iii

U.S. Department of 

Transportation

Federal Railroad 

Administration

Track Buckling Prevention: Theory, Safety 

Concepts, and Applications

Track Systems Safety

Office of Research

and Development

Washington, D.C. 20590

DOT/FRA/ORD-13/16 Final Report

March 2013

This document is available to the

public through the National Technical

Information Service, Springfield, VA 22161.

This document is also available on the FRA 

Web site at www.fra.dot.gov.
ii

NOTICE

This document is disseminated under the sponsorship of the 

Department of Transportation in the interest of information 

exchange. The United States Government assumes no liability for 

its contents or use thereof. Any opinions, findings and 

conclusions, or recommendations expressed in this material do not 

necessarily reflect the views or policies of the United States 

Government, nor does mention of trade names, commercial 

products, or organizations imply endorsement by the United States 

Government. The United States Government assumes no liability 

for the content or use of the material contained in this document.

NOTICE

The United States Government does not endorse products or 

manufacturers. Trade or manufacturers’ names appear herein 

solely because they are considered essential to the objective of this 

report.
iii

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

 March 2013

3. REPORT TYPE AND DATES COVERED

 Final Report

4. TITLE AND SUBTITLE

Track Buckling Prevention: Theory, Safety Concepts, and Applications

5. FUNDING NUMBERS

RR219/R2011; RR319/RR3011; 

RR419/AB019 

6. AUTHOR(S)

Andrew Kish and Gopal Samavedam*

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

U.S. Department of Transportation

Research and Innovative Technology Administration

John A. Volpe National Transportation Systems Center

Cambridge, MA 02142

8. PERFORMING ORGANIZATION REPORT

 NUMBER

DDTS.020104

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

U.S. Department of Transportation

Federal Railroad Administration

1200 New Jersey Ave., SW

Washington, DC 20590

10. SPONSORING/MONITORING AGENCY

 REPORT NUMBER

DOT/ FRA/ORD-13/16

11. SUPPLEMENTARY NOTES

*Foster-Miller, Inc.

350 Second Avenue

Waltham, MA 02451-1196

12a. DISTRIBUTION/AVAILABILITY STATEMENT

This document is available to the public through the National Technical Information Service, 

Springfield, VA 22161 and on the FRA Web site at www.fra.dot.gov. 

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This report is a part of the John A. Volpe National Transportation Systems Center’s Track Stability Research Program for the Federal Railroad 

Administration on thermal buckling of continuous welded rail (CWR) track and its prevention. Presented in this report are the developments of 

theoretical results and the development and application of the CWR-SAFE computer software model for prediction of CWR track buckling 

strength. This comprehensive predictive model encompasses several different modules designed to perform both deterministic and 

probabilistic buckling analyses, based on the dynamic buckling theory previously validated by tests, and predicts safe limits for buckling 

prevention. The model accounts for all the important parameters influencing track buckling, such as rail size, curvature, lateral resistance, tieballast friction, fastener torsional and longitudinal resistances, track vertical stiffness, misalignment amplitude and wavelength, and vehicle 

parameters. Applications of the model are demonstrated through analyses of parametric sensitivity, development of buckling safety limits in 

terms of safe and critical temperatures, and evaluation of annual probability of buckling occurrences for typical CWR line segments. The 

report also presents techniques to determine the input parameters for CWR-SAFE application and a practical methodology for CWR track 

safety monitoring. A risk-based approach is proposed to provide more flexibility to the industry in achieving a minimum number of annual 

buckles in a given territory and to provide science-based guidelines for improved slow order policies when operating at elevated rail 

temperatures.

14. SUBJECT TERMS

Track Buckling, Lateral Stability, CWR, Buckling Prevention, Buckling Models, CWR 

Safety and Maintenance, Neutral Temperature, Rail Temperature

15. NUMBER OF PAGES

168

16. PRICE CODE

17. SECURITY CLASSIFICATION OF

 REPORT

Unclassified

18. SECURITY CLASSIFICATION

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Unclassified

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20. LIMITATION OF 

ABSTRACT

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-9)

Prescribed by ANSI Std. 239-18

298-102
iv

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
v

PREFACE

This report provides a current state-of-the-art understanding of continuous welded rail (CWR) 

response and behavior and its impact on track safety and performance. The report is based on 

U.S. research over the past 20 years, including the development of versatile computer analysis 

software called CWR-SAFE. CWR-SAFE is a product of Volpe Center, Foster-Miller, Inc. 

(FMI), and Federal Railroad Administration (FRA) research under the latter’s Track and 

Structures Program initiatives in line with the FRA’s Five-Year Strategic Plan for Railroad 

Research, Development, and Demonstration (2002). The work reported herein was performed 

with the support of FMI under Contract DTRS-57-99-D-0003, Task Order #14, under the 

technical direction of Dr. Andrew Kish of the US Department of Transportation’s Volpe Center

and FMI’s Project Manager and Principal Investigator, Dr. Gopal Samavedam. The FRA 

sponsor was Dr. Magdy El-Sibaie of the Office of Research & Development, whose assistance 

and support is greatly acknowledged. Additional thanks also go to Mr. Wesley Mui of the Volpe 

Center for performing the CWR-SAFE parametric studies.

The report is intended to synthesize into one reference volume the various CWR relevant 

research results obtained at different periods during a 20-year program duration. The report 

covers the following research items performed from 1984 to 2004:

• Development of vehicle induced (dynamic) buckling theory for the determination of critical 

conditions leading to track buckling

• Identification and determination of track and vehicle parameters in the theory

• Development of a versatile computer software (CWR-SAFE) for application of the theory

• Parametric analyses of track buckling

• Development of buckling safety limits

• Tests to validate the theory and safety limits

• Determination of annual probability of buckling estimates for typical CWR line segments 

based on probabilistic analysis

• Methodologies to implement and monitor the safety limits for improved buckling prevention

• Development of science-based guidelines for slow order rationale for vehicle operations at 

high rail temperatures
vi

CONTENTS

1. INTRODUCTION .................................................................................................................... 6

1.1 Technical Problems.......................................................................................................... 6

1.1.1 Track Buckling................................................................................................................. 6

1.1.2 Rail Pull Apart ................................................................................................................. 8

1.2 Track Buckling Accident Severity................................................................................... 9

1.3 Background.................................................................................................................... 11

1.3.1 CWR Theory.................................................................................................................. 11

1.3.2 Buckling Parameters...................................................................................................... 11

1.3.3 Computer Programs....................................................................................................... 12

1.3.4 CWR Maintenance......................................................................................................... 12

