Railcar,” performed by the University of Nebraska at Lincoln (UNL) under an FRA 

grant. It is believed that the grant’s objective has been met. There is a strong indication of value 

in the UNL measurement. The report shows that the UNL measurement is different from

standard geometry measurements and that the UNL system can find critical maintenance 

problems not found by other inspection methods. It suggests that there is value in the UNL 

measurement that could be used to improve railroad operating efficiency and increase safety. 

Some guidelines are suggested here for implementation of the system. However, it is also clear 

that more study and tests are needed to define appropriate threshold criteria. The rationale for 

the indicated confidence in this report for the UNL measurement is based on the extensive 

validation tests which are documented in the various sections throughout the report. In addition, 

further confirmation is provided towards the end of the report with finite element modeling using 

a nonlinear track model.

Based on the prevalent opinions as expressed at the relevant industry conferences, such as the 

Transportation Research Board (TRB) and the American Railway Engineering and Maintenanceof-way Association (AREMA), there is consensus that both passenger and freight railroad traffic 

are moving to higher-speeds and higher axle loads as a means to improve efficiency. The heavy 

axle loads and high speeds of modern freight trains produce high track stresses leading to quicker 

deterioration of track condition. Fast and reliable methods are vital for identifying and 

prioritizing track maintenance needs to minimize delays, avoid derailments, and reduce costs.

The condition and performance of railroad track depends on a number of different parameters. 

Some commonly monitored parameters include internal rail defects, profile, cross level, gage, 

and gage restraint. More recently, Ground Penetrating Radar (GPR) has been used for track 

subsurface condition assessment, including track layer thickness, ballast fouling, and drainage. 

Monitoring these parameters can improve safe train operation by identifying track locations that 

produce poor vehicle performance or derailment risk. However, at the present time, no effective 

tool is available to measure one of the most important parameters—vertical track deflection 

(VTD)—at normal track speeds in real time from a moving rail car. The track deflection 

measurement could be used to estimate track stiffness by dividing the applied load by the 

measured deflection. In turn, the track stiffness value is central to the estimation of track 

modulus. This parameter, VTD, is difficult to measure because no stable reference frame is 

available on the moving rail car. The UNL approach presented here uses cameras and lasers to 

measure the shape of the rail relative to the two wheel-rail contact points in a truck assembly. 

The results herein presented include simulation analysis, extensive field testing, field 

verification, and a demonstration measurement system that has shown capability of practical 

implementation in revenue service. 

This report presents extensive field validation tests as well as analytical background for the 

measurement including a full description of the classic Winkler linear model of track deflection 

and the definition of track modulus as it relates to this model. Some field measurements of track 

deflection are presented that illustrate the limitations of this model. Two new closed-form 

analytical models of track deflection are presented that have been produced during this project. 

The first is a discrete tie analytical model that can represent independent stiffness values for each 

tie. The second is a new nonlinear closed-form analytical model presented for track deflection. 
2

This model is based on a cubic relationship between distributed load and vertical deflection. The 

advantages of the nonlinear cubic model are discussed, and this is again related to field 

measurements of track deflection. Other systems that have been proposed to measure track 

modulus are included in the discussion.

The practical aspects of the system are addressed, including the basic principle of the proposed 

measurement and corrections that can be made for pre-existing variations in track profile. This 

practical implementation of the system culminates in an autonomous system designed for 

implementation in revenue service. Also included in this discussion is a fully defined method for 

system calibration. A series of field tests that were designed and conducted to validate the 

measurement system under many conditions are presented. This effort included using several 

independent measurement techniques to confirm the UNL measurement at several locations on 

real track, and related investigations of how the measurement did or did not change with vehicle 

speed and vehicle direction. The results of these extensive field tests include site visits to 

manually validate the measurements. These tests occurred on several thousand miles of revenue 

service track on various host railroads. Included is an analysis of some specific track locations 

where variations in modulus are known problems and how these locations can change over time. 

Specifically, an analysis of various bridge approaches is shown. This work has identified several 

locations where the track support is extremely complex and the track does not behave according 

to conventional norms. More work will be needed to fully understand these potentially 

significant locations.
3

1 Introduction

This report documents the research and development work performed to measure VTD from a 

moving rail car. These deflection measurements can be used to estimate track modulus. The 

developed system uses a noncontact vision sensor system to make displacement measurements 

with respect to the wheel-rail contact point. The system is inexpensive and does not require 

significant equipment support and personnel. It is capable of autonomous operation in revenue 

service.

1.1 Statement of Problem and Task Goals

The economic constraints of both passenger and freight railroad traffic are moving the railroad 

industry to higher-speed vehicles and heavier axle loads. The heavy axle loads and high speeds 

of modern freight trains produce high track stresses leading to quicker deterioration of track 

condition. As a result, the need for track maintenance increases. Fast and reliable methods are 

needed to identify and prioritize track in need of maintenance to minimize delays, avoid 

derailments, and reduce maintenance costs.

The condition and performance of railroad track depends on a number of different parameters. 

Some of the factors that influence track quality are track support (defined by track 

modulus/stiffness/deflection), internal rail defects, profile, crosslevel, gage, and gage restraint. 

Monitoring these parameters can improve safe train operation by identifying track locations that 

produce poor vehicle performance or derailment potential. Track monitoring also provides 

information for optimizing track maintenance activities by focusing activities where maintenance 

is critical and by selecting more effective maintenance and repair methods.

Automated methods of inspection are available for most of the parameters that are included in 

track geometry (Li et al., 2002). However, at the present time, no vehicle is available to measure 

track deflection, one of the most important parameters, at normal track speeds in real time. The 

track deflection measurement could be used to estimate track stiffness by dividing the applied 

load by the measured deflection. In turn, the track stiffness value is central to the estimation of 

track modulus, which has been defined as the coefficient of proportionality between the rail 

deflection and the vertical contact pressure between the rail base and track foundation (Cai et al., 

1994) in Beam on Elastic Foundation Theory but will be extended in this report. In other words, 

track modulus is the supporting force per unit length of rail per unit rail deflection (Selig and Li, 

1994), assuming the track support is continuous and elastic. Track modulus is a single parameter 

that represents the effects of all of the track components under the rail (Cai et al., 1994). These 

components include the subgrade, ballast, subballast, ties, and tie fasteners. Track modulus is 

particularly useful for the design of new track and the evaluation of good quality track where the 

assumptions of the theory are not violated. Also, it should be noted that the track 

stiffness/modulus can dramatically vary over a very short distance such as from one tie to the 

next. Therefore, it can be appropriate to talk about an average modulus value over a 

characteristic length of track (e.g., 10 or 100 feet (ft)).

The characteristics of track support including measurement of track modulus, stiffness, or 

deflection are important because they significantly affect track performance and maintenance 

requirements. Both low track modulus and large variations in track modulus are undesirable. 

Low track modulus has been shown to cause differential settlement that subsequently increases 

maintenance needs (Ebersohn et al., 1993; Read et al., 1994). Large variations in track modulus, 
4

such as those often found near bridges and crossings, have been shown to increase dynamic 

loading (Davis et al., 2003; Zarembski and Palese, 2003). Increased dynamic loading reduces 

the life of the track components, resulting in shorter maintenance cycles (Davis et al., 2003). It 

has been shown that reducing variations in track modulus at grade (i.e., road) crossings leads to 

better track performance and less track maintenance (Zarembski and Palese, 2003). It also has 

been suggested that track with a high and consistent modulus will allow for higher train speeds 

and therefore increase both performance and revenue (Heelis et al., 1999). Ride quality, as 

indicated by vertical acceleration, is also dependent on track modulus.

The ultimate goal of this project is to provide a system capable of measuring VTD and modulus

from a moving car at a high speed. The track deflection measurement would be used to estimate 

track stiffness or modulus depending on the track conditions and intent of the survey. The 

measurement system can be used as a tool for maintenance planning, track design, and 

acceptance of newly constructed and reconstructed track.

1.2 Previous United States Track Modulus Measurement Systems

Previous localized field testing has shown that it is possible to measure areas of low track 

modulus, variable track modulus, void deflection, variable total deflection, and inconsistent rail 

deflection (Ebersohn and Selig, 1994; Sussmann et al., 2001). In the past, such systems have 

been used to identify sections of track with poor performance. These measurements have been 

useful. However, they are expensive and have only been made over short distances (in the range 

of tens of meters). The ability to make these measurements continuously over large sections of 

track is desirable (Ebersohn and Selig, 1994; Read et al., 1994).

One vehicle, called the Track Loading Vehicle (TLV), uses this approach (Thompson and Li, 

2002). This vehicle is capable of measuring track modulus at speeds up to 16.1 kilometers per 

hour (km/hr) (10 miles per hour (mph)). The TLV uses two cars, each with a center load bogie 

capable of applying loads from 4.45 to 267 kilonewtons (kN) (1–60 kilopounds per second 

(kips)). A light load (13.3 kN or 3 kips) is applied by the first vehicle, whereas a heavier load is 

applied by the second vehicle. A laser-based system on each vehicle measures the deflections of 

the rail caused by the center load bogies. The test procedure involves two vehicles that pass over 

a section of track—first applying a 44.5-kilonewton (10 kip) load and then a 178-kilonewton (40 

kip) load (Thompson and Li, 2002). The use of two loads eliminates the problems with the 

Winkler assumptions as described in Figure 2-8. Both the light load and the heavier load are 

large enough to remove the track “slack” (i.e. low stiffness at low loads as in Figure 2-5). The 

use of two loads (if properly chosen) then becomes an approximation of the modulus defined by 

the derivative in Equation 2.2-12. 

Although the TLV is operational, it does have limitations. First, tests are often performed at 

speeds below 16.1 km/hr (10 mph); therefore, it is difficult to test long sections of track 

(hundreds of miles). Second, significant expense in both equipment and personnel are required 

for operation. For these reasons, the TLV has not yet been widely implemented.

1.3 Related Track Measurements

Track geometry data includes the measurement of track gage, alignment, cross level, curvature, 

etc. The data is collected with track geometry vehicles. The geometry cars are equipped with 

instruments, measuring devices, and computers necessary to calculate track geometry. Some 

track geometry vehicles are equipped with a Gage Restraint Measurement System (GRMS) that 
5

can apply known lateral and vertical loads to the track structure and measure the gage restraint 

capacity of crossties and rail fasteners. 

Ground-penetrating radar (GPR) is a geophysical method that uses radar pulses to image the 

subsurface. This nondestructive method uses electromagnetic radiation in the microwave band 

(UHF/VHF frequencies) of the radio spectrum and detects the reflected signals from subsurface 

structures. The technology has been implemented into detecting fouling ballast and mapping 

subballast and subgrade. These are the physical changes in the roadbed often causing track 

deflection variations. The ability to detect these changes makes GPR a useful tool to diagnose 

causes of track deflection variations that are not obvious from the surface. 

None of the track geometry, GRMS, or GPR technologies can evaluate the load-deflection 

response of the track. A track deflection or stiffness measurement is important to implement to 

ensure that a complete set of track characteristic measurements is available for maintenance 

planning, safety assessment, and vehicle track interaction analyses.
6

2 Analytical Models

2.1.1 Track Mechanics

The relationship between applied loads and track deformations is an important parameter to be 

considered in proper track design and maintenance. A representative mathematical model that 

accurately describes this relationship is desirable.

2.1.1.1 The Winkler Model of Track Deflection

The Beam on an Elastic Foundation (BOEF) model, also known as the Winkler model, describes 

a point load applied to an infinite beam on an infinite elastic foundation. It assumes the 

distributed supporting force of the track foundation is linearly proportional to the vertical rail 

deflection (i.e., p(x)=uw(x)). Here, the coefficient u is defined as track modulus. The 

differential equation then becomes:

( ) ( ) ( )

4

4

uw x q x

dx

d w x EI + = Equation 1.2-1

This model has been shown to be an effective method for determining track modulus (Raymond, 

1985; Meyer, 2002), and derivations can be found in Kerr (1976) and Boresi and Schmidt 

(2003). The vertical deflection of the rail, w, as a function of longitudinal distance along the rail 

x (referenced from the position of the applied load) is given by:

( ) e [ ( x) ( x)]

u

P

w x x β β β β cos sin

2 = − + − ⋅ Equation 1.2-2

4

1

4 















= EI

u β

Equation 1.2-3

where: P is the load on the track

u is the track modulus

E is the modulus of elasticity of the rail

I is the moment of inertia of the rail

x is the longitudinal distance along the rail

When multiple loads are applied, the rail deflections caused by each of the loads are superposed 

(assuming small vertical deflections) (Boresi and Schmidt, 2003).

A plot of the rail deflection given by the Winkler model over the length of a four-axle coal 

hopper is shown in Figure 2-1. The deflection is shown relative to the wheel/rail contact point 

for five different reasonable values of track modulus (6.89, 13.8, 20.7, 27.6, and 34.5 

megapascals (MPa) corresponding to 1,000, 2,000, 3,000, 4,000, and 5,000 pounds per square 

inch (psi), respectively). The model assumes a 115-pound rail with an elastic modulus of 206.8 

gigapascals (30,000,000 psi) and an area moment of inertia of 2,704 centimeters (cm)

4 (64.97 

inches (in)

4

).
7

Figure 2-1. Relative Rail Displacement under a Rail Car

The limitations of the Winkler model are clear given the widely accepted nonlinearity of track 

structure. However, this model is often used because it does provide a clear closed-form solution 

to the relationship between load and deflection in track structure.

2.1.1.2 A Discrete Tie Model of Track Deflection

In this section, a new discrete tie model is presented that is an output of this project. This model 

has been described previously (e.g., Norman, 2004). The discrete support model describes the 

rail supported on a number of discrete springs with a single force applied. The discrete springs 

represent support at the crossties, and the single applied load represents one rail-car wheel. At 

this time, tie support is modeled by linearly elastic springs. This model is more applicable at low 

speeds, but future work will include viscoelastic behavior. It is expected that measurements of 

track modulus may vary with train speed because of damping. However, the experimental 

results in this paper suggest the influence of track damping may not be significant at speeds 

below 48 km/h (30 mph).

The discrete tie model is shown to be useful in the experimental results because track modulus 

can vary from tie to tie. The proposed model considers only finite lengths of rail and a finite 

number of ties (Figure 2-2). To reduce the model’s computational requirements (so it can be 

implemented in real time), the rail is assumed to extend beyond the ties and is fixed at a (large) 

distance from the last tie. This ensures the boundary conditions are well defined (the rail is flat 

and far away), and the rail shape is continuous. 

The deflection in each of the springs (i.e. the rail deflection) can be determined by first solving 

for the forces in each of the springs by using energy methods. The principles of stationary 
8

potential energy and Castigliano’s theorem on deflections are applied (Boresi and Schmidt, 

2003). For these methods to be applicable, small displacements and linear elastic behavior is 

assumed. The number of equations needed to determine the forces in the springs is equal to the 

number of springs (i.e., spring forces are the unknowns).

Figure 2-2. Discrete Model and Free Body Diagram

The discrete support model is similar to the Winkler model when the ties are uniformly spaced 

and have uniform stiffness, and the rail is long. Nine ties are used in this work, and experimental 

trackside measurements have shown this to be sufficient (Norman, 2004).

The moment and shear force in the cantilevered sections of the model (Figure 2-2 (A) and (C)) 

can now be calculated. Static equilibrium requires the moment and shear force, for Section A, to 

be: 

1 1 M M V x = A + A Equation 1.2-4

V1 =VA Equation 1.2-5

Likewise, for Section C:

2 2 M M V x = C + C Equation 1.2-6

V2 =VC Equation 1.2-7

Now, the forces in the springs can be determined with energy methods (Figure 2-2(B)). Section 

B is split into segments separated by the springs, and the segment’s internal moments are found 

to determine the beam’s strain. Energy from shear force is small and is neglected (Norman, 

2004).
9

Figure 2-3. Middle Segment of Discrete Tie Model

The equations for the internal moments in each segment can be written (Equations 1.2-8 – 1.2-

17), where moments M1 and M2, and the shear forces V1 and V2 are given by Equations 1.2-4 –

1.2-7 above. The lengths of each of the segments (i.e., tie spacing) in the beam are given by L1 –

L10, and the spring forces are denoted by F1 – F9.
10

1 1

1

for 0 x L

M M

≤ ≤

= Equation 1.2-8

[ ( ) ( )]

1 1 1 2

1 1 1 1 1 1 1

for L x L L

M M V x L F x L

≤ ≤ +

= − + − + − Equation 1.2-9

[ ( ) ( ) ( )]

1 2 1 1 2 3

1 1 1 1 1 1 1 2 1 1 2

for L L x L L L

M M V x L F x L F x L L

+ ≤ ≤ + +

= − + − + − + − − Equation 1.2-10

( ) ( ) ( )

( )

1 2 3 1 1 2 3 4

3 1 1 2 3

1 1 1 1 1 1 1 2 1 1 2

for L L L x L L L L

F x L L L

M V x L F x L F x L L

M

+ + ≤ ≤ + + +





 





+ − − −

+ − + − + − − = − Equation 1.2-11

( ) ( ) ( )

( ) ( )

1 2 3 4 1 1 2 3 4 5

3 1 1 2 3 4 1 1 2 3 4

1 1 1 1 1 1 1 2 1 1 2

for L L L L x L L L L L

F x L L L F x L L L L

M V x L F x L F x L L

M

+ + + ≤ ≤ + + + +





 





+ − − − + − − − −

+ − + − + − − = − Equation 1.2-12

( ) ( ) ( )

( ) ( )

10 9 8 7 2 10 9 8 7 6

7 2 10 9 8 6 2 10 9 8 7

2 2 2 10 9 2 10 8 2 10 9

for L L L L x L L L L L

F x L L L F x L L L L

M V x L F x L F x L L

M

+ + + ≤ ≤ + + + +





 





+ − − − + − − − −

+ − + − + − − = Equation 1.2-13

( ) ( ) ( )

( )

10 9 8 2 10 9 8 7

7 2 10 9 8

2 2 2 10 9 2 10 8 2 10 9

for L L L x L L L L

F x L L L

M V x L F x L F x L L

M

+ + ≤ ≤ + + +





 





+ − − −

+ − + − + − − = Equation 1.2-14

[ ( ) ( ) ( )]

10 9 2 10 9 8

2 2 2 10 9 2 10 8 2 10 9

for L L x L L L

M M V x L F x L F x L L

+ ≤ ≤ + +

= + − + − + − − Equation 1.2-15

[ ( ) ( )]

10 2 10 9

2 2 2 10 9 2 10

for L x L L

M M V x L F x L

≤ ≤ +

= + − + − Equation 1.2-16

2 10

2

for 0 x L

M M

≤ ≤

= − Equation 1.2-17

In the above equations, the shear forces, moments, and spring forces are all unknown; however, 

one spring force can be determined by a vertical force balance:

F1 = P −V1 −V2 − F2 − F3 − F4 − F5 − F6 − F7 − F8 − F9 Equation 1.2-18

where P is a known wheel load (e.g., 157 kN or 35 kips). Now, the strain energy can be written 

where ki is the stiffness of spring i:

9

2

9

8

2

8

7

2

7

6

2

6

5

2

5

4

2

4

3

2

3

2

2

2

1

2

1

2

2

2

2

2

2

2

2

0

2

2

1

2

1

2

1

2

1

2

0

1

2

2 2 2 2 2 2

2 2 2 2 2

2 2 2 2

2 2 2 2

10 9 8 7 6

10 9 8 7

10 9 8 7

10 9 8

10 9 8

10 9

10 9

10

1 2 3 4 5 10

1 2 3 4

1 2 3 4

1 2 3

1 2 3

1 2

1 2

1

1

k

F

k

F

k

F

k

F

k

F

k

F

k

F

k

F

k

F dx

EI

M dx

EI

M

dx

EI

M dx

EI

M dx

EI

M dx

EI

M

dx

EI

M dx

EI

M dx

EI

M dx

EI

M U

L L L L L

L L L L

L L L L

L L L

L L L

L L

L L

L

L L L L L L

L L L L

L L L L

L L L

L L L

L L

L L

L

L

+ + + + + +

+ + + + +

+ + + +

= + + +

∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

+ + + +

+ + +

+ + +

+ +

+ +

+

+ + + + +

+ + +

+ + +

+ +

+ +

+

+

Equation 1.2-19

Castigliano’s theorem is now used to create the number of equations needed to solve for the 

unknown spring forces and boundary conditions (moment and shear force). In this case, there 
11

are 12 unknown variables (8 spring forces, 2 reaction moments, and 2 reaction forces). From 

Castigliano’s theorem:

0, and 0

0,

2 3 4 5 6 7 8 9

= ∂

∂ = ∂∂ = ∂∂ = ∂∂

= ∂

∂ = ∂∂ = ∂∂ = ∂∂ = ∂∂ = ∂∂ = ∂∂ = ∂∂

A B A VB

U

V

U

M

U

M

U

F

U

F

U

F

U

F

U

F

U

F

U

F

U

F

U

Equation 1.2-20

With these relationships, a set of 12 equations and 12 unknowns are developed by substituting 

the moment expressions (1.2-4 – 1.2-18) into (1.2-19). These expressions can be written in 

matrix form:

MF = P Equation 1.2-21

where:

P is the load vector

M is a 12 x 12 matrix of coefficients of the external forces

F is a column vector of the external forces F2 – F9, MA, VA, MB, and VB

The solution to this matrix equation gives the forces in each of the springs. Now, the spring 

deflections are:

i

i

i k

F d = Equation 1.2-22

where:

di is the deflection of spring i

Fi is the force in spring i

ki is the stiffness of spring i

The two models are now compared. Experimental results have shown the Winkler model is a 

good representation of track deflection (Zarembski and Choros, 1980; Norman, 2004). 

Therefore, the discrete model should give results similar to the Winkler model for similar inputs. 

However, the discrete model will have the additional ability to represent nonuniform track.
12

Figure 2-4. Comparison of Winkler and Discrete Models

Figure 2-4(B) compares the deflections from the two models for uniform modulus. The 

continuous line represents the Winkler model, and the boxes indicate the tie locations in the 

discrete model. The track modulus used in the Winkler model was 20.7 MPa (3,000 poundsforce (lbf)/in/in), and the corresponding tie stiffness was 10.5 MN/m (60,000 lbf/in). Track 

modulus is equated to tie stiffness by dividing by the tie spacing (ties spacing of 50.8 cm (20 

in)). A single point load of 157 kN (3,575 lbf) (representing a single vehicle wheel) was applied 

over the center tie. The deflection predicted by both models is very similar with a maximum 

change of 6.47 percent (relative to the accepted Winkler model is correct).

Real track has nonuniform modulus and these differences can be represented by the discrete 

model. In Figure 2-4(C), the stiffness of the third tie from the left end has been decreased by 50 

percent (to 5.25 meganewtons per meter (MN/m) or 30,000 lbf/in). In Figure 2-4(D), the 

stiffness of the third tie has been increased by 100 percent (to 21.0 MN/m or 120,000 lbf/in). In 

both of these cases, the Winkler model is shown with a uniform modulus.

The track deflection with a single soft tie (Figure 2-4(C)) is no longer symmetric about the 

loading point. The rail is deflected more on the left side of the load where the soft tie is located. 

The maximum deflection of the rail was also slightly increased (by approximately 0.1219 mm
13

(0.0048 in)). Figure 2-4(D) shows the rail deflection where the stiffness of the third tie has been 

doubled to 21.0 MN/m (120,000 lbf/in). The discrete model shows that the deflection near the 

stiff tie and the maximum deflection have both decreased (by approximately 0.1829 mm (0.0072 

in)). The results from these examples show that 1) the two models give similar results for similar 

inputs, and 2) the deflection curve can be affected by a single tie.

2.1.1.3 A Cubic Model of Track Deflection

Another model of track deflection was developed as an output of this project. This model is 

based on a cubic relationship between rail deflection and the supporting distributed load. This 

model has been previously published in McVey (2006).

Field tests conducted by the American Society of Civil Engineers (ASCE) Special Committee on 

Stresses in Railroad Track (1918) clearly showed that the vertical rail deflections were not 

linearly proportional to the wheel loads. An extensive experimental study conducted by 

Zarembski and Choros (1980) also clearly documented this nonlinear response.

Figure 2-5 shows the experimental results of the track responses under various applied loads. 

This experiment was conducted by UNL at Level, NE. Rail deflection was measured at given 

locations by using linear variable differential transformers (LVDTs) as a short, slow moving 

train of known weight passed. The axles of the train weighed 150,600 N (33,850 lbf), 60,230 N 

(13,540 lbf), and 30,650 N (6,890 lbf). The LVDTs were mounted to steel rods (about 1 m (3ft)) 

driven into the subgrade to provide a stable reference. The LVDTs then measured the vertical 

motion of the flange relative to the steel rod. The results from four LVDTs are shown in 

Figure 2-5. Here, the LVDTs were placed at 1 m (3 ft) increments along the track (x = 1m, 2m, 

3m, and 4m).
14

Figure 2-5. Deflection of Track under Three Loads

These measurements, along with many others dating back to the Talbot report (ASCE-AREA 

Special Committee, 1918), clearly indicate that the vertical rail deflections are not linearly 

proportional to the wheel loads. It is also important to note that the “degree” of nonlinearity can 

change dramatically over very short distances along the track. Note the deflection of the track 

under the 30,650 N (6,890 lbf) load increased about 60 percent over a distance of 1 m. This 

nonlinearity and variability greatly complicates determining and modeling track structure and 

violates most assumptions of the Beam on Elastic Layer Theory. Several methods have been 

developed for calculating modulus with each method assuming a different definition of track 

modulus that approximates the nonlinear behavior of real track.

Here, a new model is proposed that represents the relationship between vertical rail deflection 

and the distributed rail support force, p(x), as a cubic polynomial as defined in Equation 2.2-4 

below. To support this approach, this relationship to the experimental results of (Zarembski and 

Choros, 1980) are plotted in Figure 2-6, along with a cubic polynomial curve fit between the 

distributed rail support force and displacement. The cubic polynomial fits the experimental 

results very well (R2 = 0.9987).
15

Figure 2-6. Experimental Data and Curve Fitting (Zarembski and Choros, 1980)

Using a cubic polynomial has several advantages. First, it captures the behavior of real track in 

that it provides for low stiffness at low loads and higher stiffness at higher loads, such as is 

shown in the field measurements of Figure 2-5. Also, negative displacement of the track (track 

lift) does not result in significant downward forces being applied to the rail. Unlike the previous 

models, the cubic polynomial closely represents the fact that if the track rises slightly, the ballast 

does not pull the track down.

Here, the supporting distributed load p(x) has a cubic relationship between p(x) and w(x):

3

1 3 p(x) = u w(x) + u w(x) Equation 1.2-23

Note that symmetry about the applied load requires the second order term to vanish. Substitution

into the BOEF model gives the following differential equation:

u w u w q dx

d w EI + + =3

4 1 3

4

Equation 1.2-24

Equation 1.2-24 is a nonlinear differential equation, and a closed-form analytical solution is not 

straightforward. One analytical approximation based on Cunningham’s method can be found by
16

McVey (2006). However, a numerical solution for this boundary value problem (BVP) can be 

obtained.

The BVP can be written in state space notation as:

w′ = f (w, x) Equation 1.2-25

( , )

( )

( )

( )

( )

f w x

w x

w x

w x

w x

x

w =

























′′′

′′

′

∂

∂ ′ = Equation 1.2-26

Given Equation 1.2-26, the BVP becomes:

( )



























− +

′′′

′′

′

=

























′′′

′′

′

∂

∂

( ) ( ) 1

( )

( )

( )

( )

( )

( )

( )

3

1 3 u w x u w x

EI

w x

w x

w x

w x

w x

w x

w x

x

Equation 1.2-27

As the name implies, the fourth-order BVP described above requires the value of four boundary 

conditions which are displayed in the following equations:

x o

x

x

x

w x w

w x

w x

w x

=

′ =

=

=

=

=

=−∞

=∞

0

0

( ) |

( ) | 0

( ) | 0

( ) | 0

Equation 1.2-28

Now, because the BVP can have more than one correct solution, an initial “guess” for the last 

boundary condition is needed in order for the solution to converge to an expected solution. In 

this case, the initial guess is provided by the Winkler model evaluated at x = 0 and u = u3.

( ) 4

1

3

3 4

 where :

2

0 













 = = − = EI

u

u

P

w wo β β Equation 1.2-29

The mechanics of this problem also require the solution be found subject to the additional 

constraint given by symmetry about the load:

( ) ∫

∞

+ = 0

3

1 3 2

P

u w u w dx Equation 1.2-30

The unique solution that satisfies each of these constraints will give the rail deflection. Many

numerical techniques can be used to solve this well-posed BVP. In this work, the “bvp4c” 

function in Matlab (Kierzenka and Shampine, 2001) was used.

Although the cubic model closely represents the deflection test data over the entire range of 

wheel loads, the accuracy of the linear analysis depends on the magnitude of the test load. 
17

Computed deflections for loads other than the test load will have an error of the nonlinearity and 

the fact that the linearization results in accurate predictions where the linearization intersects the 

nonlinear behavior.

Because the cubic spring is initially softer than the one in the Winkler model, the rail must 

deflect more before the base can pick up the full load. This means that the distributed load will 

be spread over a wider span than for the linear model as shown in Figure 2-7, which makes 

logical sense with expected behavior of railway track. 

Figure 2-7. Comparison of Cubic and Winkler Models

It is proposed that a good definition of track modulus is the variation in supporting distributed 

force relative to the variation in deflection near the characteristic load for a given track (Lu et al., 

2008). This characteristic load might be defined as the nominal axle load for a given freight line 

(e.g., 160 kN or 286,000/8 = 36 kips). This can be expressed mathematically as the derivative of 

the pressure-deflection curve evaluated at the characteristic load P*:

*

*

P w

p u

∂

∂ = Equation 1.2-31

where: u* is the characteristic track modulus

p is the supporting force per unit length of rail

P* is the characteristic load corresponding to a given rail line

To evaluate the derivative at the characteristic load, the load must again be transformed to a 

distributed load. This can be done with the linear assumptions as described previously (the 

Winkler model). This definition of track modulus has been used in field measurements (Arnold 

et al., 2006).

-500 -400 -300 -200 -100 0 100 200 300 400 500 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

Distance [in]

Deflection [in]

Cubic

Winkler
18

Finally, in the nonlinear cubic model described previously, the track modulus at characteristic 

load can be calculated as:

( ) *

* *

2

1 3

3

* 1 3 3

P

P P

u u w

w

u w u w

w

p u = +

∂

∂ + = ∂

∂ = Equation 1.2-32

This definition of track modulus is compared with the Winkler model as shown in Figure 2-8. In 

this figure, the load deflection curve is plotted from the experimental data of Zarembski and 

Choros (1980) shown in Figure 2-6.

Figure 2-8. Modulus Calculations in Winkler and Cubic Models

It is clear that for single data points at higher loads the Winkler model will always underestimate 

the actual track modulus (Figure 2-8). As seen in this figure, the line connecting the point of 

zero load and zero deflection with the point of actual load and actual deflection (red line with 

slope of 4,315). The Winkler model will also poorly represent changes in deflection with respect 

to changes in distributed load at these higher values because the slope of the modulus curve is 

much lower with the Winker model. 
19

3 Overall Approach

The UNL system measures the rail height relative to the line created by the wheel/rail contact 

points. This measurement of track deflection is combined with an analytical model of the track 

structure to estimate the track stiffness and track modulus. The system currently operates

continuously over long distances and in revenue service.

3.1 The UNL Measurement

3.1.1 Measurement Principle and Methodology

The geometry of the measurement system is shown in Figure 3-1. An instrumented beam is 

rigidly mounted on the side frame of a hopper car and extends a few feet away from the wheels. 

A sensor head, which includes a laser/camera system, is attached to the end of the beam. The 

sensor system has two line lasers and a camera as shown in Figure 3-2. The line lasers intersect 

the rail surface at an acute angle to create curves across the surface of the rail. Using line lasers 

allows the system to compensate for lateral movement of the rail relative to the camera and for 

changes in rail profile. The camera captures images showing two curved laser lines on the rail 

surface, and the distance between the lines, d, is obtained by an image processing program. This 

distance d is then converted to the distance between the beam and the rail surface under the 

camera h. To accomplish this conversion, a calibration method is outlined in Section 4.4.3. A 

track model is available to convert the measured track deflection to track modulus.

Figure 3-1. Diagram of Measurement Principle
20

Figure 3-2. Camera/Laser System

Figure 3-1 illustrates that the fixed distance between the wheel-rail contact point and sensor, H, 

relates the relative rail displacement, Yrel, to the measured height of the sensor above the rail 

surface, h. Here, ycamera is the deflection of the rail at the location underneath the camera/lasers,

and ywheel is the deflection of the rail at the wheel-rail contact point. The deflections are negative 

in value because the positive axis is defined upwards.

The sensor system measures the distance between the camera image plane and the rail surface, h. 

Then, the displacement of the rail surface with respect to the wheel-rail contact plane, Yrel

(Figure 3-1) can be found. The method and mathematics to transform the distance between the 

laser lines, d, to a Yrel measurement is fully described in Section 4.4.3. The displacement, Yrel, 

can then be related to the absolute rail deflection of the wheel-rail contact point (with respect to 

the unloaded rail), ywheel (Figure 3-1) by using the Winkler model. 

A mathematical model relates the measured distance between the laser lines to the track

modulus. The rail deflection measured by the sensor is dependent on the four wheel loads. The 

sensor will measure the relative rail displacement between the rail and wheel-rail contact point. 

This measurement can be made if it is assumed that the instrument beam, truck, and wheels are 

rigid. With this assumption, the distance between the sensor system and wheel-rail contact point 

can be assumed constant (H is constant). This is a reasonable assumption because the instrument 

beam, side frame, and wheels are all massive, nearly rigid elements, and these elements do not 

include the suspension of the rail car. Rotation of the side frame could cause this distance (H) to 

change, but this rotation has been experimentally shown to be insignificant (Norman, 2004). 

The sensor reading, which is the measured distance between the lasers, is geometrically related 

to the height of the sensor above the rail. The sensor in effect measures its height above the rail 

by measuring the distance between the lasers. As the sensor moves closer or farther from the rail 

surface, the distance between the lasers changes. A schematic of the sensor is shown in 

Figure 3-3.
21

Figure 3-3. Sensor Geometry

From the above figure the following equations can be written:

(L1 + l1 )tanθ 1 = h Equation 3.1-1

(L + l ) = h 2 2 2 tanθ Equation 3.1-2

1 2 d = l + l Equation 3.1-3

where L1 and L2 are the horizontal displacement of the lasers from the camera, θ1 and θ2 are the 

angles between the lasers and the horizontal, l1 and l2 are the horizontal distance between the

center of the camera and laser-rail intersection, h is the vertical distance between the 

camera/lasers and the surface of the rail, and d is the distance between the lasers on the rail 

surface. Solving these equations results in:

( ) 1 2

1 2 tan tan

L L h h d = + − +

θ θ Equation 3.1-4

Combining Equation 3.1-1 with Equation 3.1-4, a sensor reading can be calculated for a value of

track relative deflection.

Combining this information with the track model (e.g., Winkler model), the sensor reading d can 

be related to the track modulus u. On softer track, the rail will rise relative to the wheel-rail 

contact point, and the laser lines as observed by the camera will move closer together. 

Conversely, the distance between the lasers will be large for stiffer track.
22

3.1.2 The Relation between Yrel and Modulus (Winkler Model)

Figure 3-4 shows the rail deflection from multiple loaded axles. 

Figure 3-4. Superposition of the Deflections from Two Loads

In Figure 3-4, assuming the loads of wheel 1 (the left wheel) and wheel 2 (the right wheel) are 

the same (P), w1 is the deflection of the rail attributed to wheel 1, and w2 is the deflection of the 

rail attributed to wheel 2. The total rail deflection is the superposition of w1 and w2.

From the Winkler model,

( ) e [ ( x c ) ( x c )]

u c

P

w x x c = − − + − − ⋅ −

1 1

1

1 cos sin

2 ( )

1 β β β β Equation 3.1-5

where: 4

1

4

( )

1 



 



 = EI

u c β

when: E is the modulus of elasticity of the rail

I is the moment of inertia of the rail

x is the longitudinal distance along the rail

c is the position of wheel one in the x coordinate (see Figure 3-4)
23

and ( ) e [ ( x c b ) ( x c b )]

u c b

P

w x x c b − − + − − + = − − ⋅ − −

2 2

2

2 cos sin

2 ( )

2 β β β β

Equation 3.1-6

where: 4

1

4

( )

2 



 



 + = EI

u c b β

when: b is the distance between the two wheel axles (72 in)

(c+b) indicates the position of wheel two in the x coordinates

The total deflection of the rail is the superposition of the two expressions which is:

( ) ( ) ( ) 1 2 w x w x w x total = +

Then, the total deflection at the wheel/rail contact point of wheel 1 is:

( ) e [ ( b) ( b)]

u c b

P

u c

P

w x w c w c

b

total x c 2 2

1 2

1 2 cos sin

2 ( ) 2 ( ) ( ) ( ) 2 β β β β β +

+ = + = − − − ⋅

= Equation 3.1-7

And the deflection of the rail under the sensor head which is 4 ft away from wheel 1 is:

( )

[ ( ) ( )] [cos( ( )) sin( ( ))] 2 ( )

cos sin

2 ( )

( ) ( )

2 2

2 ( )

2 2

1

1 2

2 2 e a b a b

u c b

P

e a a

u c

P

w x w c a w c a

a a b

camera x c a

+ + +

+ = − + −

= − + −

− ⋅ − ⋅ +

= −

β β β β β β β β

Then, Yrel w (c) w (c a) = total − camera − Equation 3.1-8

Assuming the track is absolutely uniform (i.e., u is a constant), then:

4

1

4 1 2 



 



 = = = EI

u β β β

Therefore, 

[ ( ) ( )]

[cos( ) sin( )] [cos( ( )) sin( ( ))]}

{1 cos sin

2

( ) ( )

( ) e a a e a b a b

e b b

u

P

Yrel w c w c a

a a b

b

total camera

− + − + + +

= − + +

= − −

− ⋅ − ⋅ +

− ⋅

β β β β

β β β

β β

β Equation 3.1-9

The result of Equation 3.1-9 is shown in Figure 3-5. In this model, 132 lb rail was chosen (I = 

87.9 in4

); E is set to be 30,000,000 psi; the load on each wheel is 32,500 lb; and the distance 
24

between the two axles is 6 ft. The typical values of modulus for various main-line track 

conditions (Kerr, 2003) are listed in Table 3-1 along with the corresponding Yrel value.