1.4 Report Content/Organization......................................................................................... 13

2. THE THEORY OF TRACK STABILITY ............................................................................. 15

2.1 Buckling Mechanism ..................................................................................................... 15

2.1.1 Force Drop/Buckle Influence Zone ............................................................................... 17

2.1.2 Buckling Mode Shapes.................................................................................................. 18

2.2 Track Buckling Theories................................................................................................ 19

2.2.1 Static versus Dynamic Buckling Model ........................................................................ 19

2.2.2 Tangent versus Curved Tracks....................................................................................... 21

2.2.3 Differential Equation Formulations............................................................................... 21

2.2.4 Buckling Response Computation................................................................................... 22

2.2.5 Parameters Influencing Buckling Response .................................................................. 23

2.3 Buckling Safety Criteria ................................................................................................ 25

2.3.1 Background.................................................................................................................... 25

2.3.2 Buckling Safety Criteria ................................................................................................ 26

2.4 Buckling Safety Evaluation Methodology..................................................................... 28

2.4.1 Key Controlling Parameters........................................................................................... 29

2.5 Summary........................................................................................................................ 30

3. TRACK BUCKLING MODEL: CWR-SAFE....................................................................... 31

3.1 Basis of CWR-SAFE Modules...................................................................................... 31

3.1.1 CWR-BUCKLE............................................................................................................. 31

3.1.2 CWR-INDY ................................................................................................................... 37

3.1.3 CWR-RISK.................................................................................................................... 45

3.1.4 Summary........................................................................................................................ 52

4. TRACK BUCKLING PARAMETERS AND SENSITIVITY STUDY ................................ 54

4.1 Parametric Study Basis.................................................................................................. 54

4.2 Static Versus Dynamic Buckling................................................................................... 54

4.3 Influence of Rail Size..................................................................................................... 55

4.4 Influence of Lateral Resistance...................................................................................... 57

4.5 Influence of Torsional Resistance.................................................................................. 63

4.6 Effects of Longitudinal Resistance ................................................................................ 65

4.7 Influence of Track Curvature......................................................................................... 66

4.8 Effects of Initial Misalignments..................................................................................... 68

4.9 Effects of Track Foundation Vertical Stiffness ............................................................. 70

4.10 Effects of Vehicle Parameters........................................................................................ 71
vii

4.11 Effects of Track Structure: Concrete versus Wood Ties ............................................... 73

4.12 Summary........................................................................................................................ 75

5. PROBABILISTIC EVALUATIONS OF TRACK STABILITY ........................................... 77

5.1 Annual Number of Buckles ........................................................................................... 77

5.1.1 Expected Number of Buckles per Annum ..................................................................... 78

5.1.2 Numerical Example ....................................................................................................... 79

5.1.3 Baseline Case ................................................................................................................. 80

5.1.4 Slow Orders ................................................................................................................... 83

5.2 Parametric Studies ......................................................................................................... 84

5.2.1 5° Curve ......................................................................................................................... 84

5.2.2 Performance Comparisons............................................................................................. 87

5.2.3 Tangent Track ................................................................................................................ 90

5.2.4 Slow Orders Parametric Study....................................................................................... 91

5.3 Summary........................................................................................................................ 91

6. SAFETY ASPECTS OF CWR............................................................................................... 93

6.1 Safety Limits.................................................................................................................. 93

6.1.1 Deterministic Method .................................................................................................... 93

6.1.2 Probabilistic Approach................................................................................................... 99

6.2 Safety Limit Validations.............................................................................................. 100

6.2.1 Deterministic Limits .................................................................................................... 100

6.3 Safety Limit Implementation ....................................................................................... 101

6.3.1 Lateral Resistance ........................................................................................................ 102

6.3.2 Neutral Temperature .................................................................................................... 103

6.3.3 Misalignment ............................................................................................................... 107

6.4 Track Buckling Mitigation........................................................................................... 107

6.5 Summary...................................................................................................................... 108

7. CONCLUSIONS AND RECOMMENDATIONS ............................................................... 110

7.1 Conclusions.................................................................................................................. 110

7.2 Recommendations........................................................................................................ 112

8. REFERENCES ..................................................................................................................... 114

APPENDIX A - MATHEMATICAL FORMULATION OF CWR-BUCKLE ......................... A-1

APPENDIX B - TECHNIQUES FOR PARAMETER MEASUREMENTS............................. B-1

APPENDIX C - THEORETICAL BASIS OF CWR-RISK ....................................................... C-1

APPENDIX D - MEASUREMENT REQUIREMENTS AND SAMPLE RATES FOR 

LATERAL RESISTANCE AND NEUTRAL TEMPERATURE................... D-1

APPENDIX E - RAIL DESTRESS FORCE INDICATOR (RDFI) ...........................................E-1

LIST OF ABBREVIATIONS AND ACRONYMS ....................................................................F-1

LIST OF SYMBOLS .................................................................................................................. G-1
viii

FIGURES

Figure 1-1. Lateral Track Buckling ............................................................................................... 7

Figure 1-2. Track Buckling Statistics .......................................................................................... 10

Figure 2-1. Pre and Post Buckled Track Configurations............................................................. 15

Figure 2-2. Buckling Response Curves ....................................................................................... 16

Figure 2-3. Track Quality Influence on Buckling Response Characteristics............................... 17

Figure 2-4. Rail Force Distribution after Buckling...................................................................... 18

Figure 2-5. CWR Track Lateral Buckling Modes ....................................................................... 18

Figure 2-6. Track Model.............................................................................................................. 19

Figure 2-7. Definition of Uplift Waves........................................................................................ 20

Figure 2-8. Typical Lateral Resistance Characteristics............................................................... 23

Figure 2-9. Track Misalignment .................................................................................................. 24

Figure 2-10. Rail Beam Sectional Properties............................................................................... 24

Figure 2-11. Energy Required to Buckle ..................................................................................... 26

Figure 2-12. Illustration of Safety Criterion ................................................................................ 27

Figure 2-13. CWR Track Buckling Safety Evaluation Methodology.......................................... 29

Figure 3-1. Response of Track under Vehicles............................................................................ 32

Figure 3-2. Progressive Buckling Test (Tangent)........................................................................ 33

Figure 3-3. Dynamic Buckling of Curved Track......................................................................... 33

Figure 3-4. Strip Chart Record for 5-Degree Curve Test ............................................................ 34

Figure 3-5. Buckling Deflection versus Temperature Increase above Neutral Temperature ...... 36

Figure 3-6. Consolidation Influence - Wood Ties for Equation 3-2............................................ 40

Figure 3-7. Consolidation Influence - Concrete Ties for Equation 3-3 ....................................... 41

Figure 3-8. Schematic of Interference Principle .......................................................................... 47

Figure 3-9. Buckling Risk Evaluation Methodology................................................................... 48

Figure 3-10. Assumed Rail Neutral Temperature Distribution ................................................... 49

Figure 3-11. Assumed Lateral Resistance Distribution ............................................................... 50

Figure 3-12. Assumed Misalignment Amplitude Distribution for Class 4 Track........................ 51

Figure 3-13. Probability of Buckling vs. Rail Temperature ........................................................ 52

Figure 4-1. Static Versus Dynamic Buckling Behavior............................................................... 55

Figure 4-2. Influence of Rail Size................................................................................................ 56

Figure 4-3. Lateral Resistance Idealizations................................................................................ 57

Figure 4-4. Influence of Peak Lateral Resistance, FP .................................................................. 59

Figure 4-5. Influence of Limiting Resistance .............................................................................. 60

Figure 4-6. Buckling Response Behavior for Weak/Average/Strong Tracks.............................. 61

Figure 4-7. Influence of Tie/Ballast "Friction" Coefficient......................................................... 62

Figure 4-8. Torsional Stiffness Influence on Buckling Response ............................................... 64

Figure 4-9. Influence of Torsional Stiffness on Buckling Temperatures .................................... 64

Figure 4-10. Typical Longitudinal Resistance Characteristic...................................................... 65

Figure 4-11. Influence of Longitudinal Resistance...................................................................... 66

Figure 4-12. Effect of Curvature on the Buckling Response Behavior....................................... 67

Figure 4-13. Influence of Curvature on Buckling Temperatures................................................. 68

Figure 4-14. Influence of Misalignment Amplitude on Buckling Temperatures ........................ 69

Figure 4-15. Influence of Misalignment Wavelength on Buckling Temperatures...................... 70

Figure 4-16. Influence of Track Foundation Vertical Stiffness................................................... 71
ix

Figure 4-17. Influence of Axle Load on Buckling Temperatures................................................ 72

Figure 4-18. Influence of TCS on Buckling Temperatures ......................................................... 73

Figure 4-19. Buckling Response for “Average” Wood versus Concrete Tie Tracks. ................. 74

Figure 4-20. Buckling Response for “Weak” Wood versus Concrete Tie Tracks....................... 75

Figure 5-1. Rail Temperature Distribution .................................................................................. 80

Figure 5-2. Distributions for Baseline Case A............................................................................. 81

Figure 5-3. Buckling Probability at a Given Temperature, Case A............................................. 81

Figure 5-4. Number of Buckles versus Rail Temperature (Case A)............................................ 83

Figure 5-5. Permissible Speeds for Case A ................................................................................. 84

Figure 5-6. Distributions for Case B............................................................................................ 85

Figure 5-7. Distributions for Case C............................................................................................ 85

Figure 5-8. Distributions for Case D............................................................................................ 86

Figure 5-9. Distributions for Case E............................................................................................ 86

Figure 5-10. Number of Buckles versus Rail Temperature (Case A).......................................... 88

Figure 5-11. Number of Buckles versus Rail Temperature (Case B).......................................... 88

Figure 5-12. Number of Buckles versus Rail Temperature (Case C).......................................... 89

Figure 5-13. Number of Buckles versus Rail Temperature (Case D).......................................... 89

Figure 5-14. Number of Buckles versus Rail Temperature for Tangent Track (Case A)............ 90

Figure 5-15. Permissible Speeds for All Cases............................................................................ 91

Figure 6-1. Buckling Safety Limits Chart for Tangent with 0.5” Line Defect............................ 94

Figure 6-2. Buckling Safety Limits Chart for 3° Curve with 0.5” Line Defect........................... 95

Figure 6-3. Buckling Safety Limits Chart for 5° Curve with 0.5” Line Defect........................... 95

Figure 6-4. Buckling Safety Limits Chart for 7° Curve with 0.5” Line Defect........................... 96

Figure 6-5. Buckling Safety Limits Chart for Tangent with 1.5” Line Defect............................ 96

Figure 6-6. Buckling Safety Limits Chart for 3° Curve with 1.5” Line Defect........................... 97

Figure 6-7. Buckling Safety Limits Chart for 5° Curve with 1.5” Line Defect........................... 97

Figure 6-8. Buckling Safety Limits Chart for 7° Curve with 1.5” Line Defect........................... 98
x

TABLES

Table 3-1. Input Parameters for CWR-BUCKLE Illustrative Example ...................................... 35

Table 3-2. CWR-BUCKLE Output Buckling Results Summary................................................. 36

Table 3-3. CWR-BUCKLE Output – Buckling Safety Analysis................................................. 37

Table 3-4. Resistances Tamped Conditions................................................................................. 39

Table 3-5. β Values for Equation 3-4 .......................................................................................... 41

Table 3-6. Torsional Stiffness, τo (for two rails) ......................................................................... 43

Table 3-7. Longitudinal Stiffness (for two rails) ......................................................................... 43

Table 3-8. Input Parameters for INDY Illustrated Example........................................................ 44

Table 3-9. CWR-Indy Output – Buckling Safety Analysis ......................................................... 45

Table 3-10. Deterministic Input Parameters Required for CWR-RISK Analysis....................... 51

Table 4-1. Ranges and Nominal Values of Buckling Parameters................................................ 54

Table 4-2. Typical Rail Properties............................................................................................... 56

Table 4-3. Correlations for “Softening" Lateral Resistance ........................................................ 58

Table 4-4. Three Components of Resistance and Their Assumed Variations with μf................. 62

Table 4-5. Torsional Stiffness Values.......................................................................................... 63

Table 4-6. Typical Measured Longitudinal Stiffness Values ...................................................... 66

Table 4-7. Desired Parameters for High Buckling Strength........................................................ 76

Table 5-1. Annual Rail Temperature Data................................................................................... 79

Table 5-2. Input Distribution for Baseline Case A ...................................................................... 80

Table 5-3. Assumed Curved Track Characteristics ..................................................................... 82

Table 5-4. Comparison of Track Cases........................................................................................ 90

Table 6-1. Minimum Resistance Values for Neutral Temperature of 60 °F and Rail 

Temperature of 140 °F.................................................................................................98

Table 6-2. Minimum Resistance Requirement for Neutral Temperature of 70 °F and Rail 

Temperature of 140 °F.................................................................................................99

Table 6-3. Example Specifications on a Territory for Normal Freight (no Hazmat) ................ 100

Table 6-4. Buckling Safety Limit Test Results.......................................................................... 101
xi

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1

Executive Summary

This report provides a comprehensive review of the research performed in the area of continuous 

welded rail (CWR) track safety and performance culminating in the development of CWR-SAFE 

computer analysis software under the sponsorship of the Federal Railroad Administration (FRA). 