Figure 3-5. Relation between Yrel and Modulus (Winkler model)

Table 3-1. Modulus and Yrel for Typical Track Conditions

Track Condition Description Modulus (psi) Yrel (in)

Wood-tie track, after tamping 1,000 0.2

Wood-tie track, compacted by traffic 3,000 0.095

Concrete-tie track, compacted by traffic 6,000 0.058

Wood-tie track, frozen ballast, and subgrade 9,000 0.044

The relation between the rail deflections at the wheel/rail contact point and relative deflection 

(Yrel) is shown in Figure 3-6. This nonlinear relation is based on the Winkler model and 

superposition. For relatively small deflections (0~0.2 in), Yrel is approximately 60 percent of 

the total deflection (deflection at the wheel/rail contact point). 
25

Figure 3-6. Relation between the Total Deflection and Yrel (Winkler model)

3.2 Relationship to Existing Profile Variations

The measurement of relative deflection (Yrel) uses the wheel/rail contact line as a reference as 

shown in Figure 3-1. The measurement assumes the unloaded rail is perfectly straight. 

However, if the rail has a significant pre-existing geometry variation over a length comparable to 

the 4 ft between the measurement point and wheel/rail contact point, the system’s measurement 

will be affected. Large vertical “dips” that occur over a short length of track affect the 

measurement result. 

The relationship between modulus and geometry is complex. In real track, areas of geometry

variations often correlate with areas of modulus variations and vice versa. A case study was 

chosen to investigate this relationship. Figure 3-7 shows a section of track where there is a 

significant geometry variation and a significant modulus variation. Measurements at the site 

indicated that the unloaded rail drops by 0.5 in over a length of approximately 16 ft. A geometry 

variation of this shape is significant and easily visible. The light-colored ballast seen at this site 

also suggests tie “pumping” and low track stiffness.
26

Figure 3-7. An Example Site with Both Significant Unloaded Geometry and Low Track 

Stiffness

Relative rail deflection (Yrel) from the measurement system at this site is 1.1 in. Simulations, 

based on the Winkler model, have been conducted to quantify the effects of track geometry on 

the measurement of relative deflection.

Figure 3-8 shows an example simulation result. In this simulation, a section of track has both 

geometry and modulus variations. The unloaded track geometry is described in the top subplot 

in Figure 3-8. It has a maximum “dip” of 0.5 inch in depth, and it occurs over 200 in (between 

100 and 300 in) of track.

In the simulation, it was assumed that the modulus over this section of track varies as a cubic 

curve with a minimum at the center of the geometry variation (the middle subplot in Figure 3-8). 

The bottom subplot in Figure 3-8 shows the Yrel measurement for this site. Here, the “total” 

measurement replicates the value of 1.1 in, as it did in the real measurement when the 

measurement system passed over the location shown in Figure 3-7. To create this value, it was 

found that the modulus for this location had to drop from 3,000 psi (assumed as a reasonable 

value for “normal” track) to 800 psi in addition to the unloaded geometry profile. This 

measurement is then broken into two “elements”—a modulus element and a geometry element. 

The geometry element is the measurement that would be made if the same unloaded geometry 
27

(top subplot) existed on a perfectly rigid track. The modulus element in the remaining portion is 

the total measurement minus the geometry element.

Figure 3-8. Simulation on the Effects of Track Geometry

It can be seen that in this case the contribution of geometry (the geometry element) is about 

equal to the contribution of modulus (the modulus element). However, both are required to 

make the measurement large.

Now, the simulation can be used to study the relative contribution of geometry and modulus as 

the length of the geometry variation (L) and the depth of the geometry (d) vary. The simulation 

result is shown in Figure 3-9.
28

Figure 3-9. Effects of Unloaded Geometry of Various Length (L) and Depth (d)

It can be seen in Figure 3-9 that there is a complex relationship between modulus and geometry 

and that the effects vary depending on the length (L) and depth (d) of the geometry variation. 

The three-dimensional plot on the left shows the relative size of the geometry element and the 

modulus element. It can be seen that there is a curve where the elements are equal in magnitude.

The two graphs on the right show two cross sections of these surfaces. The top right graph 

shows the effects of variations in the length of the geometry defect (L) at a constant depth (d =

0.5 in). The bottom right graph shows the effects of variations in the depth of the geometry 

defect (d) at a constant length (L = 200 in).

Again, the conclusions that can be drawn from these simulations are that 1) only large vertical 

geometry defects occurring over a short distance significantly contribute to the Yrel 

measurement, and 2) both geometry and modulus problems are generally present where very 

large Yrel values are measured.

3.3 End-Chord Offset

As seen in the simulation and analysis in the previous section, track geometry can, in some cases, 

affect the output of the system in terms of measuring rail deflection. To eliminate the effects of 

track geometry variation and to get the real rail deflection results, track geometry profile data 

from track geometry measurement vehicles was introduced into the system. 

A track geometry vehicle is a rail vehicle used for nondestructive diagnosis of railroad tracks. It 

measures various parameters including position, curvature, and alignment of the track as it 

passes by, as well as smoothness and the cross level of the two rails, etc. The space curve 

channel of the geometry car uses multiple high-precision accelerometers onboard to produce the 

rail profile that includes effects of relatively long wavelength variations. 
29

Figure 3-10. 10-Foot ECO Calculation from Rail Profile

As shown in Figure 3-10, P(x) is the rail profile from the space curve channel of the track 

geometry data. The longitudinal position of the track is defined as x (unit: foot). ECO(x) is the 

10-foot end-chord offset (ECO) when the leading wheel’s longitudinal position is x (ECO is 

positive if the string is above the rail.) Here, the 6- and 4-foot lengths were chosen because they 

are the distance between the two wheel axles and the distance from the sensor head to the 

inboard wheel axle, respectively. 

From the geometry relation in Figure 3-10:

4

6

( ) [ ( 4) ( )]

( 6) ( ) = − + +

− −

P x P x ECO x

P x P x

Equation 3.3-1

Therefore, [ ( ) ( 6)] ( ) ( 4) 3

2 ECO(x) = ⋅ P x − P x − + P x − P x + Equation 3.3-2

3.4 Relative Track Deflection

Figure 3-11 shows how this calculation can be made. In Figure 3-11(a), a schematic 

representation is shown. Here, the unloaded rail is shown along with the loaded profile. For the 

calculation of ECO, it is assumed that the leading and trailing wheel track the same profile. The 

actual shape of the rail is shown as the loaded rail. Finally, the 10-foot chord is shown with the 

graphical definitions of ECO, Yrel, and relative deflection.

Figure 3-11(b) is calculated from actual rail measurements made by the test car. P(x) is the 

vertical position of the inboard wheel/rail contact point when the inboard wheel’s longitudinal 

position is x. P(x-6) is the vertical position of the inboard wheel/rail contact point when the 

inboard wheel’s longitudinal position is x-6. If it is assumed the two wheels always have the 

same space curve, then P(x-6) is the vertical position of the trailing wheel/rail contact point when 

the inboard wheel’s longitudinal position is x. P4(x) is the rail’s vertical position 4 ft ahead of 

the inboard wheel when the inboard wheels longitudinal position is x.
30

a) Deflection Schematic

b) Deflection for Actual Measurements

Figure 3-11. Deflection Calculation
31

Let A(x)=P4(x)-Yrel(x), then from geometry relation in Figure 3-11, we have:

4

6

( ) ( )

( 6) ( ) = −

− −

P x A x

P x P x

So, [5 ( ) 2 ( 6)] 3

1 A(x) = ⋅ P x − P x − Equation 3.4-1

Therefore, [5 ( ) 2 ( 6)] ( ) 3

1 P4(x) = A(x) +Yrel(x) = ⋅ P x − P x − +Yrel x Equation 3.4-2

The vertical position of the rail at location x+4 may be determined when the inboard axle is at 

location x. The vertical rail position at the same location may be determined again when the 

inboard axle is actually at location x+4. Then, the difference between these two measurements 

(P4(x)-P(x+4)) may be calculated by:

“Deflection”

= P4(x)-P(x+4)

[5 ( ) 2 ( 6)] ( ) ( 4) 3

1 = ⋅ P x − P x − +Yrel x − P x +

[ ( ) ( 6)] ( ) ( 4) ( ) 3

2 = ⋅ P x − P x − + P x − P x + +Yrel x

= ECO(x) +Yrel(x)

Equation 3.4-3

It should be noted that the so-called “Deflection” calculated above is not the maximum 

deflection of the rail attributed to the loads. Rather, it is a relative deflection of the rail from 

partially loaded (when the axle is 4 ft away) to fully loaded (when axle is right at the point).

Therefore, Yrel can be accounted for by two parts. One part is ECO, attributed to the track 

geometry variations, and the other part is the deflection related to track modulus variations.
32

4 The UNL Measurement Car

4.1 Early Deployment

During the early years of this project, experiments were conducted on wheel sets and trucks as 

shown in Figure 4-1. These experiments, which were performed without rolling stock, helped to 

better understand the load-deflection relation of the wheel/rail interaction. 

Figure 4-1. Experiments in Early Deployment

A hopper car was later introduced into the project and a frame was designed, built, and installed 

on the side frame of the hopper car as shown in Figure 4-2. A few short-distance tests were 

conducted using this hopper car to help improve the measuring sensors.

Figure 4-2. The Hopper Car Used in Early Deployment

4.2 Caboose

Before the measurement system was upgraded to its autonomous capability, all tests were 

conducted with onboard crews located in a caboose shown in Figure 4-3. The caboose served as 

a platform to accommodate the testing personnel. It also hosted the computers and other 

equipment and provided a power supply to the sensors and other equipment.

12,000 

32,000 

42,000 
33

Figure 4-3. Exterior and Interior of the Caboose

4.3 Tank Car

After further development, a tank car filled with water was used as the rolling stock carrying the 

sensor heads as shown in Figure 4-4. The tank car generated the heavy loads needed to create 

relatively large deflection under the wheels. The tank car was coupled with the caboose during 

the tests. A number of tests were conducted in this configuration. However, because of the 

liquidity of the content in the tank car, the dynamic load of the vehicle was distributed more 

unevenly among the wheels. This kind of imbalance negatively affects the measurement system. 

Therefore, the system was later upgraded to a refurbished hopper car donated from Union Pacific 

Railroad (UPRR) and formerly used by FRA in the development of the Gage Restraint 

Measurement System (FRA GRMS T-6), filled with sand, which is the current testing vehicle in 

use.

Figure 4-4. The Tank Car Used in Early Stage of the Project

4.4 UNLX002 Autonomous Measurement System

4.4.1 Instrumentation

The measurement system is installed on a refurbished hopper car. The hopper car is filled with 

sand such that the total weight of the car and sand is approximately 260,000 lb. As shown in 

Figure 4-5, the system includes two rigid beams attached to the side frames (structural members

that connect the axles of a truck) of the hopper car, two sensor heads holding cameras and lasers 
34

at the ends of the rigid beams, a solar panel array, a global positioning system (GPS) antenna on 

the top of the car and an enclosed box containing the computers, data acquisition (DAQ) boards, 

and a GPS receiver for onboard image processing and data computation.

Figure 4-5. System Instrumentation

All of these components will be described in detail in the following sections.

4.4.1.1 Sensor System

The sensor head is attached to highly rigid steel beams that are fastened to the side frames. The 

beams are bolted to the side frames without modification of the side frames. These beams 

suspend the sensor heads out in front of the wheel/rail contact point and over the top of rail. 

Investigations into rigidity have shown that the beams are sufficiently rigid to be considered 

stationary relative to the side frames (Norman, 2004). Therefore, the measurement will not be 

affected by the bending or movement of the beams. An actual view of the assembly is shown in 

Figure 4-6.
35

Figure 4-6. The Rigid Beam on the Side Frame

A sensor head holds a digital vision system and two line lasers as shown previously in Figure 

3-2. The camera and line lasers are rigidly attached to an enclosed sensor assembly, which is 

mounted at the end of the rigid beam on the side frame. These lasers are projected at an acute 

angle (~40 ̊). They cross and create curves (because of the curved profile of the rail head) across 

the surface of the rail. 

The assembly of the sensor head is shown in Figure 4-7. The two laser holders are adjustable to 

ensure that the lasers project at an appropriate angle. The height of the camera is also adjustable. 

This ensures that well-proportioned images will be captured. 

Figure 4-7. Sensor Head Assembly

Ambient light, specifically sunlight, ruins the image of the laser lines across the rail and makes it 

impossible for the image processing program to distinguish the laser lines. Therefore, a shroud 

assembly was made to shield the sunlight. The shroud is a light steel frame bolted to the bottom 

of the rigid beam. Rubber landscaping material is attached along the bottom edge to add further 
36

shading. This material is flexible, making it resistant to rocks and debris, but is also

continuously solid making it resistant to wind effects. 

A typical sensor image is shown in Figure 4-8. The video image shows the laser beams

intersecting the top of the rail. Images are captured and processed in real time to output the 

distance between the laser lines (d in Figure 3-2).

Figure 4-8. Typical Test Image

This measurement is geometrically related to the height of the sensor above the rail (h in Figure 

3-1). As the sensor moves closer or farther from the rail surface, the distance between the laser 

lines changes. With the calibration technique, Yrel can be calculated as the systems output. Yrel

is the relative displacement between the rail surface under the camera and the wheel/rail contact 

line. Yrel can then be mathematically related to track stiffness and modulus (Norman, 2004; 

McVey, 2006; Lu et al., 2007).

The real-time image processing and data management are performed by onboard computers. 

These computers are installed in enclosed boxes as shown in Figure 4-9. A GPS receiver and 

two data acquisition boards are also installed in these boxes. All of the cables and wiring for the 

equipment are routed through flexible conduits. The sealed boxes protect the computers and 

other electrical equipment from harsh environments. 
37

Figure 4-9. Enclosed Box for Computers

4.4.1.2 GPS and Encoder

The measurement system includes a GPS receiver. The latitude and longitude are reported in 

real time and recorded in the output data. The GPS data is used to get milepost information by 

comparing it with Precision Measurement Vehicle (PMV) data and provides accurate coordinates 

for exception locations. 

The GPS receiver used in the system is a NovAtel OEM4 model, which is capable of absolute 

single-point positioning accuracies of 1.8 meters of Circular Error Probable (CEP) (Geometric 

Dilution of Precision (GDOP) < 2; no multipath). However, this level of accuracy is only an 

estimation and may vary widely depending on numerous GPS system biases, environmental 

conditions, as well as the GPS receiver design and engineering quality.

Because of the GPS error, there may be some stretches in the data in terms of mileposts. An 

encoder has been introduced into the system to eliminate this stretching problem caused by the 

GPS error.

4.4.1.3 Power Supply and Management System

Two rugged computers are used to process images in real time and save the data. To reduce 

power consumption when the testing vehicle is not moving, the whole measurement system 

enters a “sleeping” mode in which the lasers and cameras are turned off and the computers 

remain in standby mode. The total power consumption is approximately 50 watts when the 

system is in full-on testing mode and 10 watts in sleeping mode.

An onboard power supply and management system was developed to make autonomous testing 

possible. As shown in Figure 4-10, four solar panels installed on the top of the testing vehicle 

provide the power source for the measurement system. The solar panels are rated at a maximum 
38

of 400 watts. A battery pool consisting of eight deep-cycle marine batteries is used as energy 

storage and as a buffer to provide stable and consistent power to the system during both day and 

night. The eight batteries have a total capacity of 400 Amp-hours, which can supply 4 days of 

continuous testing or 20 days of “sleep” mode without inputs from the solar panels. The 

batteries are enclosed in the black box as shown in Figure 4-10 along with a solar panel voltage 

regulator, which manages the battery recharge process and prevents the batteries from being 

overcharged. Inside the battery box, two watt meters were also installed to provide information 

about the measurement system’s energy usage and energy input from the solar panels.

Figure 4-10. Power Supply System

The power supply system has proven to be sufficient and reliable during autonomous tests in 

several tests over many years under various conditions. In these tests, the batteries were returned 

fully charged after the 6-day, 1300-mile-long journey. 

4.4.1.4 Remote Supervision and Data Downloading

The voltage across the batteries, the current input from the solar panels, and the ambient 

temperature are monitored by a computer in the white boxes shown in Figure 4-10. This 

information can be recorded and sent back to a server on the Internet through wireless 

communication, which enables the power supply system to be remotely monitored.

The data logged and uploaded in real time from the computer onboard to a remote server through 

the Internet during the test in April 2008 is displayed in Figure 4-11. As shown by the current 

and voltage traces in this figure, the batteries were charged during the daytime. The output of 

the solar panels was measured as high as 150 watts. The highest output from the solar panels 
39

usually occurred in the morning to compensate for the power consumption during the nights. 

While in sleeping mode, the batteries can be fully charged before noon and the voltage of the 

batteries will remain constant for the rest of the day. 

Figure 4-11. Power Supply System Monitoring Information in the April 2008 Test

4.4.1.5 Testing Results Database

The testing vehicle has been tested on thousands of miles of track. As a result, huge amounts of 

data have been produced from these tests. Therefore, a database was set up to organize and 

manipulate this data. A Web site was created to search through the database and to develop 

plots. A screenshot from this Web site is shown in Figure 4-12. The Web site can also produce 

lists of exception locations from the database and plot them on maps as shown in Figure 4-13. 
40

Figure 4-12. Database Web Site Screenshot 
41

Figure 4-13. Exception Locations List from the Web Site

4.4.2 Real-Time Image Processing

In the typical image captured during the test shown in Figure 4-8, the laser lines are easily 

identifiable on the top of the rail. The image processing program scans through all the pixels on 

each horizontal line of the image and locates the peaks of the pixel intensities, which represent

the locations of the laser lines. Subsequently, the laser lines can be reproduced as shown in 

Figure 4-14. After scanning through the image and obtaining the valid points on each laser line, 

some points may be considered as valid but out of the curve because of the reflection on the rail 

surface or some distractions captured in the images as shown in Figure 4-15. These outliers are 

filtered out, and cubic curves are applied to fit the remaining valid points. 

Figure 4-14. Reproduced Laser Curves
42

Figure 4-15. An Imperfect Image Example

4.4.3 Calibration Approach and Procedure

As shown previously in Figure 3-1, Yrel is the relative displacement between the rail surface 

under the sensor and the wheel/rail contact line. Yrel is the measurement systems output.

The system processes images in real time and obtains the number of pixels between the two laser 

lines in the images. This number of pixels is the system’s direct measurement. To convert this 

number of pixels into the value of Yrel in inches, a calibration must be conducted.

The purpose of the calibration procedure is to obtain the relation between the system’s direct 

measurement (n, the number of pixels between the laser lines in images) and the expected output 

(Yrel). 

Figure 4-16 shows how to convert the number of pixels into the actual distance in inches. An 

image of a ruler on top of the rail was captured. The 6-inch ruler corresponds to 208 pixels in 

the captured image. Therefore, 1 inch in the image corresponds to 208/6 = 34.67 pixels (i.e., if 

the number of pixels between the two lines is n, the actual distance d is n/34.67 in).
43

Figure 4-16. Converting Number of Pixels into Distance in Inches

Now the problem consists of finding the relation between d and Yrel. To do this, imagine a 

situation where a single measurement of Yrel (that will be called Yrel*) can be taken that 

corresponds to a known value of d (that will be called d*). Here, the “*” indicates one specific 

incidence of Yrel and d. Therefore, if at one moment, Yrel = Yrel* and d = d* are known, and 

the ratio of ΔYrel and Δd ( d

Yrel R

∆

∆ = ) is also known then, the relationship between ΔYrel and Δd 

may be determined as follows:

First, ΔYrel and Δd may be represented as:

ΔYrel = Yrel - Yrel*

Δd = d - d*

Then it follows that, 

Yrel = Yrel *+∆Yrel = Yrel *+R⋅∆d = Yrel *+R⋅(d − d*) Equation 4.4-1

Therefore, the calibration problem consisted of finding Yrel*, d*, and R, which involves the 

following steps.

4.4.3.1 Finding the Ratio of Δd and ΔYrel (R)

A special plate was fabricated for this calibration procedure. The device consists of seven steps 

as shown in Figure 4-17 and Figure 4-18. The lowest step is one-fourth-inch high and the other 

six steps are one-eighth-inch high. When doing the calibration, this plate is placed on top of the 

rail under the camera, and the two laser lines project onto the plate. Therefore, an image similar 

to Figure 4-19 is captured by the camera. 
44

Figure 4-17. Calibration Plate on the Top of the Rail

Figure 4-18. Calibration Plate on Top of the Rail (side view)

Figure 4-19. Captured Image of the Calibration Plate

The image in Figure 4-19 is then processed. The distances between the laser lines on each step 

of the plate were obtained and plotted in Figure 4-20 with respect to the step’s height above the 
45

rail. As shown in Figure 4-20, a linear line fits the data points very well (R2 = 0.9988). From the 

line fit, one pixel in the image represents 0.01178

84.857

1 = in of ΔYrel.

Figure 4-20. Calibration Results

This result verifies the calibration results very well. On the basis of the geometric relation, one 

pixel in the image was calculated to represent 0.01176 in of Yrel. The difference between the 

results from the two different methods is (0.01178 – 0.01176)/0.01176 = 0.17%, which is 

negligible.

This difference may be attributed to many factors including the limited resolution of the captured 

images, the wide laser lines in the images, measurement error during the calibrations, etc. Some 

of these factors will be discussed in detail in Section 4.4.4.

4.4.3.2 Finding Yrel* and d*

Again, the goal of calibration is to find a single instance where Yrel is known, and the 

corresponding value of d is known. We are calling these known values Yrel* and d*. This 

section describes how these known values can be obtained through measurement.

After calculating the ratio of Δd and ΔYrel, all that is needed is to find a reference point where 

both the relative deflection (Yrel) and the distance between the laser lines (d) may be 

determined.

Assuming that a section of track is perfectly uniform and the modulus over the section is a 

constant, the deflection of the rail under the axles will be identical over the entire section. If a 

train moves over this section at a very slow speed, the rail deflection at one fixed point as a 
46

function of time can be mapped from the time domain into the space domain so that the static rail 

profile under the axles can be obtained. The dynamic load factor can be ignored because the 

train speed is slow. 

Figure 4-21 demonstrates the setup for capturing the rail deflection with a video camera. A 

sticker is placed on the side of the rail head as a marker, while a video camera away from the 

track captures this sticker in its view. Another marker is placed on the ballast 4 ft away from the 

sticker. 

Figure 4-21. Capturing the Rail Deflection with a Video Camera

The measurement vehicle is pulled at a constant speed and passes by the video camera. 

Meanwhile, the video camera captures and saves the video for postprocessing. An image from 

the captured video is demonstrated in Figure 4-22. As the wheels of the measurement vehicle 

move closer to the position of the marker, the deflection of the rail increases and the marker in 

the camera image goes down. When the wheels move away from the marker, the marker goes up 

in the video.
47

Figure 4-22. Captured Video Showing the Rail Deflection

An image processing program was developed to process the video and determine the vertical 

position of the top edge of the marker as shown in Figure 4-22. Assuming the deflection of the 

rail is zero when the locomotive and measurement vehicle are far away from the marker, the 

vertical position of the marker in the video can then be converted into the actual deflection of the 

rail, which is plotted in Figure 4-23. Therefore, Yrel*, the vertical height difference between the 

rail surface under the camera and the wheel/rail contact plane, is shown in Figure 4-23. From 

this calibration data, Yrel* is 0.045 in. 
48

Figure 4-23. The Deflection Curve of the Rail from Calibration

During the calibration procedure, the measurement camera in the sensor head also captures the 

video of the laser lines on the top of the rail. This video is postprocessed to capture the image of 

the marker placed on the ballast as shown in Figure 4-24. This marker is simply used to indicate 

at what point in the video capture the sensor head is directly over the measurement point. From 

this image, the distance between the two laser lines (d*) is obtained.
49

Figure 4-24. Captured Image when Sensor Head Passing by the Marker

This calibration procedure assumes that the section of track (approximately 40 ft around the 

sticker’s position) is uniform. However, track conditions may change from tie to tie. Even 

though this procedure was done on a section of specially selected quality track, there is no

guarantee that modulus over the section is a constant. Therefore, the accuracy of the results is 

worthy of further examination.

4.4.3.3 Verifying Calibration Results

After obtaining the ratio of Δd and ΔYrel, Yrel*, and d*, the distance between the two laser lines 

in the images can be converted into the measurement results, Yrel, by applying Equation 4.4-1. 

However, as stated above, it is difficult to accurately determine Yrel* and d*. Therefore, another 

procedure was developed to verify Yrel*, d*, and the entire calibration results. 

A mechanical shop for railroad vehicles is involved in this procedure. The track in this shop is 

assumed to be absolutely stiff such that no rail deflection will be considered when trains move 

across this section of track. This assumption is reasonable because the track in the shop is in 

good condition, and the rail is supported by a structural concrete foundation that limits track 

deflection. 

52,000 
50

Figure 4-25. Yrel Data from the Mechanical Shop

The results of the Yrel data for this section of track are plotted in Figure 4-25. The average Yrel 

value is shown (horizontal red line) for the readings in the shop between entering the shop and 

exiting the shop. The average of Yrel over this section of track is 0.0018 in. This value is 

extremely close to the expected value of zero. The variations of Yrel, which are relatively small 

(standard deviation is less than 0.1 in), are attributed to the pre-existing geometry variations of 

the track.

4.4.4 Error Analysis

4.4.4.1 Errors Caused by Limited Sampling Rate

When the system was first implemented, the cameras used analog video signals in a National 

Television System Committee standard, which is 30 frames per second. As a result, individual 

data points were spaced approximately every 2 ft when the measurement vehicle travels at 

40 mph. However, Yrel measurements may change considerably within 2 ft because certain 

track modulus and geometry variations occur over extremely short distances. Therefore, this 

limited sampling rate led to measurement errors.

For example, 50 ft of data using this older system is presented in 1-foot increments in Figure 

4-26. The highest peak within the 50-foot range is point B, which is 0.77 in. If the test is 

conducted at a speed of 40 mph, the system used to record readings every 2 ft. As a result, the 

peak at point B could be missed, while only the data points at A and C would be recorded. 

Hence, an error of 0.04 in (the difference between points A and B) would result. Because the 

errors, based on limited sampling rate, depend on how the actual data varies over short distances, 

they were difficult to quantify. Some locations, such as joints, are much more sensitive to this 

sampling rate issue because the actual Yrel data can change dramatically over a few ties. On the 

basis of an examination of the data from past tests, the error could be as much as 0.2 in at some 

locations. 
51

Figure 4-26. Limited Sampling Rate Causing Measurement Errors

To address the problem caused by the limited sampling rate, new higher speed cameras have 

been implemented to upgrade the system’s sampling rate to a maximum of 120 frames per 

second. The current prototype system uses a sampling rate of 90 frames per second that leads to 

a measurement at approximately every 8 in at a testing speed of 60 mph. This greatly improved 

the system’s measurement repeatability and accuracy and minimized all the potential errors 

discussed in this section. 

4.4.4.2 Measurement Resolution

The measurement system is based on image processing. The most direct measurement output is 

the number of pixels between the two laser lines in the images. Therefore, the measurement 

resolution is directly determined by the resolution of the captured images. 

The current system captures images at 90 frames per second with a resolution of 782 × 582 

pixels. On the basis of the calibration results, for images of 782 × 582 pixels, one pixel in each 

image represents 0.00482 in of Yrel. Hence, the measurement system’s resolution is less than 

0.005 in. 

4.4.4.3 Laser Line Width

The image processing program identifies locations of the laser lines by finding peaks of pixel 

intensities. However, as shown in Figure 4-27, a laser line can be as wide as three or four pixels 

in a captured image. Therefore, it is difficult for the image processing program to precisely 
52

distinguish which pixel should represent the location of the laser lines, especially when two 

adjacent pixels have the same intensity. Although the curve fitting algorithm used in the image 

processing program greatly reduces the error caused by this factor, it is reasonable to conclude 

that an error of ±1 pixel (±0.0118 in Yrel for this camera system) may result. 

Figure 4-27. Laser Line Width

4.4.4.4 Laser Beam Drifting

The laser line generator’s Boresight accuracy is rated at a maximum of 2.5 millimeters per meter. 

Boresight accuracy, known as pointing accuracy, is a measure of the angular difference between 

the beam propagating axis (where the laser beam is pointing) and the mechanical axis (where the 

laser housing is pointing). Because of the nature of the measurement principle and the 

calibration procedure, the Boresight accuracy has no effect on the measurement result because 

the measurement is not related to the mechanical axis of the laser housing. The calibration 

described in Section 3.4.3 shows that the absolute angle of the laser, defined by α in Figure 4-28, 

is not critical because the value of the angle is found, and confirmed, through the calibration 

process (from Section 3.4.3). As a result, no measurement error is caused by the Boresight 

inaccuracy; however, changes in this angle, defined as ∆α or pointing stability, can lead to 

errors. 

Pointing stability, a measure of how much the laser beam alignment drifts over a period of time, 

can lead to minor measurement errors. The laser line generator’s pointing stability is rated at 

<50 μrad. The geometry relation between the lasers and the rail is shown in Figure 4-28, where 

H is the vertical height of the laser generator above the rail, α is the angle between the laser beam 

and the vertical direction, Δα is the drifted angle, and Δd is the drifted distance of the laser line 

on the rail. 

From geometry relation, we know:

 ∆d = H ⋅ tan(α) − H ⋅ tan(α − ∆α) Equation 4.4-2

From the calibration, we know H = 13.6; α = 50°; and the maximum Δα is 50 μrad. Therefore, 

Δd is calculated to be 0.0016 in. On the basis of the geometric relation from the calibration, 

0.00165 inch in Δd will result in a ΔYrel of 0.00165 × 0.853 = 0.0014 in. 
53

Considering the worst-case scenario in which both laser beams are drifting either toward or away 

from the camera at the same time, the maximum error will be doubled. Therefore, the maximum 

error of Yrel that the laser beams drifting can cause is 0.0028 in, which is negligible. 

Figure 4-28. Laser Beam Drifting
54

5 Validation of the UNL System

A validation test was conducted in October 2008 on the UPRR’s Yoder Subdivision between 

Egbert, WY, and Yoder, WY. The purpose of this validation test was to confirm the 

measurements collected by the system. 

5.1 Measurement of Deflection

To validate the deflection measurements of the UNL’s system, three verification methods were 

performed. The results from the three methods show that the measurement is accurate and 

repeatable as outlined in the following sections.

5.1.1 Stringline Measurements

As described previously in Section 3.1, Yrel is the distance from the rail surface under the 

camera to the wheel/rail contact plane. Therefore, the method depicted in Figure 5-1 to measure 

Yrel is very straightforward. Here, a string is pulled to pass the bottoms of the two wheels. 

Then, the distance from the top surface of the rail under the camera to the string is the Yrel

reading at this location.

Figure 5-1. String Measurement Diagram

As shown in Figure 5-2, a specially made magnetic ruler is attached to the rail during the field 

measurement to make the reading process easier. The string is tightly held from the two ends as 

it barely touches the bottom of the two wheels. The distance from the top surface of the rail 

under the camera to the string is read to compare against the Yrel reading from the measurement 

system.
55

Figure 5-2. Field String Measurement

The string measurement was performed at three locations during the validation test. The 

measurement results and the Yrel measurements from the testing vehicle are listed in Table 5-1. 

The field string measurements and the Yrel measurements each differ by less than 0.1 in.

Table 5-1. String Measurement and Yrel Measurement

Locations Yrel (in) String Measurement (in) Difference (in)

MP 192.84 1.000 1.004 0.004

MP 196.94 0.703 0.610 0.093

MP 217.22 0.703 0.669 0.004

The string measurements from these three locations closely matched the Yrel measurements 

from the testing vehicle, which suggests that the system’s Yrel measurements are correct. 

However, this type of stringline measurement is not very accurate. A significant practical 

limitation is ensuring that the string perfectly passes the bottom of each of the two wheels—a 

rather difficult task. This limitation can easily contribute a relatively large error in the 

measurement results. To provide more accurate results, further methods were used, and they are 

described in the following sections. 

5.1.2 Survey Measurements

This method uses a surveyor’s total station as shown in Figure 5-3 to independently measure the 

vertical deflection of the rail. Such instruments are commonly available with an indicated 

accuracy of less than one millimeter. In this scenario, rulers used as targets are attached to the 

side of the railhead by strong magnets as shown in Figure 5-3.
56

Figure 5-3. Instruments Used in Survey Measurements

Figure 5-4 demonstrates the measurement scenario. Thirteen rulers are placed every 3 ft on the 

side of the rail so that the total measurement range is 36 ft. This distance is usually enough to 

cover the entire deflection basin. First, the unloaded profile of the rail is measured by using the 

total station and a glass Porro prism commonly used in surveying. Next, the total station is used 

to measure the height of each ruler on the side of the rail while the rail is unloaded. After the 

measurements are recorded, the locomotive parks the measurement vehicle on this section of the 

track such that the inboard axle is directly on top of the center ruler as shown in Figure 5-4. The 

total station is then used to take the height measurement of each ruler a second time. The 

difference between the two height measurements for the same ruler is the rail deflection at that 

ruler’s location. Adding the deflection measurement to the unloaded rail profile, the loaded rail 

profile can then be determined. Analyzing the resulting measurements provides for confirmation 

of the measured Yrel reading.

Figure 5-4. Measurement of Vertical Rail Position by Surveying

Survey measurements were conducted at two locations during the validation test. Measurement 

results from one of these locations are presented in Figure 5-5. The curve demonstrates the 

loaded profile of the rail with one wheel positioned at the origin and the other at a position of 

70 in. The profile points at these two locations are connected with a line that is then extended 

by –48 inches in the horizontal direction. The distance from the end of the line to the rail at 
57

position –48 in is the Yrel measurement at this location. As shown in the figure below, Yrel 

measures 0.732 in at this location.

Figure 5-5. Survey Measurements

The testing vehicle’s measurements are compared with the survey measurements from each site 

as shown in 

Table 5-2, the two methods produced almost identical results at both sites. This suggests that the 

testing vehicle’s measurements are accurate and reliable.

Table 5-2. Comparison between System Measurement and Survey Measurement

Measurement 

from testing 

vehicle (in)

Measurement 

from surveying 

(in)

Difference 

(in)

Site A 0.738 0.732 0.006

Site B 0.150 0.150 0.000

5.1.3 Absolute Deflection Measurements by Cameras

Similar to the calibration procedure described in Section 4.3.3, a common camcorder is used to

observe the vertical displacement of the rail as shown in Figure 5-6. A white magnetic strip is
58

affixed to the rail to use as a target. The camera records the video of the white strip as the 

moving train passes the location. The video is then postprocessed to determine the vertical 

location of the strip over time. This process is much simpler than using LVDT’s as described in 

Section 2.3 and will provide complete information on the passing train. A section of sampled 

deflection data processed from a video of a coal train is presented in Figure 5-7. In this section 

of data, a series of coal hoppers causes the rail to deflect about one-half inch and three heavier 

locomotives at the end of the train to deflect the rail over 0.9 in.

Figure 5-6. Wayside Camera Measurements Setup
59

Figure 5-7. Sample Data of Absolute Deflection from Wayside Cameras

During the validation test, this type of absolute deflection measurement was conducted at three 

locations where survey measurements were also performed. The raw data of deflection over 

time from one of the locations is plotted in Figure 5-8. Here, the deflection caused by the 

locomotive, the test vehicle (UNLX002) and the geometry car are identified, respectively. The 

maximum deflection of the rail caused by the test vehicle load occurred at point A, which is the 

moment when the axles pass by the position of the magnetic strip. The maximum absolute 

deflection was determined to be 1.1 in for this location. Similar plots were made for the other 

two locations and the absolute deflections were evaluated.

Trailing Locomotives

Coal 

Hoppers
60

Figure 5-8. Deflection Data from Camera Measurement

Table 5-3 contains all of the deflection data from the surveying measurements and the camera 

video at the three locations. The three deflection measurements from the camera are very close 

to the corresponding ones from the surveying technique. The differences are less than 0.1 in at 

each location, so the two types of measurements support one another.

Table 5-3. Comparison between Measurements from Surveying and Camera

Deflection from 

Surveying (in)

Deflection from 

Camera (in)

Difference 

(in)

Location #1 0.35 0.34 0.01

Location #2 1.14 1.1 0.04

Location #3 0.7 0.8 0.1

5.1.4 Summary

The string measurements and the survey measurements confirm the systems Yrel measurements 

accurately. The maximum error from the three locations in string measurements is less than 0.1 

in and the survey measurements from the two sites were almost identical to the systems Yrel 

measurements. Results from these methods confirm that the system’s measurement is accurate 

and reliable. 

By comparing the survey measurement to the measurements by the wayside camera and then 

comparing both to the Yrel measurement, demonstrated accuracy is less than 0.1 in at each of 

these locations.
61

5.2 Measurement of Strain

Stress in rail can result from a complex combination of loads. Longitudinal loads can result from 

temperature-based expansion of the rails, while shear loads can result from tie supports and 

wheel loads. Rail cant and gage widening can also contribute to unusual loading conditions. 

However, axial stresses, and particularly cyclic axial stresses, often dominate rail stress and lead 

to crack growth and fatigue failure. Major contributions to the axial stress come from 1) 

temperature-based longitudinal rail stress, 2) bending stress associated with the wheel loads, and 

3) contact stresses associated with the wheel-rail contact patch, as indicated in Figure 5-9.

Figure 5-9. Rail Stresses and Fatigue

Each of these three factors can be significant and difficult to measure. Of the three, rail contact 

stress can be the most predictable because the wheel loads are generally a function of the applied 

loads and material properties of the rail. However, both temperature-based longitudinal stress 

and bending stress are difficult to determine. Both can dominate the stress field in the rail and 

both can independently lead to rail failure. A new method is presented to estimate one of these 

two important stresses, the rail bending stress. The measurement system is explained, and the 

mathematical background is presented that allows the measurement to be mapped into rail stress. 

Field results are presented from two sites on the Union Pacific’s Yoder subdivision. These 

results show that the measured bending stress match the predicted stress to within approximately 

12–14 percent.

5.2.1 Mapping Yrel into Bending Strain

The UNL track deflection system measures three points on the rail that partially describes the 

deflected shape of the rail under load. Because the deflected shape is mainly determined by the 

applied load, a method was devised for estimating the rail strain induced from the applied load, 

which could be used to estimate rail stress. Several models can be used, but the classical 

approach is the Winkler model (Cai et al., 1994; Ebersohn and Selig, 1994). The Winker model 

is briefly presented here, and it is shown how it can be mapped into rail stress given a value of 
62

Yrel. The model is then confirmed in the following section using field testing and strain gages. 

Future work may use the cubic model described in Section 2.1.1.3.

When multiple loads are present, the rail deflections caused by each of the loads are 

superimposed (assuming small vertical deflections) to produce the rail shape under the wheel 

loads (Lu et al., 2007).