The report also attempts to synthesize the various research items performed between 1984 and 

2004 and compiles the results into one reference volume addressing the following key areas:

• Development of vehicle-induced (dynamic) buckling theory for the determination of 

critical conditions leading to track buckling

• Identification and characterization of track and vehicle parameters in the theory

• Development of the CWR-SAFE computer software for application of the theory

• Parametric analyses of track buckling

• Development of buckling safety limits

• Tests to validate the theory and safety limits

• Methodologies to implement and monitor the safety limits for improved buckling 

prevention

• Determination of probabilistic approaches to buckling evaluations

• Applications of probabilistic models to risk-based safety and maintenance issues

The theory embodied in CWR-SAFE includes both deterministic and probabilistic approaches; 

the latter was also used to develop risk-based safety limits for rail vehicle operations at elevated 

rail temperatures, including mitigation measures, such as risk index-based slow-order 

procedures.

Theoretical Aspects

The report reviews and presents the dynamic buckling theory for CWR track buckling 

evaluations. Track buckling under a moving train is found to be more prevalent than static 

buckling (i.e., without any train action). This is attributed to resistance loss resulting from uplift 

tendency of the track between the trucks of cars. A number of parameters including lateral 

resistance, longitudinal resistance, torsional resistance, rail section properties, track modulus, and 

vehicle parameters (such as truck center spacing and axle loads) have been identified as the key 

factors governing track buckling response. With large deflection theory, the appropriate 

nonlinear differential equations governing lateral deflection in the buckled zone and the 

longitudinal displacement in the adjoining zones are formulated. A Fourier series technique has 

been developed for a fast converging solution of the differential equations. The solution 

typically represents a nonlinear stability relationship between the lateral deflection and the rail 

temperature increase, which identifies two salient temperature points. One is the upper critical 

temperature, TBmax, at which the CWR track buckles explosively with no external energy. The 

other is TBmin, the lower critical temperature, at which the track can buckle with some external 

disturbance such as due to the moving train. The actual buckling temperature regime is between 

these two temperatures. In some instances (defined as progressive buckling), such a regime does 

not exist, in which case the critical temperature is defined as TP (a temperature at which 
2

increased lateral displacements begin to occur with small increases in rail temperature). Such 

progressive buckling, typified by a much slower or more gradual increase in lateral alignment, 

usually occurs at tracks with weak lateral resistance.

A number of full-scale static and dynamic buckling tests conducted at the Transportation 

Technology Center (TTC) validated the theory. In these tests, the track was instrumented to 

monitor rail longitudinal force and to measure uplift; then, the rails were artificially heated to 

induce the thermal load, and various buckling characteristics (such as explosive and progressive) 

were simulated on tangent and curved test segments. This report briefing summarizes some of 

the salient test results.

Buckling Parameters Characterization

Knowledge of key parameters, their characteristics, and their determination are very important 

aspects of CWR safety and performance. Parametric studies have identified that the peak value 

of the resistance influences the upper critical temperature, whereas the steady or limiting value 

influences the lower critical temperature. Nonlinear characteristic were determined in the field 

for various wood and concrete tie track and are described in the report. Also, by loading the ties 

vertically, the friction coefficient between the ballast and the tie can also be determined. This 

loaded resistance is also an important parameter affecting the buckling strength of CWR tracks 

because it influences the uplift regime, which controls the buckling process. The report also 

describes measurement techniques used to determine the other resistance parameters of 

longitudinal and torsional resistances.

Determination of neutral temperature (or the rail’s stress free temperature), which controls the 

longitudinal force in the rail, has been a challenging research problem to date. The Rail Uplift 

Technique has been developed as a means of determining the rail force and the neutral 

temperature, in addition to the use of strain gages. The latter requires rail cutting for zero 

reference condition determination. The rail uplift technique requires unfastening the rail from 

the ties and the application of an uplift force to produce a vertical deflection. Various neutral 

temperature tests have identified many causes for its variation from the initial installation 

condition, including excessive rail creep because of ineffective fasteners/anchors, inadequate 

readjustment in conjunction with rail repair and destressing, curve pull-in/pull-out movement, 

and various maintenance practices, such as curve realignment, tie renewal, and surfacing. Some

of these tests and analyses of buckled track data indicate occurrences of neutral temperature 

reductions down to the 50–60 °F range from initially installed values of 90–110 °F, and buckling 

analyses confirm that such low values can be highly instrumental in producing buckling prone 

conditions.

Computer Programs

This report discusses the computer program CWR-SAFE, which consists of three modules, 

CWR-BUCKLE, CWR-INDY, and CWR-RISK, developed under this research effort to evaluate 

buckling strength and perform safety analysis of CWR track. The program and all the modules 

operate in the Microsoft Windows environment and come with easily accessible User’s Guides 

in the software’s Self-Help Menu.

CWR-BUCKLE is a program for a deterministic evaluation of buckling strength and safety. The 

input parameters are user specified based on the specific track conditions being analyzed. These 

include lateral resistance, torsional and longitudinal resistances, tie-ballast friction coefficient, 
3

track curvature, tie type, rail size, misalignment amplitude and wavelength, rail neutral 

temperature, maximum expected rail temperature, and vehicle type. Default values are provided 

for the parameters to guide the user in performing average or baseline calculations. The output 

includes a buckling curve in the form of a lateral deflection versus temperature increase, which 

provides information on the track’s possible equilibrium (or buckled) states. Pre- and postbuckling longitudinal rail forces and buckling energies are predicted, which are important in 

some buckling evaluations. The output of CWR-BUCKLE includes two critical temperatures, 

TBmin and TBmax bounding the buckling temperature regimes, allowable or safe rail temperature 

for buckling prevention, and the buckling margin of safety (i.e., the reserve strength against 

buckling). Recommendations are provided where the margin of safety is inadequate on methods 

to increase it.