Figure 5-10 shows the rail shape for two axle loads (and other axles of a trailing car to the right 

not shown in the figure) of the UNL test car. The individual deflections from the Winkler model 

for each axle (axles 3 and 4) are shown in dashed lines with the solid (blue) line showing the 

loaded rail shape. A line is then drawn to create the wheel-rail chord and a value of Yrel can be 

calculated (0.099 inch in the figure). Given that the wheel loads are 70 in apart and the Yrel 

measurement is 48 in from axle 3, Yrel is calculated as:

( 70) (48) 70

48 (0) 70

118 Yrel = y − y − − y Equation 4.2-1

Figure 5-10. Winker Shape of Rail with Yrel Value

The bending moment, M(x), which results from the deflection estimated by the Winkler model, is 

shown below.

M(x) = P

4β

e−β |x|[cos(β | x |−sinβ | x |)] Equation 4.2-2

The bending moment, M, is calculated for x = 0, which corresponds to the location of axle 3. 

Knowledge of the bending moment allows for the calculation of the bending strain present in the 

rail due to the application of the axle loads. The strain resulting from the applied bending 

moment is: 

Locomotive

Axle 3

UNLX

Axle 4

UNLX
63

ε = Mz

EI Equation 4.2-3

where ε is the rail bending strain, E is the modulus of elasticity for the rail material, I is the 

moment of inertia, and z is the vertical distance from the neutral axis of the rail to the strain 

location.

The preceding calculations used the Winkler model to calculate theoretical Yrel and bending 

strain values for a known set of loading conditions. For a particular rail profile and set of axle 

loads, the only variation in the calculations is the track modulus. Varying the track modulus and 

calculating the corresponding Yrel and bending strain values allows for the creation of a lookup 

table, which is displayed graphically in Figure 5-11.

Figure 5-11. Relationship between Yrel and Rail Strain

5.2.2 Field Strain Measurement

Field tests were conducted on the UPRR’s Yoder subdivision to verify the theoretical mapping of 

Yrel measurements into bending strain values. The tests consisted of mounting strain gages on 

top of the rail’s bottom flange and comparing the strain gage value to a value calculated from 

Yrel with the UNL measurement system. The strain gages were mounted on the unloaded rail, 

and all measurements of strain were relative to the pre-existing strain state. For example, after 

being mounted, the strain gages would indicate zero strain, even though there is undoubtedly 

strain present in the rail (i.e., rail is not at neutral temperature, rail is not perfectly straight, and 

rail has residual stress from manufacturing). All measurements from the strain gages indicate a 

change in strain from this initial state. This removes effects from changes in neutral temperature 

(as measurements were made at approximately the same temperature as the gages were mounted) 

and the effects of residual stresses so the measurements can focus on the change in bending 

strain.

The strain gages were placed symmetrically on each side of the rail in a Wheatstone bridge 

configuration as shown in Figure 5-12, where Ei is the input voltage and Eo is the output voltage. 

This two-gage bridge configuration is designed to cancel any off-axis transverse loading effects 

that might be present.
64

Figure 5-12. Two Gage Wheatstone Bridge Configuration

The output from the bridge was input into a Vishay Model 2150 strain indicator and signal 

conditioner and then recorded with a standard analog-to-digital USB input card. The strain 

output from the strain indicator is expressed as:

( 2 )

2

2 i o

q q o

b F E E

E

k − ∆

∆ = = = ε ε

ε Equation 4.2-4

where:

εb = strain output from strain indicator

εq = strain output for a quarter-bridge Wheatstone bridge

ΔEo = change in output voltage recorded from strain indicator

Ei = excitation voltage input to bridge (10 volts)

F = gage factor (~2.1 for all gages used in tests)

k = bridge factor (2 for the bridge configuration of Figure 5-12)

The measured strain was then correlated to the Yrel measurement made, whereas axle 3 of the 

hopper car was spotted on the strain gages. Figure 5-13 shows an example strain measurement 

as the test car is spotted at a measurement location. Increases in strain are seen as the locomotive 

passes over the strain gage followed by axles 1 and 2 of the test car. As axle 3 is placed directly 

over the strain gage, it can be seen that the strain increases from 0 to 290 με.
65

Figure 5-13. Strain as Test Car is Spotted

5.2.3 Field Profile Measurement

Figure 5-13 shows the unloaded strain (before the train is spotted) to be zero. This is because the 

strain gage was mounted to the unloaded rail. The strain measured by the gages (290 με in 

Figure 5-13) indicates the change in strain relative to the unloaded profile and cancels out effects 

of temperature-based axial strain (since temperature didn’t change significantly as the train was 

spotted) and residual stresses. However, it is important to know the initial bending strain present 

in the unloaded profile since the unloaded profile is not perfectly straight. To accomplish this, a 

measurement of the unloaded (and loaded) profile was made using an independent measurement 

system described previously in Figure 5-4.

5.2.4 Strain Measurement Results

Two sites were chosen to compare the strain calculated from the UNL Yrel measurement with 

strain measured from the strain gage. Both sites were on the Union Pacific’s Yoder subdivision 

in southeastern Wyoming. Measurements were made on October 15, 2008. The track at both 

locations had 132-pound rail and wood ties. This subdivision receives approximately 2 million 

gross tons (MGT) per year in traffic.

The first site tested (MP 231.6) was a tangent section of track with good rail support from a 

visual inspection. The second site (MP 228.6) was near an at-grade road crossing with muddy 

ballast and visually obvious rail pumping. The strain at the first site was expected to be low with 

the strain at the second site expected to be higher.

5.2.4.1 Tangent Track with Good Support (MP 231.6)

Figure 5-14 shows the measurement results from the well-supported tangent track at MP 231.6. 

The Yrel reading at this location was 0.15 in indicating stiff track. Therefore, an absolute 

bending strain of 332 με is predicted from the lookup graph in Figure 5-11. The strain gage 

reading relative to the unloaded profile was 290 με. However, there is clearly some initial 

bending strain as a result of the initial unloaded profile (again, the strain gage was mounted on 
66

the unloaded profile that was not perfectly straight). To match the Yrel indicated strain with the 

strain gage measurement, the unloaded bending strain must be removed from the Yrel 

measurement.

To make this correction, Figure 5-14 also shows the results of the survey of both the loaded and 

unloaded profile. Both the loaded and unloaded profiles are displayed with the values of 

displacement shown next to the arrows in inches. These profiles are used to calculate a change 

in Yrel called ∆Yrel. This new ∆Yrel is the loaded Yrel from the measurement car minus a Yrel 

value calculated from the unloaded profile. The strain predicted from the ∆Yrel value and the 

lookup graph in Figure 5-11 can now be compared to the change in strain measured by the strain 

gage.

In this case, the loaded Yrel value is 0.015 in, and the “unloaded Yrel” is 0.001 in. Therefore, 

the ∆Yrel is 0.149 in, and the change in bending strain from loaded to unloaded is 330 με. This 

calculation differs from the strain gage measurement of 290 μ-strain by only 13.7 percent. In 

this location, the unloaded Yrel is not significant (only 0.001 in), and the loaded Yrel is nearly 

identical to the ∆Yrel.

Given all this, it is believed that the original value of loaded Yrel from the measurement test car 

is a realistic indicator of bending strain. However, the ∆Yrel value is used to compare to the 

strain gage readings and validate the measurement only because the strain gages are not mounted 

when the rail is strain free. This becomes more important at the following site where there is 

significant bending strain in the unloaded rail (significant unloaded geometry).

5.2.4.2 Pumping Track (MP 228.6)

Figure 5-15 shows the measurement results from the poorly supported track with muddy ballast 

and visually obvious track pumping at MP 228.6. The Yrel reading at this location was 0.73 in,

indicating softer track. This would predict an absolute bending strain of 868 με. The strain gage 

at this location measured a 580-με change in strain between the unloaded and the loaded state.

In this location, the unloaded profile has a significant dip. A Yrel reading calculated from the 

unloaded profile indicates a value of 0.254 in. Repeating the method described earlier, the 

calculated unloaded Yrel value is subtracted from the measured Yrel value and the result is used 

to determine a strain value in the look-up table. The ∆Yrel in this case is 0.478 in predicting a 

strain of 651 με.

The calculated strain of 651 με differs by 12.2 percent from the measured strain of 580 με. 

Again, it is believed that the absolute bending strain is best represented by the original Yrel 

measurement of 868 με. However, the ∆Yrel value is used to compare to the strain gage readings 

and to validate the measurement only because the strain gages are not mounted when the rail is 

strain free. The results from both tests are displayed in Table 5-4 below.
67

Table 5-4. Comparison for Strain Measurement

MP231.6 MP228.6

Strain from ∆Yrel 330 651

Strain from gage 290 580

% Error 13.7% 12.2%

Figure 5-14. Measurements from Tangent Track (MP 231.6)

Figure 5-15. Measurements from Pumping Track (MP 228.6)
68

5.2.5 Summary and Conclusions

The UNL system measures three points of the rail shape beneath the loaded wheels including the 

location of highest bending stress below the loaded wheels. This direct measurement of the rail 

shape can then be mapped into rail stress through the curvature of the rail and beam theory. 

As verification of the UNL measurement system, results from tests conducted on the UPRR’s 

Yoder Subdivision are discussed. In these tests, bondable resistance strain gages were mounted 

to the lower flange of the rail at several locations. The track was then loaded by spotting the 

measurement car over the strain gages and by moving the car over the gages at various speeds. 

The loaded and unloaded profiles were measured using surveying equipment, and the 

relationship between the UNL deflection measurement and the measured rail stress was 

explored.

The field results show a strong correlation between the strain calculated from Yrel and the strain 

gage measurement. The differences at the two sites of 13.7 and 12.2 percent are well within the 

modeling errors and the measurement errors present in the field tests. In both instances, the 

calculated strain was slightly above that of the measured strain. The lower measured value may 

have resulted from slight misalignment of the strain gages with the axis of the rail. Furthermore, 

the axle may not have been parked exactly above the strain gages so that the gages did not record 

the maximum strain present directly beneath the axle.

These results suggest the UNL system is capable of accurately measuring real-time bending 

stress in the rail given the assumption that the rail would be in a zero bending stress state when 

the rail is straight. The validity of this assumption is unclear, and the entire process of measuring 

bending stress needs further examination.
69

6 Testing on the Union Pacific South Morrill Subdivision

Since 2006, the measurement system has conducted revenue service tests in many locations 

including the BNSF Railway’s St. Joseph subdivision in Kansas, as well as the South Morrill 

subdivision, Sidney subdivision, Ravenna subdivision, and Powder River subdivision, all of the 

UPRR. Some short distance tests such as between Lincoln and Ashland, NE, and between 

Lincoln and Tecumseh, NE, were also conducted. 

Recent tests focused on the 160 miles of heavy-axle load freight line of UPRR on the South 

Morrill subdivision of the North Platte division. The tests were performed on Main 1 (primarily 

unloaded, empty coal trains returning to the mine with approximately 50 MGT/year) and Main 2 

(primarily loaded coal trains with approximately 250 MGT/year) of the subdivision. Repeated 

tests were conducted on this subdivision in October 2006, December 2006, February 2007, April 

2007, January 2008, April 2008, and June 2008. Tests were done at speeds up to 60 mph in a 

work train consist. The autonomous testing ability of the system made these tests easy to 

conduct and cost-efficient. These repeated tests were performed in different seasons, under 

different weather conditions, and at different speeds, providing tremendous amounts of data for 

analysis. Figure 6-1 shows the measurement vehicle in the consist of a coal train during a test.

Figure 6-1. System in Revenue Service Testing

An example of data recorded over a distance of one mile is shown in Figure 6-2. The plot shows 

the relative deflection of the rail, Yrel (as defined in Figure 3-1), as a function of GPS 

coordinates given in degrees of longitude and latitude. The data is overlaid on a satellite image 

(Google Map). In this figure it is possible to qualitatively trace changes in relative deflection to 

specific track events such as grade crossings, culverts, and bridges.
70

Figure 6-2. Yrel Data Overlaid on a Satellite Map

Figure 6-3 shows data from 1 mile of track. The plot shows the relative rail deflection between 

the measurement point and the wheel-rail contact point (Yrel as defined in Figure 3-1). Also 

shown are the mean and standard deviation of the data. The mean represents the average value 

over the past 0.1 miles of track and the standard deviation corresponds to that mean over the 

same distance. This distance was chosen arbitrarily as a characteristic length. The mean and 

standard deviation are therefore dependent on the direction of travel (right to left in this plot).

This section is an interesting example because it corresponds to a high-speed crossover between 

MP 0.6 and 0.9. A noninsulated joint bar (near MP 0.9) is located at the right end of the 

crossover. The relative deflection at this point was very large (over 1 in), indicating a very low 

track modulus. The joint bar failed and caused a derailment only 2 wk after the track modulus 

measurements were taken. This location represented the second highest relative deflection 

measurement recorded over the nearly 350 miles of track tested for this portion of the test. This 

spike in relative displacement obviously caused a jump in both the mean and standard deviation 

of the data. The other end of the crossover (near MP 0.65) also displayed a rise in the standard 

deviation, indicating a rough section of track.
71

Figure 6-3. A Rough High-Speed Crossover

In contrast to Figure 6-3, an example of 1 mile of quality track is shown in Figure 6-4. Yrel and 

therefore modulus were both very consistent over this section of track. The consistent mean and 

relatively low standard deviation emphasize the track’s quality.

The results of these tests suggest that information in these measurements may be useful in 

indicating unsafe sections of track in need of repair. The contrast between the two figures 

suggests a quantitative method to more rationally schedule and prioritize track maintenance.

Figure 6-4. Track with Consistent Modulus
72

6.1 Derailments following the 2007 Test

Postprocessing of the data indicated a correlation with two of the top 10 deflection locations and 

the location of two derailments. These were within 30 days of the track modulus measurement 

date.

The exception list was generated, based on the large values of Yrel and large defections (Yrel –

ECO) criterion, and was independent of the knowledge of the derailments. The site with the 

second highest difference exception, MP B.89, coincided with the location of a defective field 

weld, which caused an eight-car derailment 14 days after the test. The site with the 10th highest 

exception at MP J.58 coincided with the location of a broken joint bar derailment 30 days after 

the test.

The site with the second highest deflection was MP B.89. The raw data from that location is 

shown in Figure 6-5 for 1 mile of track. The exception at MP B.89 is clearly visible with a large 

peak in relative displacement. The standard deviation is also larger around this turnout. 

Although no one observed the failure, the assumption of railroad staff is that the poor track 

support contributed additional stress to the weld under each passing wheel load and the weld 

continued to degrade until failure 14 days later resulting in a derailment.

Figure 6-5. Site of Broken Field Weld 14 Days after Test

The site with the 10th highest deflection was at MP J.58. The raw data from that location is 

shown in Figure 6-6 for 1 mile of track. The exception at MP J.58 is also visible with a large 

peak in Yrel measurements. The standard deviation is also larger around this turnout with both 

ends of the turnout clearly visible. Although no one observed the failure, the assumption of 

railroad staff is that the poor track support contributed additional stress to the joint under each 

passing wheel load, and the joint continued to degrade until failure of the noninsulated joint.
73

Figure 6-6. Failed Noninsulated Joint 30 Days Posttest

6.2 System Measurement under Various Conditions

6.2.1 Results from Different Train Speeds

The results from multiple tests in South Morrill subdivision suggest that the measurements are 

not strongly related to the train speed. The average train speed for the 0.1 miles shown in Figure 

6-7 are 48, 22, and 35 mph, respectively. 
74

Figure 6-7. Measurements from Different Speeds

Repeated tests over the same section of the track at different speeds were performed during a 

special validation test on the Yoder subdivision. Four different speeds (20, 30, 40, and 50 mph) 

were tested. For most of the sections, the measurements from the different speeds are highly 

repeatable, which indicates that the speed is not a significant factor in the measurements at those 

locations. However, at some locations, differences do exist between the different speeds. The 

section shown in Figure 6-8 is an example of these locations. The values of the peaks at MP 

224.4685 corresponding to the four different speeds within this short section (21, 30, 40, and 48 

mph) are 0.607, 0.687, 0.692, and 0.77 in, respectively. The Yrel measurements increased 0.163 

in when the train speed increased from 21 to 48 mph. This can possibly be explained by an 

increased dynamic load when the train moves at higher speeds. 
75

Figure 6-8. Dynamic Loads Affecting Measurements at Different Speeds

6.2.2 Results from Different Seasons

Although most of comparisons between tests showed significant repeatability as shown in and 

Figure 6-7 and Figure 6-8, there were some variations in certain sections of track. The most 

noticeable variation is the comparison between the December 2006 test and the other tests. At 

multiple locations of the track, the relative deflection data (Yrel) from the test in December 2006 

has an offset compared with the results from other tests as shown in Figure 6-9. The test in 

December 2006 was special because it was conducted in extreme weather conditions with falling 

snow and an ambient temperature around a high of 20°F. Figure 6-9 shows similarity in the 

shape of the measurements but a shift in the Test 2 measurement. The December 2006 test has a 

lower mean value, indicating the track is stiffer (smaller Yrel). This could indicate a change in 

track modulus associated with frozen track that may contain moisture. These results suggest the 

system could be useful to quantify seasonal variations in track modulus caused by factors such as 

variations in subgrade moisture.
76

Figure 6-9. Variations of the Measurements

Interestingly, the offset of the measurements shown in Figure 6-9 only occurred at some track 

sections. Figure 6-10 shows a section of track where only the measurements on the left half of 

the figure has this offset, whereas on the right half of the figure the results from the three tests 

are very similar. This could indicate variations of subgrade moisture can lead to greater (or 

lesser) seasonal variations in track modulus. 

The conclusion here is that seasonal variations are significant and measureable with this system. 

Such absolute seasonal measurements cannot be made with dead-reckoning measurement 

systems that only measure position relative to the previous position (e.g., acceleration-based 

profile). These measurements show areas of track that contain significant moisture and therefore 

have changes in stiffness under frozen and nonfrozen conditions. Areas of track that do not 

contain significant moisture do not have as much of a change in stiffness or areas that are 

dominated by geometry variations may not show changes.
77

Figure 6-10. Variations in Some Sections of Track

6.3 Trending Analysis

6.3.1 Data Preprocessing

Because of GPS misalignment, data from different tests may not exactly coincide with each other 

in terms of milepost. Figure 6-11 shows the two sets of data from the same section of track. An

offset exists between the two curves. 

Measurements from two tests over approximately 0.05 miles of track are shown in Figure 6-11. 

The figure clearly shows that the measurements from both tests have similar shape. However, 

the two sets of data have an obvious horizontal offset. The offset represents errors in the 

milepost location associated largely with GPS error in localizing the data. Despite errors in GPS 

measurement, the shape of the curve is maintained because the relative GPS measurement (one 

data point with respect to the next) is much more accurate than two independent absolute GPS 

measurements.
78

Figure 6-11. The Original Data from Two Tests

For the purpose of trending, it is desirable to remove the offsets so that relative comparisons can 

be made over short sections of track. The relative comparisons would evaluate one measurement 

relative to a previous measurement made at the same location at an earlier time.

To remove the offset in milepost, the cross-correlation function is introduced to mathematically 

quantify the offset. Cross correlation is a standard method of estimating the degree of correlation 

between two sets of measurements. Consider two series x(i) and y(i) both of length N where 

i = 0,1,2...N – 1. The cross correlation, Rxy

ˆ , at delay m is defined as :

∑

−

=

= +

1

0

( ) ( ) 1 ( ) ˆ N

n

xy x n y n m

N

R m Equation 6.3-1

where m = –(N – 1),...,-2 –1,0,1,2,...,N – 1

For various values of m, Rxy

ˆ is in the range 1 ˆ −1 ≤ Rxy ≤ . The bounds indicate maximum 

correlation, whereas 0 indicates no correlation. A high negative correlation indicates a high 

correlation but where one series is the inverse of the other series.