CWR-INDY is a program for a quick evaluation of CWR track buckling strength and intended 

for an easier on-the-spot industry use. The inputs for this program are simpler than those 

required for CWR-BUCKLE. The CWR-INDY inputs are based on information, such as the 

ballast depth, shoulder width, tie types, track class, and traffic tonnage, which can be more easily 

obtained and does not require complex measurements. The program converts these inputs 

internally into their scientific counterparts and uses the CWR-BUCKLE’s computational engine 

to perform the calculations and to evaluate buckling strength and safety. The conversions from 

CWR-INDY inputs into their scientific CWR-BUCKLE counterparts are based on empirical 

equations developed through extensive test programs on revenue service lines and tracks at the 

U.S. Department of Transportation’s TTC. The outputs of CWR-INDY are critical temperatures, 

allowable rail temperature, and a track’s (numerical) lateral resistance value. Thus, CWR-INDY 

also serves as a track lateral resistance estimator, without an actual physical test conducted.

CWR-RISK is a computer program that evaluates the probability of buckling on the basis of 

statistical principles and input parameters. Since lateral resistance, lateral misalignment, and 

neutral temperature have been identified as the primary parameters with significant influence on 

track buckling and which tend to be highly variable quantities, statistical distributions are used to 

describe them. With these statistical inputs and other deterministic parameters, CWR-RISK 

evaluates the probability of buckling at a given rail temperature. The analysis identifies a critical 

temperature, Tc, above which a finite probability of buckling exists.

Probabilistic Approach

The probabilistic approach of CWR-RISK is extended to evaluate the expected number of 

buckles per year on a given territory. This helps to relate the probability of buckling as a percent 

occurrence (which is hard to interpret in industry applications) to a more usable parameter, such 

as the number of expected buckles per year on a specified line segment. This evaluation requires 

annual temperature and lateral resistance frequencies, misalignment amplitudes, and neutral 

temperatures in the territory. The application of CWR-RISK shows that the expected annual 

number of buckles increases with the increase in rail temperature beyond the critical limit Tc. A 

limiting temperature, TL, can be chosen by the industry to limit the annual number of buckles to 

an accepted value.

The probabilistic approach can be used in a decisionmaking process on the imposition of slow 

orders (i.e., restrictions on vehicle speed during periods of elevated rail temperatures). Presented 

is a method for selecting the rail temperature up to which full speed may be permissible. Beyond 
4

this temperature, a speed reduction regime is specified in which the operating speed is reduced to 

minimize the possible consequences of a buckle.

Parametric Analyses of Track Buckling

The parametric study, using CWR-BUCKLE, revealed information of practical significance by 

identifying key variables and evaluating their sensitivity on buckling strength. For example, the 

peak value of lateral resistance has a significant influence on the upper critical temperature, 

TBmax, as has the track’s lateral alignment condition. The lower buckling temperature, TBmin, is 

influenced by the nonlinear part of the lateral resistance (referred to as the limiting resistance), 

torsional resistance, and lateral misalignment. Track curvature strongly influences both 

temperatures. Vehicle truck center spacing and axle loads also impact buckling temperatures 

through their influence on the central uplift. Study results indicate that for CWR buckling safety 

considerations, the TBmin values (hence parameters influencing it) are the more important ones 

because they drive the safety criterion. Special cases addressing conditions and features such as

static versus dynamic buckling, buckling comparisons for weak versus strong lateral resistance 

track, and wood versus concrete tie track have also been presented with important implications 

on their respective buckling behavior and the key parameters influencing it.

Safety Limits

Safety limits for buckling prevention using deterministic (CWR-BUCKLE) and probabilistic 

(CWR-RISK) approaches have been proposed. The deterministic limits are reasonably well 

developed, whereas the probabilistic approach limits need further evaluation because of a lack of 

data on parameter variation for the required statistical representations. Deterministic safety 

limits are based on TBmin for explosive buckling and TP for progressive scenarios. The allowable 

rail temperature, Tall (sometimes referred to as the safe rail temperature), is chosen to have a 

margin of safety of at least 10 °F. The report provides typical safety limit charts to illustrate the 

concept and the development process.

The safety limits concepts were evaluated and validated by testing various CWR track segments 

after heating the rails artificially up to the allowable temperature. Two tangent tracks, two 5°

curved tracks and one 7.5° curve, were used in the validations, and the results are briefly 

summarized in the report.

Safety Implementation

To facilitate industry use and application, the safety limits on the allowable rail temperature are 

translated into limits for the lateral resistance and neutral temperature. For ease of 

implementation, the requirements on lateral resistance and neutral temperature are uncoupled 

because the lateral resistance parameter must be above certain minimum values for buckling 

prevention at a prescribed rail temperature. Based on this approach, this report presents 

methodologies for monitoring the lateral resistance and neutral temperature.

To monitor the lateral resistance condition, for example, direct measurements on weak or 

substandard locations are recommended. Under some conditions (such as for recently 

maintained and stabilized tracks), monitoring can also be accomplished using CWR-INDY’s 

lateral resistance predictor, thereby requiring no physical testing.

Monitoring rail neutral temperature is more difficult because of the lack of an accurate, 

nondestructive measurement technique. In the interim, the application of existing measurement 
5

technologies based on Rail Uplift Device (RUD) in conjunction with strain gage is 

recommended. The RUD concept, which was field demonstrated in the late 1980s, can be used 

to calibrate the strain gage without cutting the rail (which is required to obtain the zero 

calibration value). In situations where the rails are being cut as part of the maintenance action 

(such as when inserting rail plugs when performing rail repair or destressing), the applications of 

strain gages is recommended for monitoring the neutral temperatures at those locations. In the 

event of no measurement being feasible, the assumption of a “neutral temperature safety factor” 

is recommended for CWR buckling safety evaluation and assurance.

In addition to monitoring methods, this report discusses control of the parameters. For example, 

the lateral resistance can be controlled to a minimum desired value through several means, 

including consolidation either through traffic/tonnage or through dynamic track stabilization, 

using good quality ballast with full ballast section including wide shoulders, and through limiting 

seasonal curve lateral movement. Better control on neutral temperature variation could be 

attained through more effective rail break repair and destressing procedures, as well as through

minimizing rail movement by providing and maintaining effective longitudinal restraint.

The report also presents conclusions of practical interest and recommendations for further 

research. Appendices provide additional information on buckling theory formulation/equations 

and on key parameter measurement techniques and measurement requirements.
6

1. Introduction

Over the past two and a half decades, significant research work was performed in the field of 

CWR safety and performance because of its increasing use as a replacement for jointed track. 

Improved ride quality, increased rail and rolling stock fatigue life, and reduced track 

maintenance costs were the principal motivations to changing to CWR. However, with the 

elimination of the joints, the constrained thermal expansion and contraction-induced compressive 

and tensile forces created the possibility of track buckling in the summer and tensile pull-apart 

failures in the winter often causing catastrophic derailments. For this reason, substantial research 

has been conducted to address the CWR safety and performance through the development of 

analytical models, test investigations, parameter characterization studies, measurement technique 

development, and improvements in track maintenance practices.

The research work on CWR, sponsored by FRA, has been published in several technical papers 

and reports. Computer models developed for the buckling safety analyses have been assembled 

into a program called CWR-SAFE. A user’s guide available in the program’s HELP menu 

describes the model’s application details. Much of the important technical supportive material 

has been presented in different sources and therefore must be synthesized into a single 

comprehensive report to provide an easy updated reference for the research community and to 

facilitate the understanding and use of CWR-SAFE for buckling safety assessments. This report 

will fulfill this specific need and further address the most recent developments in probabilistic 

approaches for estimating track buckling potential, which can be applied for risk-based estimates 

on buckling safety.

Briefly described in this section are safety-related issues. Present technical problems of CWR 

review the problem severity from the accident rate perspective and summarize historical 

background of research work performed to date under the FRA sponsorship.

1.1 Technical Problems

The increased use of the CWR prompted significant research for understanding and controlling 

failure modes caused by thermal loads in the rails. Two of these failure modes, with a great 

influence on the safety of vehicle operation, include the following:

• Loss of lateral stability (track buckling, track shift, and radial breathing)

• Rail pull apart (rail break under high tensile forces)

1.1.1 Track Buckling

Use of CWR will generate large thermal rail loads that are typically compressive when the rail 

temperatures are high and can induce track buckling. The amplitude of the buckling deflections 

can be large (reaching 30 inches (in) in some cases), and the associated buckle lengths can be on 

the order of 40–60 feet (ft). These misalignments can cause vehicle derailments because they 

cannot be negotiated at typical operating speeds.

Track buckling can also occur in the vertical plane, although this mode is rare because of high 

rail stiffness in the vertical direction. The weight of the track also resists the upward buckling 

movement. Invariably, whenever the CWR track buckles, it does so in the lateral plane as in 

Figure 1-1. Even if vertical buckling were to occur, the track would tend to eventually move in 
7

the lateral plane because of lower rail stiffness in this plane. The shape of the initial 

misalignments (lateral alignment defects) is instrumental in determining the resulting buckling 

mode shape. Rails outside the buckling zone (referred to as the adjoining zones) move 

longitudinally only with no lateral deflection in these zones. From these adjoining zones, the 

rails extend and pull into the buckled zone providing the extra rail required for the buckle to take 

place. The adjoining zones can be as long as 600–1000 ft on either side of the buckled zone, so 

in a buckling situation, the impacted track length for repair and readjustment can be quite large.

(a) (b)

(c) (d)

Figure 1-1. Lateral Track Buckling

A critical issue in the subject of CWR track safety is the influence of vehicle loads on track 

stability. However, thermal loads alone can precipitate a buckling called static buckling 

(Figure 1-1(b) and (d)). Buckling due to the combination of thermal and vehicle loads is called 

dynamic or vehicle-induced buckling (Figure 1-1(a) and (c)). The majority of buckles in revenue 

service occur under the train; hence, vehicle loads play an important role in the buckling 

mechanism. This report will present this dynamic buckling theory and its implications on 

buckling prevention of CWR tracks.
8

In addition to large deflection buckling, track lateral shift and radial breathing of curves can also 

develop and present problems in CWR tracks.

1.1.2 Track Shift

Apart from thermal and vehicle loads and other parameters to be discussed later, an important 

contributor to the buckling phenomenon is the initial lateral misalignment. In the case of track 

buckling, the lateral misalignment influences the buckling strength significantly. The 

misalignments can have a variety of shapes and can be characterized by a wavelength and its 

amplitude (offset). FRA has specifications for allowable limits of the misalignment amplitudes. 

However, these misalignments can trigger buckling especially when they grow. The growth 

occurs because of repeated application of wheel loads (especially high axle lateral forces), and 

this growth is usually acerbated by the presence of thermal loads in the rails. The formation and 

growth of misalignments in CWR is defined as track shift. Track shift can occur and reach a 

stable limit without leading into full-blown buckling. Conversely, sudden buckling can occur 

without any visible evidence of track shift. A principal distinction between track shift and track 

buckling is that track shift safety typically requires the determination of allowable net axle lateral 

loads (NAL), whereas track buckling safety requires the determination of the allowable thermal 

loads. Chapter 8 of this report references reports on track shift research performed under FRA 

sponsorship. The remainder of this report will be confined to the evaluation of CWR track 

buckling potential and the methods to minimize it.

1.1.3 Radial Breathing of Curves

Curved CWR tracks can experience radial movements or breathing because thermal loads. 

Compressive loads in the rails generated in summer tend to move the track outward. Likewise, 

tensile loads in winter will pull the curved track inward. These movements can be large for high 

degree curves, especially in the presence of high thermal forces and weak lateral restraint. 

Because radial breathing is seldom uniform, additional misalignments can develop in the track, 

as well as pockets of lowered lateral resistance. As discussed later, radial breathing is 

instrumental in the change of the rail’s stress-free (or neutral) temperature. As far as the 

buckling mechanics are concerned, these radial breathing displacements must be accounted for in 

the proper formulation of the theory for curved track lateral buckling.

1.2 Rail Pull Apart

CWR will not only experience compressive loads because of temperature increase but also 

tensile forces due to the rail temperature decrease in winter. The maximum tensile load in the 

rail is determined by the difference in the installation temperature (actually the neutral 

temperature as defined later) and the lowest rail temperatures. The maximum tensile force can 

be of the same order of magnitude, or larger, than the maximum compressive load generated in 

summer. High tensile force is instrumental in causing rail fractures at locations of internal defect 

and or weak welds. This mode of failure is referred to as pull apart failures. Because of vehicle 

and thermal stress cycles, the internal defects can grow into sizeable cracks, which propagate 

rapidly under high tensile load conditions.

The resulting rail gap due to pull apart failures is generally small (usually less than 4 in if the rail 

is properly anchored or held longitudinally to the ties by effective fasteners). The train wheels 

may traverse these breaks more safely than they can the large buckled displacements. From the 

consideration of safety (i.e., derailment potential and consequence), pull apart are considered less 
9

severe than buckles. Also, rail breaks may be detected by the signaling system. Hence, the 

installation temperature of CWR is generally high and sometimes biased to produce smaller 

compressive loads in summer at the expense of higher tensile loads in winter. Pull apart failures 

are damaging to track structure and need expensive repairs; therefore, they must be avoided by 

careful design and maintenance of CWR tracks. The repair of pull apart, especially in cold 

temperatures, can lead to eventual buckling prone conditions when not properly restressed or 

adjusted to the correct desired neutral temperature. The industry usually inspects rails for defects 

by ultrasonic or other means and makes timely replacements/repairs to avoid pull apart and other 

fracture failures. The tensile pull apart failure prediction is a subject of future research and will 

not be treated here.