The results of applying the cross correlation function to the two series in Figure 6-11 at 

numerous values of m (between –100 and 100) is shown in Figure 6-12. This figure shows the 

cross correlation as a function of offsets. Clearly, the cross correlation reached a maximum 

when the offset was 8 ft. The value of the cross correlation at this offset is 0.8, indicating a high 

correlation between the data (this also suggests the measurements are highly repeatable).

Loaded Rail

Shape
79

Figure 6-12. Cross Correlation

Based on the cross correlation plot, the second series in Figure 6-11 was shifted by 8 ft, and the 

new plot is shown as Figure 6-13. Now the two series of data line up well, which is helpful and 

convenient for further data comparison and analysis.

Figure 6-13. The Shifted Data from Two Tests

6.3.2 A Trending Example

Some trending results from the South Morrill subdivision are now presented. Several tests were 

conducted over the same approximately 160 miles of this subdivision at 3-month intervals. 

Figure 6-14 shows the relative deflection measurement (Yrel) over 0.1 miles of track from MP 
80

A.70 to A.80 (exact mileposts are removed). Here, the third axis (into the page) illustrates the 

time interval between the tests in months. Figure 6-14 clearly demonstrates the changes in this 

section of track as a function of time. Specifically, two locations are singled out over time. It 

can be seen that MP A.76 is not changing quickly over time, whereas MP A.74 corresponds to a 

peak in the measurement (soft spot) that is increasing over time (becoming softer).

Figure 6-14. Data from Three Tests at MP A.74

From the measurements shown in Figure 6-14, two sections at MP A.74 and A.76 are extracted 

as an example and shown as a function of time in Figure 6-15. Since three tests were performed, 

three data points are shown in each of the plots for these milepost locations. Given this data, a 

prediction can be made based on the trends. In this case, a line is fitted to the data and used as 

the prediction. A correlation can be produced to indicate how well the line fits the three data 

points. Having only three points (three tests) of course may not accurately predict the trend and 

clearly the prediction has uncertainty. More testing will improve the prediction. With more 

available tests, it may be desirable to use other curves (rather than a line) to improve the 

prediction.
81

Figure 6-15. Trending at MP A.74 and A.76

To further illustrate the possible usefulness of this technique, MP A.74 is shown again in Figure 

6-16. Here, an assumption is made that a given threshold of relative displacement, Yrel, would 

be undesirable. On the basis of the previous measurements, a value of 1.2 in is chosen and 

indicated by the red dashed line. Now, the linear prediction can be used to estimate the time 

required to reach this threshold. In this case, 5.5 months from the last test is the window for 

maintenance. The accuracy of this prediction is difficult to quantify; however, this is a tool that 

can be used to prioritize maintenance that is based on actual track data. It is also possible to 

apply this technique to other track measurements such as gauge, gauge restraint, cross level, and 

other standard measurements.

Figure 6-16. Test Data at MP A.74 as a Function of Time
82

The above approach can now be applied over the entire section of track between MP A.70 and 

A.80. This range is broken into 12-foot bin lengths, and a curve fit is created for each bin. The 

Yrel measurements are shown in Figure 6-17 for the three tests. A subplot is created that 

represents the slope of the trending line (linear curve fit in Figure 6-16). Here, the slope, given 

by ∆Yrel

∆Time in units of inches per month, is approximated as the difference between the two tests. 

When both differences are equal, it represents a linear change over time. The figure shows that 

MP A.74 is changing approximately linearly over time at a relatively fast rate, while the other 

part of the track in this section shows little change. The location of MP A.74 corresponds to a 

muddy road crossing, and the needed maintenance schedule can now be estimated (given some 

amount of uncertainty).

Figure 6-17. Trending from MP A.70 to A.74

6.4 Comparison between Different Measurement Systems

6.4.1 Comparison of VTD with Vehicle Track Interaction

A track quality measurement system known as vehicle track interaction (VTI) is currently in use 

by UPRR. This system has been successfully integrated into revenue service. The VTI system 

uses accelerometers mounted on various locations of a railroad vehicle to measure vertical 

accelerations. The system is used on both locomotives and hopper cars and has been fully 

integrated into revenue service with sophisticated communication and data processing 

infrastructure.

Results presented here are from a test conducted on a locomotive that passed over the same 

section of track (the Powder River and South Morrill subdivisions) 2 days after the vertical rail 

deflection measurements presented above (Section 5). The relevant VTI results come from an 

accelerometer mounted to the axle of the locomotive (below the suspension system). The 
83

acceleration data is used (with knowledge of the locomotive’s mass and suspension 

characteristics) to estimate loads between the wheel and rail.

The VTI measurements and the vertical deflection measurements are fundamentally different. 

For example, a smooth section of track with a low modulus will show no significant difference in 

VTI measurements as compared to a smooth section of high modulus track. However, this 

difference will be captured by the VTD system. 

6.4.2 Comparison of Revenue Service Results from VTD and VTI

A test of the same section of track (270 miles of the Powder River and South Morrill 

subdivisions) was conducted using the VTI system on August 11, 2006—2 days after the VTD 

measurements were made. Some results of this test are shown in Table 6-1 as a comparison 

between the two measurements. Selections of the ranked results from VTI measurements are 

shown in the left four columns. The right two columns show how the same measurements 

appear in the deflection criterion exceptions. Only the locations common to both tables are 

shown in Table 6-1.

Table 6-1. Prioritized Exceptions of VTI Data

VTI 

Rank MP Acceleration (g) Estimated Axle Force (lbf) Deflection (in) Rank of ∆

1 X.5 28.15 104,500 NA NA

: : : : : :

5 G.97 24.97 84,800 0.828 7

: : : : : :

27 J.58 27.27 69,650 0.083 10

: : : : : :

62 B.89 27.78 64,600 0.89 2

: : : : : :

123 A.47 22.09 59,000 1.424 1

: : : : : :

249 C.97 17.1 52,300 0.973 3

The most important conclusion to reach from Table 6-1 is that the two measurements are 

fundamentally different. The highest force estimated from the VTI data (at MP X.5) does not 

appear in the top 12 deflection criterion exceptions. The top VTD exception (at MP A.47) 

appears at #123 in the VTI measurements. The two derailment locations described above appear 

on both lists (highlighted in both tables). The derailment that occurred 14 days after the vertical 

modulus measurements (MP B.89) appears as #2 on the VTD list and #62 on the VTI list. 

Similarly, the derailment that occurred 30 days after the test (MP J.58) was #10 on the VTD list 

and #27 on the VTI list. However, an enormous impact load at MP X.5 is not indicated in the 
84

top 12 of the VTD measurements. Clearly, these are different, but complementary, 

measurements.

6.4.3 Comparison of Revenue Service Results of Yrel and ECO

The relation between Yrel and ECO has been explained in Section 3.4.2. Although high 

correlations exist between Yrel and ECO, they are fundamentally different measurements. 

Results from revenue service tests further confirm the fundamental differences. 

A test was conducted on the South Morrill subdivision in April, 2007. The VTD system and 

UPRR’s track geometry car (EC5) collected data over the same 160 miles of track on the same 

day. Exception lists were produced based on Yrel measurements and calculated ECO from the 

geometry cars space curve data.

Table 6-2 lists the top 20 locations in the VTD list along with their rankings in the ECO list. 

Among the sites, #4 and #17 in the list are the locations where a derailment occurred 1 week 

before the testing day. The track was under construction at these locations on the day of the test. 

The #1 item in the list is the location where another derailment occurred 2 weeks before the test. 

Again, the track was under construction on the day of the test. 

It can be observed that nine of the top 20 sites in the VTD list did not show up on the ECO list 

(ECO list includes 200 sites). Besides the derailment locations, only four locations (#5, #9, #18, 

and #20 in VTD list) were identified by both top 20 lists. In addition, these locations rank 

differently on each list.
85

Table 6-2. Comparison between VTD and ECO (ranked by deflection)

VTD 

Rank Deflection(in) ECO Rank ECO (in) MP Description

1 1.18 1 1.1954 58.813 Derailment 

reported

2 1.13 N/A N/A 11.802 Unknown

3 1.06 N/A N/A 93.489 Signal

4 1.02 4 1.02 105.106 Derailment 

reported

5 0.94 7 1.0196 56.286 Road crossing

6 0.92 N/A N/A 14.37 Signal

7 0.9 N/A N/A 9.628 Unknown

8 0.88 93 0.625 55.307 Road crossing

9 0.87 28 0.8514 115.447 RR XING

10 0.86 12 1.0038 23.102 Signal

11 0.85 N/A N/A 38.938 Signal

12 0.84 N/A N/A 116.774 Unknown

13 0.83 N/A N/A 147.589 Unknown

14 0.83 NA N/A 31.294 Signal

15 0.8 43 0.7383 39.228 Road crossing

16 0.78 N/A N/A 100.149 Switch

17 0.76 25 0.8672 105.152 Derailment 

reported

18 0.76 9 1.0157 19.508 Road crossing

19 0.75 33 0.793 24.594 Road crossing

20 0.75 3 1.172 53.56 Road crossing
86

A list ranked by ECO data was also produced as shown in Table 6-3. In this table, 18 of the top

20 locations in the ECO list were also in the VTD list (VTD list includes 90 sites). However, 

most of these locations have lower rankings in the VTD list. Except for the derailment locations 

and five unknowns, almost all of the top 20 locations in the ECO list are road crossings.

Table 6-3. Comparison between VTD and ECO (ranked by ECO)

ECO 

Rank ECO (in) VTD Rank

Deflection 

(in) Description

1 1.1954 1 1.18 Derailment reported

2 1.1406 33 0.61 Road crossing

3 1.1172 20 0.75 Road crossing

4 1.0976 4 1.02 Derailment reported

5 1.0351 48 0.58 Crossover Switch

6 1.0312 72 0.52 Unknown

7 1.0196 5 0.94 Road crossing

8 1.0196 29 0.63 Culvert

9 1.0157 18 0.76 Road crossing

10 1.0156 22 0.7 Road crossing

11 1.004 82 0.51 Road crossing

12 1.0039 10 0.86 Road crossing

13 1.0001 81 0.51 Unknown

14 0.9961 46 0.58 Unknown

15 0.9687 56 0.55 Road crossing

16 0.9649 52 0.57 Road crossing

17 0.9414 59 0.55 Unknown

18 0.9336 N/A N/A Road crossing

19 0.9218 30 0.63 Unknown

20 0.8828 N/A N/A Road crossing
87

6.5 Field Investigations

The VTD measurement system conducted a test in a coal train consist at the South Morrill 

subdivision in June 2008. About 1 month earlier, UPRR’s EC5 geometry car collected track 

geometry data on the same subdivision. The track geometry data provided by UPRR was then 

implemented into the system’s Yrel measurements so that Yrel and ECO measurements could be 

compared and relative deflection measurements could be calculated. 

On the basis of both the proposed criterion and the calculated relative deflection measurements, 

15 sites were selected, and field investigations were completed on July 1, 2008. The track visits 

identified the reasons for large Yrel and deflection measurements at all sites with varying levels 

of maintenance urgency. Among them, one site was taken out of service immediately, and two 

additional sites were scheduled for maintenance. A variety of track problems were identified 

including poor joints, broken ties, muddy ballast, and crushed rail head, etc. Some of the sites 

will be discussed in detail in the following sections. The actual mileposts for these sites are 

concealed in the discussions.

6.5.1 Site 1: A Crushed Rail Head

The data from the first site is plotted in Figure 6-18. The peak in the Yrel data on the north rail 

measures 0.89 in, whereas the ECO measurement is 0.34 in. As a result, the relative deflection is 

calculated as 0.55 in. 

Figure 6-18. Data at the Crushed Rail Head Site

The pictures of this location are displayed in Figure 6-19. The white-colored ballast indicates 

that strong pumping has occurred. The track director estimated the pumping to be up to three-
88

quarters of an inch. The rail profile dropped down one-half inch at the crushed spot. This site 

was investigated 2 weeks after the test. The track director suggested that the weather was damp, 

and a flash flood warning was issued around the testing time so that the deflection was expected 

to be large.

Figure 6-19. A Crushed Rail Head

Although the large track deflection itself would not be expected to crush a railhead, the dynamic 

wheel load and increased rail stress tend to expose any potential weakness. If the rail would tend 

to fail by railhead crushing, then the additional stress at the large track deflection location will 

expose this tendency faster than in other track locations.

6.5.2 Site 2: A Muddy Crossing

A number of road crossings are listed in the VTD exceptions list. Most of them exhibited fouled 

ballast and broken ties. Data from one particular site is shown in Figure 6-20. On both of the 

south and north rails, two peaks of Yrel data may be observed. They represent the two 

approaches to the road crossing. The approach at the east side of the crossing appears to be 

much worse than the west side based on the Yrel data. The deflections at both rails on the east 

approach are almost 0.6 in.
89

Figure 6-20. Data at the Muddy Crossing Site

At least four ties were center cracked, and some of them appeared to have horizontal cracks at 

the bottom. The muddy ballast and the tie cavities indicated large movements of the ties. On the 

basis of this observation, the ties were estimated to have moved by over 0.5 in. This muddy area 

was located at an approach to a road crossing. The other side of the road crossing was also 

muddy. This explains the two peaks in the Yrel data. Again, the track was much wetter on the 

day of the test.

Figure 6-21. The Muddy Crossing
90

6.5.3 Site 3: A Failing Joint

Approximately half of the sites in the VTD exception list are joints. The measurement system 

demonstrated a notable ability to identify poorly supported joints, which often deteriorate rapidly 

once the joint begins to deflect substantially. Figure 6-22 shows the data at one particular joint. 

In this case, Yrel measured 1.21 in at the joint with an ECO reading of only 0.3 in. Therefore, the 

deflection is over 0.8 in.

Figure 6-22. Data at the Failing Joint Site

Figure 6-23 shows the condition of the insulated joint. This supported joint is rare in North 

America where suspended joints are dominant. The supporting tie in the center was in a very 

poor condition. The tie was split, and a large portion (left part in the picture) was nearly 

separated from the rest of the tie. Applying pressure with a foot was enough to deflect the tie. 

The bolts on the joint bars were loose, and one of them would actually rotate as a train passed. 

The track director commented that this joint was probably among the worst 10 percent of all 

joints. A wayside camera measurement was taken as a coal train passed the site. The video 

showed that the deflection at the joint was over 1.2 in under the loads of the locomotives.
91

Figure 6-23. The Failing Insulated Joint

6.5.4 Site 4: A Series of Broken Ties

The peak in the Yrel data from this site reads 0.95 in, whereas the ECO is 0.41 in. The site is 

located on a portion of tangent track with no joints or crossings. Field measurements of the midchord offset were consistent with the geometry car measurements. As shown in Figure 6-24, the 

curve of Yrel data from this site has a special shape when compared with the data from other 

sites. Unlike the sharp peaks in Figure 6-22, the peak here is broader, and high deflections 

occurred over a relatively long distance. 
92

Figure 6-24. Data at the Broken Ties Site

A picture of this site is displayed in Figure 6-25 in which a series of broken ties are clearly 

visible. The ties barely constrained the south rail. The clips were either missing or unattached to 

the rail because of the cracks in the ties. The broken parts of the ties could easily be lifted by 

hand. Recent geometry car tests and VTI tests did not identify an exception at this location. The 

track director explained that defects like these were difficult for track inspectors to notice. The 

track director immediately stopped service on the track as shown in Figure 6-26.

This location is one of several sites where measurements from acceleration-based systems do not 

indicate as significant a reading as measurements by the VTD system (i.e. Yrel >> ECO). The 

track support conditions where the VTD measurement does not match measurements from other 

systems are extremely complex. The track behavior in these locations does not follow traditional 

models and will require site specific observations to fully understand the complex track 

mechanics leading to the large deflection associated with the Yrel measurement. More 

importantly, these locations are potentially the most significant locations of track performance 

problems observed during this research project. To evaluate the track conditions in site 

inspections associated with this project, many sites were selected where Yrel was larger than 

ECO. In every instance a specific failure in a track component was identified ranging from 

cracked and broken ties, to poorly performing switches, to poorly supported joints. These all 

represent track support related deterioration that is a risk to track safety, but that are not 

explained using simple track deflection models like beam on elastic foundation theory or even 

generalized finite element solutions.

It should also be noted that there are several locations where a significant reading is obtained 

from acceleration-based measurement systems and the VTD system does not show significant 
93

relative deflection. Follow-up visits to these sites confirmed that the track condition was 

reasonably good and confirmed the Yrel measurement, but raised questions about how to 

threshold the acceleration data. This confirms the general challenge of thresholding 

acceleration-based measurements and the associated problems with attempting to interpret 

acceleration data in terms of deflection or load.

The authors feel it is important to further understand track locations where Yrel values 

significantly exceed associated ECO. Data that have been associated with the potential for track 

deterioration should be evaluated to more fully understand the track conditions and associated 

safety risks. This will require more detailed follow up during site visits to make careful 

measurements of track, component, support conditions, and, perhaps, instrumentation. 

Figure 6-25. Six Broken Ties in a Row
94

Figure 6-26. Track Taken Out of Service

6.6 Analysis of Bridge Approaches

Further analysis has been made of certain bridges and at-grade road crossings where changes in 

track stiffness have been problematic. As stated above, the use of track modulus can be 

problematic because the term is only well defined in situations of linear and uniform track 

support. Real track is always nonuniform and nonlinear, and therefore, modulus is a difficult 

term to define. However, modulus is an often used in the rail industry and can be useful in 

discussions as in the analysis here. Given this, changes in track deflection are discussed below, 

but these deflections are also related to modulus measurements. The values of modulus are not 

accurate absolute numbers but instead provide insight and are useful when compared in a relative 

sense (e.g., how much higher or lower is the modulus on the bridge when compared with the 

neighboring tangent track or the approach).

To make relative comparisons, modulus values are taken from Kerr (2003). These estimates are 

shown in Table 6-4.

Table 6-4. Modulus Values from Kerr (2003)

Track Condition Description Modulus (psi)

Wood-tie track, after tamping 1,000

Wood-tie track, compacted by traffic 3,000

Concrete-tie track, compacted by traffic 6,000
95

Wood-tie track, frozen ballast and subgrade 9,000

On the basis of a relative deflection measurement from Yrel measurements and ECO 

measurements, the deflection can be translated into a modulus value using the values from the 

above table. For example, the difference between wood and concrete ties can be seen in 

Figure 6-27. Here, the relative deflection measurement, when averaged over long distances (1/2-

to 1-mile segments), shows a significant deflection difference using the two types of ties.

Figure 6-27. Deflection Change Between Wood and Concrete Ties

The first site described here is a concrete box culvert. This culvert has pads bonded to the 

bottom of the concrete tie over the culvert in an attempt to better match the culvert vertical 

stiffness to that of the surrounding track. The relative deflection measurements (difference 

between Yrel and ECO) are shown in Figure 6-28 and summarized in Table 6-5. Here, it can be 

seen that the deflection over the culvert is on the order of 0.145 in, and the deflection of the 

surrounding track is lower (maybe 0.105 in). The change in deflection is approximately 0.04 in. 

This means that the culvert is less stiff than the surrounding track. Again, interpolating from a 

relative comparison based on the Kerr estimates (Table 6-4), this suggests a culvert modulus of 

≈4,400 psi and the culvert is ≈1,600 psi softer than surrounding track.

Wood 

Difference 

between wood 

and concrete 

ties: 0.05”
96

Figure 6-28. Concrete Boxed Culvert

Table 6-5. Average Deflection over and around Culvert

Average 

Deflection 

over 0.1 mile 

surrounding 

culvert (in)

Average 

Deflection over 

the Culvert 

(in)

Increment (in)

South Rail 0.101 0.148 0.047 } δDeflection 

North Rail 0.106 0.142 0.036 ~0.04 in

The second site described here is a three-span bridge. This bridge has a “ballast mat,” bonded to 

the steel deck pan of the bridge (under 12 in of rock) in an attempt to better match the bridge 

vertical stiffness to that of the surrounding track. The relative deflection measurements 

(difference between Yrel and ECO) are shown in Figure 6-29 and summarized in Table 6-6. 

Here it can be seen that the deflection over the bridge is on the order of 0.095 in and the 

deflection of the surrounding track is lower. The change in deflection is approximately 0.02 in. 

This means that the bridge is less stiff than the surrounding track. Again for a relative 

comparison based on the Kerr estimates, this suggests a bridge modulus of ≈5,200 psi and the 

culvert is ≈800 psi softer than surrounding track (again, interpolating from Table 6-4).
97

Figure 6-29. Three Span Bridge

Table 6-6. Average Deflection over and around Three-Span Bridge

Average 

Deflection 

over 0.1 mile 

surrounding 

bridge (in)

Average 

Deflection over 

the bridge (in)

Increment (in)

South Rail 0.091 0.110 0.019 } δDeflection ~ 

North Rail 0.097 0.119 0.022 0.02”

6.7 Identifying Thresholds for Maintenance

The above analysis and field observations are summarized here as a beginning toward 

understanding how to interpret the measurements and how to use this knowledge to better 

schedule maintenance operations. The discussion presented here is empirical and preliminary.

At each of the site visits, host railroad experts provided interpretation of the location and the 

priority for a maintenance action at the site. These were categorized as red, yellow, or green 

with red representing an immediate maintenance concern that should be attended to in a short 

time, and green meaning it is not an immediate maintenance concern with yellow in-between. 

An example of a test from June 2008 on UPRR is shown in the table below.
98

Table 6-7. Maintenance Summary

MP DESCRIPTION DEFLECTION

A.971 very bad joint, worst 5% of all joints in NPSU 0.91

B.906 pumping joint, but not immediate concern 0.807

C.942 temp fix where rail was cut out, needs work soon 0.79

D.279 big pumping, failing joints 0.725

E.878

Deland: “joint looks like #$%$#” – will replace within 30 

days 0.683

F.961 Deland called nearby crew, temp fix where rail was cut out 0.66

G.138 pumping joint, but not immediate concern 0.606

H.054

very muddy crossing-probably much worse during rains 

when measurements were taken 0.59

I.948 bad joint, loose bolts 0.58

J.04 pumping joint, but not immediate concern 0.57

K.29

crushed head at weld, pumping 3/4 in, probably much worse 

during rains when measurements were taken 0.545

L.24 taken out of service immediately – data a different shape 0.54

M.458 pumping joint, but not immediate concern 0.496

N.703 has recently been repaired – trending site 0.4082

A second summary from a two different tests on both wood and concrete track and on two host 

railroads is shown in Figure 6-30. Here, the red, yellow, and green dots each represent a site. 

The sites are plotted by the magnitude of Yrel and show that most of the red locations have a 

larger Yrel measurement and most of the green sites have a lower measurement.
99

Figure 6-30. Summary of Maintenance Priority of Sites

These results are again extremely preliminary. A problem with the presentation in Figure 6-30 is 

that it does not show sites that correspond to good track conditions with low Yrel readings. This 

is because few good sites were visited since the goal of the visit was to identify track conditions 

at locations of high Yrel to determine whether maintenance or other problems existed. Small 

Yrel readings represent the majority of the Yrel data, and it is strongly believed that the 

numerous green dots that this would add at the bottom of this figure would make a strong 

distinction between green and red to better select a threshold.

Even though this analysis is not yet complete, it does suggest that high Yrel readings and high 

deflection readings are less desirable. Any future guidelines may include thresholds for wood 

and for concrete as well as for deflection and for Yrel readings (and of course class of track).

Another exception is that the criteria being proposed and have been used above (Section 6) are 

the deflection defined by the difference between Yrel and ECO. A table similar to that above 

can be made using these criteria.

For the measurements shown above, an extremely preliminary suggestion might be that 

deflection measurements greater than 0.6 in might represent a significant concern (5 of 7 as red 

in the measurements above) and small measurements would be less of a concern (5 of 7 not red 

in the measurements above). For Yrel, it might be suggested that measurements greater than 0.9 

in (25 of 34 being red) or 1 in (15 of 19 being red) might represent a significant concern with 

lower readings being less of a concern (2 of 2 not green with readings less than 0.9 in).

Several other site visits have been made (~200) but these are summarized here. Clearly, more 

work needs to be done to solidify these thresholds.
100

7 Finite Element Analysis from System Perspective

As described earlier, several analytical models have been developed for use in railroad track 

design and to characterize the track response under load. Despite the usefulness of some of these 

models, they lack the complexity needed to fully characterize real track. Most of these models 

assume uniform support from homogeneous materials, whereas evaluation of problematic track 

conditions most often occurs in locations of track support variations with variable materials 

properties. Advancements in finite element analyses (FEA) have provided more realistic 

representations of the track response. These FEA programs often need to be formulated with a 

specific objective to minimize computational costs. Therefore, a FEA program suitable for 

solving one problem may not adapt well to another goal.

Custom FEA programs offer realistic solutions to well-defined problems. One particular FEA 

program titled GBEAM was developed to better understand track dynamics (Carr, 1999). This 

program analyzed the motion of a single mass rolling along the track. The track was modeled as 

a Euler beam resting on a damped, elastic foundation. The results highlighted the importance of 

measuring track deflection and concluded that track dynamics become important in high-speed 

simulations (Carr, 1999).

The following sections describe the development and results of a new custom FEA program 

created to characterize track response from the perspective of the UNL system. Nonlinear 

foundation characteristics are examined with the commercial FEA software, ALGOR®. The 

purpose of this analysis is to provide better understanding of the data collected by the UNL 

system. Improved evaluation of this data would enhance its potential as a resource for 

maintenance planning.

7.1 FEA Model Development

7.1.1 FEA Objectives and Specifications

The main objective of the FEA analysis in this report is to characterize track response based on 

the track characteristics that influence the output of the UNL measurement system. This system 

measures the vertical deflection of the track from a moving reference frame attached to a threepiece truck on a modified hopper car. Variations in the track support and in short-wavelength 

track geometry affect the UNL measurement. Understanding how these track characteristics 

relate to the UNL data is important for interpreting the data with the goal of improving 

maintenance planning.

The UNL system, described previously yields relative track deflection data defined as Yrel. This 

measurement is used in conjunction with a 10-foot ECO calculated from the vertical track 

geometry rail profile to compute the track deflection that can identify areas of poor track support. 

The difference between Yrel and ECO is used as one exception criterion for identifying locations 

needing further inspection (Lu, 2008). This criterion is used along with relative thresholds for 

the magnitudes of the Yrel and ECO measurements.

Certain track locations yield identical shapes and magnitudes for both Yrel and ECO as shown in 

the top of Figure 7–1. Other locations result in large peaks in Yrel data with relatively minor 
101

ECO peaks as illustrated in the bottom of Figure 7–1. This phenomenon is frequently observed 

in the UNL measurements. Corresponding site visits have shown in almost every case a 

problematic track support condition or identification of a track defect. Since the source of track 

support irregularities are generally below the surface, the surface inspection of track involves 

identification of clues as to the surface manifestation of subsurface problems. In locations where 

no surface clues are obvious, observation of track load-deflection behavior under passing trains 

generally shows the nonuniform support that the UNL measurement indicated even though the 

subsurface failure had not yet progressed to the surface.

It is believed that these special locations have significantly complex track behavior and that FEA 

analysis, including the work presented here, can help improve the understanding of these 

locations. However, additional field tests and site visits will also be needed to fully understand 

the risks associated with these locations.

Figure 7-1. Different Shapes in Yrel and ECO Data

Using the FEA program to characterize the track response based on the Yrel and ECO

measurements may provide insights into which track features lead to certain measurement results 

and will be particularly useful for locations where the surface manifestation is limited and might 

provide the opportunity to conduct preventative maintenance that might otherwise be impossible 

to efficiently specify. For example, different combinations of poor track support, poor track 

geometry, and failing joints may lead to either case shown in Figure 7-1 as well as intermediate 
102

situations. Therefore, the particular program developed should have the ability to vary each of 

these items individually and analyze the corresponding effects on the UNL measurement results.

Several specifications simplify the structure and focus of the FEA program. The FEA program 

assumes a Winkler foundation. However, the program allows for variation in support modulus 

along the track. More complicated, nonlinear foundations are examined with ALGOR®. The 

model is constructed in two dimensions and considers only one rail. Fixed boundary conditions 

are used, and the length of the model ensures that the boundaries do not affect the deflection 

results. The dimensions correspond to a standard coal train traveling over 132-pound rail. The 

elements are only 1 inch in length so that enough nodal deflections are calculated to reproduce 

the Yrel measurement. Dynamic effects are ignored, and the deflections needed to reproduce the 

Yrel data are obtained using a quasi-static approach.

7.1.2 Governing Equation and Program Development

The assumption of a Winkler foundation leads to the governing differential equation defined in 

earlier sections. This equation is rewritten in a form more suitable for the derivation of the 

necessary finite element equations as displayed in Equation 5.1-1. The equation represents a 

beam supported by a series of continuous, independent springs. Equilibrium of a differential 

length of the beam yields:

Equation 5.1-1

The derivation of the finite element equations follows a variational formulation based on the 

principle of minimum potential energy and the calculus of variations. The following simplifying 

techniques based on the method in Thompson (2002) are used in the derivation:

1. The governing equation is placed in its “weak” form (from the calculus of variations) 

rather than directly into the potential energy functional.

2. The general element approximating functions are defined by a local coordinate system 

with the origin at the element’s left end.

3. Equations and calculations are completed in matrix notation more suitable for 

programming.

4. Known boundary values are included in the calculations as if they were unknowns and 

only accounted for after the final matrix equation is created.

The governing equation defined in Equation 5.1-1 requires that the fourth derivative of w exists. 

In order to reduce these requirements, the equation is placed in its weak form giving:

Equation 5.1-2

where

δw represents a small variation in w

L is the length of a general element

Placing the governing equation in its weak form reduces the requirements for higher order 

derivatives in the solution. Integrating by parts twice and substituting the expressions for shear 

and moments found in elementary beam theory yields Equation 5.1-3 (Thompson, 2002).

d2

dx 2 EI

d2

w

dx 2

 

  

 

  + uw − q = 0

δw

d2

dx 2 EI

d2

w

dx 2

 

  

 

  + uw − q

 

  

 

  dx = 0

0

L

∫
103

Equation 5.1-3

where Equation 5.1-4

Equation 5.1-5

On the basis of the weak formulation of the governing differential equation, only the second 

derivative of w is required to exist, as illustrated by Equation 5.1-3. The condition for existing 

second derivatives requires that the approximating shape functions be continuous and have 

continuous first derivatives. Hermite cubics are used as the approximating functions to 

accomplish these objectives. Each function is defined in terms of the nodal values for w and 

dw/du at the ends of each element. These functions are expressed in terms of their Gaussian 

coordinates as displayed in the following equations.

Equation 5.1-6

Equation 5.1-7

Equation 5.1-8

Equation 5.1-9

where: u defines the Gaussian coordinates

With respect to the nodal values and matrix notation, the approximation becomes:

Equation 5.1-10

where: a and b represent the left and right nodes, respectively

u defines the Gaussian coordinates

Gaussian coordinates are used since Gaussian quadrature is the numerical integration method 

used to solve the finite element equations. Gaussian quadrature is a method of determining an 

δwLVL −δwOVO − dδw

dx

 

   

  

L

ML +

dδw

dx

 

   

  

O

MO +

d2

δw

dx 2 EI

d2

w

dx 2

 

  

 

  +δwkw −δwq

   

   dx = 0

0

L

∫

M = EI

d2

w

dx 2

V = d

dx

EI

d2

w

dx 2

 

  

 

  

N1 = 1

4

 

   

  (u + 2)(u −1)

2

N2 = 1

4

 

   

  (u +1)(u −1)

2

N3 = − 1

4

 

   

  (u − 2)(u +1)

2

N4 = 1

4

 

   

  (u −1)(u +1)

2

w(u) = N1(u) N2 (u) N3(u) N4  (u)

Wa

dWa du

Wb

dWb du

 

 

  

 

 

 

 

 

  

 

 

 
104

integral by using a weighted average of the integrand evaluated at specific sampling points 

(Thompson, 2002). In this case, four-point Gaussian quadrature is used since four points are 

needed to exactly integrate a cubic polynomial.

The approximating shape functions are depicted graphically with respect to the Gaussian 

coordinates as shown in Figure 7-2.

Figure 7-2. Hermite Cubic Shape Functions

Since the shape functions are expressed in Gaussian coordinates, the integration limits of ±1 do 

not correspond with the x-axis of the elements. A change of variables is needed to map the 

functions and their derivatives into the x-axis of the element coordinate system. This mapping is 

completed with the following linear transformation:

Equation 5.1-11

Equation 5.1-12

Equation 5.1-13

where: a and b represent the left and right nodes, respectively

u represents the Gaussian coordinates

The actual transformation for each shape function and its derivatives is formulated by Thompson 

(2002) and used in the FEA program code. The solution and its variation can then be expressed 

in terms of the shape functions as follows:

x = a + b

2

 

   

  +

b − a

2

 

   

  u

dx = b − a

2

 

   

  du

du = 2

b − a

 

   

  dx
105

 

 

 

Equation 5.1-14

where: W and δW represent the arrays of nodal values

Substituting the expressions in Equation 5.1-14 into the integral term of Equation 5.1-3 yields the 

following integral expressions for each element:

Equation 5.1-15

where: [S1]e, [S2]e, and {f}e correspond to the first, second, and third 

terms of the integral in Equation 5.1-3

These expressions are evaluated and assembled for each element. The shear and moment terms 

appearing in Equation 5.1-3 are used to define the boundary conditions. The [S1]e and [S2]e

terms define the stiffness matrices and the {f}e term represents the loading. After collecting the 

element matrices into a global matrix, the resulting expression shown in Equation 5.1-16 can be 

solved for the nodal deflections. The matrices that result from this model are banded and 

symmetric allowing for a compact storage arrangement. The FEA program written in 

MATLAB® uses Gaussian elimination to solve the overall matrix equation.

Equation 5.1-16

where: [SK] is the global stiffness matrix

{W} is the array of nodal deflections

{Q} is the forcing vector

The FEA program constructed in MATLAB® completes the integrations specified in Equation 

5.1-15 over the array of elements, assembles the global matrices and solves for the nodal 

deflections in Equation 5.1-16. A flowchart illustrating the execution of this program is 

displayed in Figure 7-3. 

w = N{W } δw = N{δW }

w ′ = N ′ {W } δw ′ = N ′ {δW }

w ′ ′ = N ′ ′ {W } δw ′ ′ = N ′ ′ {δW }

S[ 1]e = {N ′ ′ }

T

EIN ′ ′ 

0

L

∫ dx

S[ 2 ]e = {N}

T

kNdx

0

L

∫

{f }e = {N}qdx

0

L

∫

[SK]{W }= {Q}
106

Figure 7-3. Flowchart of Custom FEA Computer Program

The FEA program is an extension and adaptation from similar programs developed in Thompson 

(2005). The actual code is divided into four separate MATLAB® files designated as FEM.m, 

INITIAL.m, SF.m, and sGAUSS.m. The main file, FEM.m, controls the program flow, 

integrates the element matrices, and plots the results. The execution of the program begins by 

loading the quadrature weights and points from a separate text file, QUAD, stored in the program 

directory. The loading is defined in the file INITIAL.m, which also specifies the model 

dimensions and initializes all variables. A program loop is used to traverse through a series of 

static loads as specified by the diagram in Figure 7-3. This method of quasi-static loading is 

described in the next section and allows for a simple calculation of the Yrel measurement found 

with the UNL system.

Within each loading cycle, the element matrices are constructed and then integrated within the 

main program using the SF.m file. This file defines the approximating shape functions and their 

derivatives. The element matrices are successively assembled into the global matrix. Once the 

global matrix is constructed, the boundary conditions are taken into account, and the nodal 

deflections determined using the sGAUSS.m file. The main program displays deflection plots of 

the track for each loading cycle. These plots include graphics of the Yrel and ECO

measurements and the data that would be found by the UNL system. After the specified section 

of track has been traversed, a movie is created by merging together the individual plot frames.

7.1.3 Quasi-Static Loading and Yrel Computation

The main goal of the FEA program is to analyze the track response from the perspective of the 

UNL measurement system. The data is collected from a moving rail car and expressed by the 

Yrel measurement defined previously. To simulate the movement of the railcar along the track,

a quasi-static, rather than a dynamic, approach is used. Placing static loads at specific locations 

along the track and computing the corresponding nodal deflections completes this method. The 
107

static loads are then translated a specified amount (12 in), and the deflections are recomputed. 

This process is repeated until the static loads have traversed far enough to cover a particular track 

feature allowing for the calculation of the corresponding Yrel and ECO data.

The quasi-static approach significantly simplifies the FEA model. To capture the resolution 

needed to compute Yrel and ECO data, the elements need to be small (1-in length). In addition, 

a relatively large number of elements are needed to capture realistic track features. Therefore, 

the quasi-static simplification is expected to significantly decrease computation time compared 

with a dynamic model. 

7.1.4 Model Parameters and Visual Representation

The FEA model is visually represented by the diagram shown in Figure 7-4. Deflections occur 

along the y direction while the track extends along the x direction. 

Figure 7-4. Visual Diagram of FEA Model

Each beam element is 1 inch in length. The element length was chosen through experimentation 

to ensure a fine enough resolution in the deflection curve to capture short-wavelength peaks in 

the Yrel and ECO data. The model encompasses 193.5 ft of track resulting in a total of 2,322 

beam elements. This length makes certain that the loads are applied far enough away from the 

boundaries so that the fixed boundaries do not influence the deflection results. The beam 

element properties represent the 132 RE rail often found on heavy-haul coal lines as shown in 

Figure 7-5.
108

Figure 7-5. 132 RE Rail Section Properties

The static loads are configured to represent two adjacent hopper cars with the dimensions 

displayed in Figure 7-4. Constant load magnitudes of 32,500 lb are applied for each static load. 

To collect enough data to traverse typical track features, 72 static load groups are applied. The 

loads are translated 12 in between each cycle. Therefore, the closest any load will be to a fixed 

end boundary is 60 ft.

The support elements shown in Figure 7-4 are located 18 in apart to simulate the tie spacing of 

real track. Stiffness, rather than modulus, is defined in the FEA model. To simulate the correct 

stiffness, the desired modulus is multiplied by the distance between the support elements. The 

support stiffness for the beam elements between the support elements is set to zero. Each 

support element can have a unique stiffness value along the track.

The custom FEA program written in MATLAB® is applicable for linear elements only. More 

complicated nonlinear support elements and support elements used to represent voids are 

simulated in the commercial FEA software ALGOR®. The advantage of the custom FEA 

program is a significant reduction in computation time as well as automatic calculation and 

plotting of the Yrel and ECO results for each simulation. When ALGOR® is used for a 

simulation, the deflection results must be manually exported to text files. These text files are 

then imported into MATLAB® for the analysis of Yrel and ECO. The overall process is much 

less cumbersome with the custom program.