1.3 Track Buckling Accident Severity

The severity of track buckling can be deduced from the statistics of buckling-induced 

derailments and the resulting dollar damage [1] as Figure 1-2 shows. The data is based on FRA 

reportable accidents and does not include buckling incidents that did not result in a train 

derailment. The dollar damage includes the following:

• Loss and damage to the track and the cost of repair

• Loss and damage to the rolling stock and the freight

The damage does not include the loss because of suspended or rerouted traffic.

As can be seen from the data, the past 10 year’s average was 34 derailments a year with a 

decreasing trend over the past 3 years. However, the dollar damage has been increasing over the 

past 10 years, reaching a peak of $17 million in 2002. The reasons for this trend might be due to 

better cause-finding and reporting, inflation, and more costly derailments resulting from higher 

speeds and larger tonnage levels.
10

Calendar year

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

0

50

100

150

200

Number of derailments

(a) Frequency of buckled track derailments

Average: 34

Dollars (million)

0

5

10

15

20

Calendar year

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

(b) Damage from track buckle derailments

Figure 1-2. Track Buckling Statistics
11

To keep the derailments and the dollar damage down, FRA and the industry continue to work 

toward improved buckling prevention practices. This includes a better understanding of the track 

buckling mechanics and parameters, developing better techniques for improved maintenance, 

detecting incipient buckling prone conditions, and better managing the risks associated with 

buckled track. The research presented in this report is aimed at furthering this knowledge toward 

a better understanding of track buckling prevention.

1.4 Background

Under FRA’s sponsorship, significant research and development work has been performed in the 

area of CWR buckling. The work involved development of the theory for predicting the critical 

forces and temperatures causing buckling, identification of critical parameters in the theory, 

development of a safety criterion, conducting full-scale static and dynamic buckling tests for 

model and safety criterion validation, and development of methodologies for the assurance of 

safety against buckling. A comprehensive computer tool (CWR-SAFE) is a major achievement 

under this FRA program, as are new techniques and hardware for measuring key parameters 

required in the computer program and methods for implementing improvements in the 

maintenance practices by the industry.

1.5 CWR Theory

The early theories of buckling, which do not include vehicle load effects, are known as static 

theories. The early version of static theory was developed in reference [2] for tangent and 

curved track with misalignments. The theory was validated with tests at Plains, VA, in 1982 on 

the Southern Railroad [3]. Because track buckling is predominately a train-induced event, the 

vehicle loads were considered to be important. This resulted in a dynamic theory of CWR track 

buckling by Kish et al. in 1984 [4]. Several tests on tangent and curved tracks were conducted 

[5–7] at TTC in Pueblo, CO, with the Association of American Railroads (AAR)/TTC test 

support to validate the dynamic buckling theory. New safety concepts were also developed 

based on these works. Reference [8] summarizes the dynamic buckling theory, test validations, 

and safety concepts. To apply the theory, several computer programs were developed, 

culminating in CWR-SAFE.

Using the dynamic buckling theory, an extensive study was conducted to identify critical 

parameters and evaluate the sensitivity of the parameters on buckling strength [9]. The 

following section briefly reviews theses parameters, and a later chapter presents a comprehensive 

discussion on them.

1.6 Buckling Parameters

Extensive field tests were conducted to evaluate and characterize track parameters having large 

influences on buckling strength. These included the following:

• Tie-ballast lateral resistance

• Rail fastener longitudinal and torsional resistance

• CWR neutral temperature

New specialized, portable hardware was designed and developed for field evaluation of single tie 

lateral resistance for wood and concrete ties. The tie-ballast resistance data was developed for 

revenue service conditions and TTC track. The data encompasses a variety of track conditions 

and parameters, such as shoulder width, different crib levels, pre- and post-tamping, different 
12

consolidation levels, and different ballast materials. The results and the description of the 

hardware are available in references [10, 11].

The longitudinal and torsional resistance of rail fasteners and ties in the ballast was evaluated at 

TTC, under laboratory and field conditions. The resistances were measured for various wood 

and concrete fasteners and conditions [9].

Using rail force measuring strain gages, the CWR neutral temperature was monitored on several 

revenue lines and at TTC for tangent and high degree curves. Conditions under which the 

neutral temperature can significantly drop from the installation value were evaluated [12]. These 

measurements were instrumental in providing information on the causes of neutral temperature 

variation and on typical values. Subsequent chapters in this report will address further 

discussion on these and other parameters, including their influences on track buckling.

1.7 Computer Programs

CWR-SAFE is a PC-based analysis program that calculates the buckling response of CWR 

tangent and curved track because of thermal and vehicle loads. The software embodies three 

analysis modules called CWR-BUCKLE, CWR-INDY, and CWR-RISK. CWR-BUCKLE 

performs buckling analysis, determines the allowable temperature increase for buckling 

prevention, performs safety analysis to determine buckling safety margin (BSM), and serves as a 

platform for the CWR-INDY and CWR-RISK modules. Embedded in the software is a track 

quality-based safety criterion, which is used to determine the allowable temperature increase or 

the safe temperatures for buckling prevention.

CWR-INDY is a simpler version of CWR-BUCKLE, customized for easy industry use by 

allowing for simple design parameter inputs such as:

• Tie, fastener, and ballast type

• Crib and shoulder width

• Consolidation type and level

These input parameters are converted to their scientific numeric counterparts within the program 

using empirical equations based on field test data. With the converted scientific data, CWRINDY predicts the same type buckling estimates as CWR-BUCKLE.

Although CWR-BUCKLE and CWR-INDY are deterministic models, the CWR-RISK module 

provides a probabilistic analysis of track buckling. Statistical descriptors are used to define the 

three important parameters lateral resistance, neutral temperature, and lateral alignment defect. 

With these input parameters, CWR-RISK predicts the probability of buckling as a function of the 

rail temperature and allows for the evaluation of various risk acceptance levels to be used in 

buckling prevention. These risk levels can then be used for the development of risk-based 

performance standards for CWR or the determination of risk-based slow orders for hot weather 

operation [13].

1.8 CWR Maintenance

In addition to the development of theoretical models, hardware, and applications programs for 

field measurement of parameters, the research has also produced important advances on 

determination of longitudinal rail force and its influence on longitudinal restraint of CWR, aimed 

at helping improve CWR maintenance and inspection practices.
13

1.8.1 Rail Longitudinal Force Evaluation

A significant aspect of buckling prevention is the effective control of the longitudinal force in 

CWR to safe levels. The allowable or safe longitudinal force level can be determined through 

CWR-SAFE, and these then can be monitored through accepted measurement diagnostics.