All of the parameters defined in Figure 7-4 are adjustable within the custom FEA program. The 

element dimensions, element properties, load magnitudes, and support stiffness can all be varied 

between individual simulations. A schematic of all possible simulation variables and 

combinations using either the custom FEA program or ALGOR® is illustrated in Figure 7-6.
109

Figure 7-6. Schematic Representation of Model Input Variables

The diagram in Figure 7-6 illustrates all of the simulation possibilities but does not characterize 

one particular simulation. This type of diagram is used to visualize each simulation in the 

sections that follow. The blue springs signify linear stiffness elements. The dark brown ties 

correspond to a nominal modulus of 3,000 psi, whereas the lighter ties indicate areas with a 

softer supporting foundation. The orange springs represent nonlinear support elements. The 

gaps between some of the springs and ties denote areas of track with voids where some 

deflection occurs before the foundation provides support. The modulus profile is shown at the 

bottom of the figure with the specific values listed.

Another feature that can be included in the model is the pin joint shown in Figure 7-6. This 

element represents a joint that transfers shear forces but cannot transfer any bending moment. 

The pin joint is modeled using a feature known as a “beam end release” in ALGOR®. In the 

custom FEA program, the pin joint is modeled by creating a very small beam element (1.0E–10 

in). Then, the flexural modulus and stiffness for this element are set to zero. The deflection of 

the nodes are kept equal by assigning a large value (1.0E+12) to the (1,1), (1,3), (3,1), and (3,3) 

elements of the [S1] beam-element matrix. The other values in the [S1] matrix are set to zero, 

which allows the slopes of the adjoining elements to be decoupled. This procedure is outlined in 

Thompson (2002).

7.2 Verification of FEA Model

Before proceeding with more complicated simulations, the finite element programs are verified 

by comparing them with the traditional Winkler model. First, the simulation of a single load in 

the custom FEA program is compared with the Winkler model as illustrated in Figure 7-7. In 

this case, a 32,500-pound load was exerted on a foundation modulus of 3,000 psi. As shown by 

the main figure and in the enlarged inset, the FEA results closely match the Winkler model.
110

Figure 7-7. Single-Load Simulation with FEA Program Compared with Winkler Model

Next, two 32,500-pound loads were simulated in the custom FEA program, representing adjacent 

axles of a hopper car. Again, the results correspond very well with the deflections found using 

the Winkler model as displayed in Figure 7-8.

Figure 7-8. Two-Load Simulation with FEA Program Compared with Winkler Model

Verification of the custom FEA program’s more advanced features is not possible using the 

Winkler model. For example, the Winkler model is unable to simulate varying modulus, preexisting geometry, or joints in the rail. However, these features are precisely why the model was 

created. More complex simulation results are presented in the following section.
111

Another simulation was performed using the commercial FEA software ALGOR® and the 

results compared with the Winkler model. In this case, nonlinear gap elements, rather than linear 

elements, were used as the supporting elements in the foundation. These elements provide linear 

support in compression only and cannot be loaded in tension. The effect is that the foundation 

does not “pull down” on the rail in the areas where the track lifts up from the foundation. The 

results of the simulation are shown in Figure 7-9.

Figure 7-9. Gap Element Simulation in ALGOR® Compared with Winkler Model

Within the vicinity where the load is applied, the results match closely with the Winkler model. 

However, beyond approximately 5 ft on either side of the load, the track lifts up with a much 

steeper slope than in the Winkler model. An important fact to note is that the weight of the rail 

has not been included in this simulation. Rather, the simulation was completed to observe how 

the nonlinear gap elements compare with a linear foundation. Because of the slightly larger peak 

deflection and the increased track lift, more advanced simulations using these elements are 

expected to help explain differences observed between Yrel and ECO in field testing with the 

UNL track deflection measurement system. Simulations using these elements are described in 

the following section.

7.3 FEA Analysis and Results

This section describes several simulations performed with the custom FEA program written in 

MATLAB® as well as some nonlinear simulations performed in ALGOR®. The purpose of 

each simulation is to examine how a particular track feature or combination of features affects 

the shapes and magnitudes of Yrel and ECO data. The features simulated include pin joints, soft 

foundation support, pre-existing geometry, voids, and several combinations of the parameters.
112

7.3.1 Pin Joint

The custom FEA program was used to simulate a single pin joint as illustrated in Figure 7-10. In 

this case, the pin joint transfers shear loads but not bending moments. Therefore, this model 

represents the properties of a failing joint (Kerr, 2003). The foundation is modeled using linear 

support elements with a modulus of 3,000 psi. The length of the model (193.5 ft) is not 

represented in the diagram of Figure 7-10 or any of the other simulation schematics to follow. 

These diagrams are intended for visualization of the specific track feature modeled, in this case 

the joint, and these schematics are not of the complete model.

Figure 7-10. Schematic of FEA Model with Pin Joint

As mentioned earlier, the custom FEA program loops through a series of static loads to obtain 

the data needed to compute Yrel and ECO. The nodal deflections due to each load are plotted in 

individual frames and compiled into a movie for visualization as shown in Figure 7-11. The 

movie illustrates the loads moving from right to left along the track and plots the Yrel, ECO, and 

loaded-profile curves. The Yrel data is computed from the instantaneous rail deflection, whereas 

the ECO data is determined from the loaded profile. The loaded profile is calculated from the 

deflection of the leading axle (the inboard axle beneath the red beam on the UNL hopper car). 
113

Figure 7-11. Movie Frame from FEA Simulation with Pin Joint

The final Yrel and ECO results are shown in Figure 7-12. The ratio of ECO to Yrel is 0.54 with 

a maximum Yrel peak of 0.25 in and the maximum ECO at 0.15 in. The shapes of each curve 

closely match those measured in the field by the UNL system (e.g. Figure 6-23).

Figure 7-12. Yrel and ECO Results from FEA Simulation with Pin Joint

The results in Figure 7-12 reveal an interesting point concerning failing joints. Field 

measurements with the UNL system include large Yrel peaks (1 in or more) at the location of 

very poor joints. Certainly, a joint that does not transfer any bending moment would be 

considered to be in very poor condition. However, this feature that represents a joint bar with 
114

loose bolts alone did not produce the magnitude of peak expected. To analyze this phenomenon

further, the following simulation was completed.

7.3.2 Pin Joint with Bad Ties

The custom FEA program was again used to simulate a pin joint but with the addition of two 

poorly supported ties. The linear foundation support was reduced to 100 psi for each tie on 

either side of the joint as displayed in Figure 7-13.

Figure 7-13. Schematic of FEA Model with Pin Joint and Two Bad Ties

As before, a movie was plotted during the simulation to visualize the results. The movie frame 

shown in Figure 7-14 was plotted shortly after the leading axle traversed the pin joint. This 

image reveals the usefulness of the custom FEA program. The deflection of each axle is 

apparent in the movie along with the chord used to compute Yrel. The image provides a way to 

correlate the axle deflections and Yrel chord with the Yrel and ECO data for a particular 

location.

Figure 7-14. Movie Frame from FEA Simulation with Pin Joint and Two Bad Ties
115

The final Yrel and ECO results are shown in Figure 7-15. The ratio of ECO to Yrel is 0.91 with 

the maximum Yrel peak of 1.0 in and the maximum ECO peak of 0.91 in. Obviously the 

addition of two poorly supported ties around the joint led to significantly higher peaks in the Yrel

and ECO data. The sharpness and magnitude of the Yrel peak is similar to data measured in the 

field at the location of failing joints. However, the peak in ECO, which nearly matches the one 

in Yrel, is not always expected as displayed in the bottom portion of Figure 7-15. In fact, as 

described in earlier sections, pre-existing unloaded geometry is expected to be the main 

contribution in the calculation of ECO. Yet, this simulation did not include any unloaded 

geometry effects. This result is discussed in more detail at the end of the section.

Figure 7-15. Yrel and ECO Results from FEA Simulation with Pin Joint and Two Bad Ties

7.3.3 Pin Joint with Voids

The complexity of the simulation is further increased with the inclusion of voids and nonlinear 

elements as shown in Figure 7-16. The voids and nonlinear elements are modeled with gap 

elements in ALGOR®. The size of each void is listed beneath the corresponding element in 

Figure 7-16. A maximum void of one inch occurs beneath the pin joint. The support modulus is 

also slowly decreased to a minimum beneath the joint. 
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Figure 7-16. Schematic of FEA Model with Pin Joint, Bad Ties, and Voids

The use of gap elements prevents the foundation from “pulling down” on the track where it lifts 

up from the foundation. The configuration of the gap elements with voids leads to a nonlinear 

deflection curve as shown in Figure 7-17. This sort of nonlinear deflection curve is similar to the 

one measured by Zarembski and Choros (1980). Despite the traditional use of linear models, the 

actual response of the track is expected to be somewhat nonlinear (Sussmann et al., 2001; Lu, 

2008). The response shown in Figure 7-17 represents the softer support associated with the 

seating load followed by the stiffening of the track as the various components compress together.

Figure 7-17. Nonlinear Deflection Curve for FEA Model with Voids

The final Yrel and ECO results are shown in Figure 7-18. The ratio of ECO to Yrel is 0.93 with 

the maximum Yrel peak at 1.4 in and the maximum ECO peak at 1.3 in. The decrease in support 

led to an increase in the Yrel peak as might be expected. However, the ratio between Yrel and 

ECO remained essentially the same. Therefore, the discrepancy between Yrel and ECO, found 

in real measured data and shown in the bottom of Figure 7-18, must result from some other effect 

not yet present in the simulations.
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Figure 7-18. Yrel and ECO Results from FEA Simulation with Pin Joint, Bad Ties, and 

Voids

7.3.4 Pre-Existing Geometry

To study how pre-existing geometry influences Yrel and ECO data, a 1-inch rise over 30 ft was 

simulated in the track as displayed in Figure 7-19. Again, the diagram is not drawn to scale. The 

foundation modulus was kept constant at 3,000 psi.

Figure 7-19. Schematic of FEA Model with Pre-Existing Geometry

The final Yrel and ECO results from the simulation are shown in Figure 7-20. The ratio of ECO

to Yrel is 0.8 with the maximum Yrel at 0.46 in and the maximum ECO at 0.37 in. The data is 

symmetric about the geometry feature. The geometry was created with two sloped and one 

horizontal track segments as shown in Figure 7-19. These regions are evident in the data as both 

the Yrel and ECO curves peak as the loads move into and out of the sloped regions.
118

Figure 7-20. Yrel and ECO Results from FEA Simulation with Pre-Existing Geometry

The simulation results verify that relatively short wavelength geometry can affect the Yrel

measurement. As shown in Figure 7-20, the Yrel and ECO data match very closely. Therefore, 

these results also validate the method of eliminating the effect of pre-existing geometry as 

described in previous chapters. Since both curves are nearly the same, subtracting ECO from 

Yrel should remove the geometry component in the Yrel measurement. However, the interaction 

between these two measurements with respect to modulus and geometry is complex. For 

example, ECO closely matches Yrel in the other simulations as well despite the absence of preexisting geometry.

7.3.5 Large Mud Hole

The last simulation is completed to examine the effect of poor support over an extended region 

of track. The diagram for this simulation is shown in Figure 7-21. In this case, 10 ties have a 

reduced support modulus of 250 psi.

Figure 7-21. Schematic of FEA Model with 10 Bad Ties
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The Yrel and ECO results from the simulation are shown in Figure 7-22. The ratio of ECO to 

Yrel is 0.91 with the maximum Yrel at 0.53 in and the maximum ECO at 0.48 in. Wider peaks 

in the data result from the long section of low support modulus. Of particular interest is that the 

ECO peak is shifted with respect to the Yrel peak. This effect is present in all simulations when 

the section of low support modulus is longer than the 10-foot chord used to calculate Yrel and 

ECO.

Figure 7-22. Yrel and ECO Results from FEA Simulation with 10 Bad Ties

7.4 Suggestions for Further Development

The results from all of the simulations discussed in the previous section are displayed in Table 

7-1 below.

Table 7-1. Results of FEA Simulations

Simulation Scenario Yrel (in) ECO (in) Ratio: 

ECO/Yrel

Pin joint only (linear elements) 0.28 0.15 0.54

Pin joint with two, 100 psi modulus ties on 

each side (linear elements) 1.0 0.91 0.91

Pin joint with 1-inch gap between joint and 

supporting foundation 1.4 1.3 0.93

Continuous rail with pre-existing geometry 

(3,000 psi modulus) 0.46 0.37 0.8

Continuous rail with 10 ties of 250 modulus 

(linear elements) 0.53 0.48 0.91
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The simulation results provide good insight into how specific track features might affect the 

shape and magnitude of Yrel. Of particular interest, however, is that the ECO data nearly 

matched the Yrel values in each case. This effect is certainly observable in real measured data 

with the UNL system and has been shown at several locations, along with the deteriorated track 

conditions. Yet, these results do not indicate why certain track features lead to large peaks in 

Yrel with much smaller peaks in ECO, as shown in the lower portion of Figure 7-1. Therefore, 

some parameter or combination of parameters present during the actual measurement of this data 

is not represented in the simulations.

Considering how the FEA model is created in comparison to how the real Yrel and ECO data is 

measured may provide a basis for further development of the simulations. One possible 

discrepancy is the exclusion of dynamic interactions between the measurement vehicle and the 

deflection of the track. Dynamic effects were not included in the model to reduce its complexity 

and corresponding computation time. However, previous research has shown that at higher 

speeds, the dynamics should be included in the model (Carr, 1999). The dynamic response of 

Yrel and ECO to short wavelength perturbations may be different, and this may contribute to the 

missing peak in the ECO data. Therefore, it is recommended that further development of these 

FEA simulations include the dynamics of the rail car interacting with the deflection of the track. 

However, discrepancies between Yrel and ECO are observed at speeds as low as 20–25 mph (the 

minimum speed often supported by track profile measurement systems) where dynamic loads 

would not be expected to cause the observed differences in Yrel and ECO. It is believed that 

dynamics can account for some small portion of these differences, but that improved 

understanding of the mechanics of the observed track behavior is necessary to fully explain the 

observed differences.

Another possible contribution to the discrepancy between Yrel and ECO peaks may be the 

method by which ECO is calculated. The ECO measurement is calculated from space curve data 

provided to UNL from track geometry vehicles. Track geometry vehicles use multiple highprecision accelerometers to measure geometric properties of the track. The vertical accelerations 

can be integrated to determine the vertical rail profile. This profile is then used to calculate the 

10-foot ECO. This method raises the possibility that sampling and frequency issues may result 

in missing data peaks for short-wavelength track features. The accelerometers used to measure 

the rail space curve have a specific sampling rate. The data is then filtered to remove lowfrequency (<0.3 hertz) and high-frequency (>60 hertz) noise. The resulting signal is then 

integrated, filtered, and integrated again to obtain the vertical displacement of the rail profile. 

Finally, the ECO data is computed from the resulting space curve profile. This procedure may 

result in lower peaks of ECO than Yrel at the location of short-wavelength track perturbations (at 

failing joints, for example). Again, these limitations may contribute in part to the difference 

observed between Yrel and ECO.

To study how the real computation of ECO compares with the ECO measurement found in the 

simulations, accelerometers have been added to the UNL system as described in Section 3. 

These accelerometers are mounted to the bearing adapters above the inboard axle (axle 3 of the 

UNL car). These accelerometers have been used in a few tests, but as of yet, have not produced 

conclusive results. Ongoing development is underway to match the UNL accelerometer output 

with data provided by track geometry vehicles. The goal is that these instruments will eliminate 
121

the need for data provided by the track geometry vehicles and verify the computation of ECO 

used in the UNL deflection measurement system.

The sites that show a difference between Yrel and ECO are likely the most significant finding of 

this research. These sites are the focus of research to understand the complex track behavior and 

dynamics that occur in deteriorated track sites. These sites are significant and often exist on real 

track and have been commonly observed during tests. It is believed that these locations have 

nonuniform support and the track behavior at these locations is complex. It is also believed that 

these differences cannot be fully explained with vehicle dynamics and measurement differences 

or sampling issues. These sites not behaving according to expected standards may pose a risk to 

the safe maintenace of track. More analysis, and most importantly, field tests and site 

instrumentation data are needed to fully understand these sites so that the risks can be more fully 

understood and the data from the UNL measurement system can be properly thresholded to 

identify and prioritize these potentially significant problem sites. 
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8 Summary and Conclusions

UNL has been conducting research sponsored by FRA’s Office of Research and Development to 

create a system to measure VTD/modulus from a moving rail car. Previous work has suggested

that the system can identify critical maintenance problems not detected by other inspection 

methods including standard track geometry.

This report presents the theoretical basis for the UNL-developed measurement system, as well as

a review of mathematical representations of track stiffness and resulting track support analysis. 

FEA analysis is presented that represents quasi-static track analysis with a complex track model 

with nonlinear and varying stiffness. Also, a new theoretical model for track, along with a 

nontraditional definition of track modulus, is presented that represents a cubic relationship 

between the applied distributed load and the track vertical deflection.

The system has been implemented and significant field testing has been performed. The 

prototype field system is described along with a calibration procedure. Results of a field 

validation test are presented that show the measurement is accurate and repeatable. Finally, 

results of field tests are presented along with some interpretation and analysis of various sites 

including the relative changes in stiffness over joints, bridges, and approaches.

The work presented here has been closely analyzed by personnel at FRA and the John A. Volpe 

National Transportation Systems Center (Volpe Center). FRA, Volpe Center, and UNL will 

continue to consider the following issues:

1) There is indication that the UNL measurement, designated as Yrel, can locate and 

systematically quantify track problem sites.

2) The UNL measurement (Yrel) is different from standard geometry measurements (e.g.,

ECO calculated from loaded profile information).

3) The UNL system can identify critical maintenance problems not found by other 

inspection methods including short wavelength processed track geometry or VTI data.

4) Some guidelines are suggested here to form a basis of thresholds for interpreting Yrel and 

associated track deflection data, but more study and tests are needed to define threshold 

criteria.

5) More work including field tests and site specific measurements are needed to better 

characterize the complex behavior of locations and evaluate the risk at sites where 

advanced track deterioration has been associated with the trend of the UNL measurement

(Yrel) significantly exceeding comparable ECO data from track geometry.
123

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Abbreviations and Acronyms

ASCE-AREA American Society of Civil Engineers/

BNSF BNSF Railway

BOEF beam on an elastic foundation

BVP boundary value problem

CEP circular error probable

cm centimeter

DAQ data acquisition

ECO end chord offset

FEA finite element analysis

ft feet

GDOP geometric dilution of precision

GPa gigapascal

GPR ground-penetrating radar

GPS global positioning system

GRMS Gage Restraint Measurement System

in inch

km/hr kilometers per hour

kn kilonewton

lb pound

lbf pound-force

LVDT Linear Variable Differential Transformers

m meter

MGT million gross ton

mm millimeter

MN/m millinewton per meter

MPa megapascal

mph mile per hour

N Newton

PC personal computer

psi pounds per square inch
126

PVM precision measurement vehicle

TLV track loading vehicle

UHF/VHF ultrahigh frequency/very high frequency

UNL University of Nebraska-Lincoln

UPRR Union Pacific Railroad

Volpe Center John A. Volpe National Transportation Systems Center

VTD vertical track deflection

VTI vehicle track interaction