Under FRA’s research program, the Rail Uplift Technique was developed to measure rail force 

[14]. The technique is based on relating the CWR tensile or compressive force in the rail to the 

uplift deflection of the rail when freed from fasteners and pulled up by a car-mounted uplift 

device. The method was successfully demonstrated at TTC and was used on the Burlington 

Northern Railroad [15]. It is the only known method to give the absolute force of the rail 

without cutting the rail. However, because the fasteners needed to be removed to facilitate the 

rail uplift, the method received minimal acceptance by the U.S. railroad community.

Techniques, based on strain gage concepts, have been used to measure rail force, although with 

difficulty in application because of the need to have a zero reference, which requires cutting the 

rail. Other techniques explored based on ultrasonics, acoustics, magneto-elasto-mechanics, 

Barkhausen-noise, vibro-elastics, x-ray diffraction, and fiber optics were proven ineffective, 

except in some specific laboratory applications. Hence, to date, a nondestructive, accurate, and 

easily deployable longitudinal force measurement system, either vehicle or track borne, is not 

available, making the development of CWR force measurement a major worldwide research 

need.

1.8.2 Destressing of CWR

One way railroads attempt to control large compressive stresses in the summer is to cut, destress, 

and reweld the rail after adjusting it to the desired level of neutral temperature. In winter 

conditions, large tensile forces may fracture the rail, requiring repair and readjusting to the 

desired neutral temperature. Under FRA’s research program, a longitudinal restraint model has 

been developed for a more effective destressing of rails in summer and restressing in winter [16]. 

A prototype for a portable device called the Rail Destressing Force Indicator (RDFI) has been 

designed to help the track crew to obtain the required information for a more effective rail 

destress/readjustment procedure. An RDFI prototype has been field tested and is currently being

finalized for industry trial.

1.9 Report Content and Organization

Chapter 2 presents the theory of track stability that is required in buckling safety analyses, which 

forms the basis of the CWR-SAFE buckling model. The theory considers the vehicle and 

thermal loads on tangent or curved CWR with lateral misalignments and nonlinear lateral 

resistance characteristics. Appendix A presents the actual mathematical formulations, but this 

section presents the mechanisms and assumptions in the development of the theory. This section 

also presents the procedure used to determine the buckling strength (in terms of critical forces 

and temperatures), along with a fundamental safety criterion for buckling prevention and the key 

parameters influencing the criterion.

Chapter 3 presents an overview of CWR-SAFE and its three modules. The model’s applications 

are illustrated with numerical examples. Chapter 3 discusses assumptions and correlations of the 

CWR-BUCKLE scientific parameters to simpler design counterparts (such as the ballast and 

fastener type, crib content, and shoulder width), which are the required inputs to CWR-INDY. 

This chapter also briefly introduces the CWR-RISK fundamentals.
14

Chapter 4 presents detailed parametric and sensitivity studies on the basis of the CWR-BUCKLE 

program. Chapter 4 describes the effects of track resistance, misalignments, curvature, and 

vehicle parameters on track buckling strength. Appendix B presents the techniques to measure 

the parameters.

Chapter 5 presents the CWR-RISK module that deals with the probabilistic buckling theory. 

Chapter 5 describes the assumptions, the details of the methodology, and the probabilistic 

descriptors for the parameters, including implications on buckling safety. As an example of the 

model’s application, a risk-based approach for the slow orders is also provided. Appendix C 

presents the mathematical formulations involved in CWR-RISK. Appendix D presents the 

measurement requirements including sampling criteria for lateral resistance and neutral 

temperature parameters required in CWR-RISK.

Chapter 6 deals with the safety aspects of CWR. The chapter presents applications of the 

buckling safety assurance methodology, including buckling prevention safety limits, as well as 

other CWR safety applications, such as track consolidation, hot weather slow orders, and neutral 

temperature maintenance.

Chapter 7 details the conclusions on all aspects of CWR studied to date in the United States. 

This chapter also provides recommendations and future research needs.

Chapter 8 lists all the references for this document.

Finally, Appendix E addresses neutral temperature issues during CWR maintenance operations, 

such as rail repair via cutting, destressing, and readjusting. This appendix also discusses the 

RDFI device for a more effective destressing procedure.
15

2. The Theory of Track Stability

This chapter describes a track buckling mechanism, theories for static and dynamic buckling, and 

governing parameters. The chapter also introduces the buckling energy concept, buckling safety 

criteria, and the analysis methodology for conducting buckling safety evaluations.

2.1 Buckling Mechanism

Consider a long CWR track, Figure 2-1, which is straight but for a small initial lateral sinusoidal 

type misalignment described by an amplitude δo and a wavelength 2L0. With increase in rail 

temperature, the compressive force P will increase, which may produce some growth in the 

initial misalignment. Experiments and field observations have shown that as temperature (and 

the corresponding rail force) increase to a maximum (critical) level will increase the initial 

misalignments to wB, an unstable equilibrium state. At this state, the track can buckle out 

suddenly into a new lateral position, wC, spanning a length of 2L. The magnitude of wC is 

typically large, on the order of 6–30 in, while its wavelength can be on the order of 40–80 ft. 

This process, in terms of temperature increase above neutral versus the maximum deflection, is

graphically shown as in Figure 2-2(a). The neutral temperature (as referred to on the vertical 

axis) is that temperature at which the net longitudinal force in that rail is zero and is a reference 

condition for the force build up in the rail; as shown later, this is a key parameter for buckling 

prevention. The displacement wB is referred to as the pre-buckling displacement, the 

temperature increase at B when buckling takes place is the buckling temperature, and wC is the 

post buckling displacement.

2L

2Lo

WC

WB δ0

P P

Figure 2-1. Pre- and Postbuckled Track Configurations

Figure 2-2(b) shows the same process with the dashed overlay representing all the possible 

buckled positions (equilibrium configurations) given by stability theory (to be discussed later). 

Two distinct temperature increase values ∆TBmax (sometimes referred to as the bifurcation 

temperature) and ∆TBmin, in between which multiple positions of equilibrium can exist. A unique 

property of buckled or equilibrium positions is the nature of their stability (i.e., being stable or 

unstable). Thus, the dashed prebuckled branch up ∆TBmax can be shown as stable, the dotted 

branch down to ∆TBmin unstable, and the dashed branch increasing up to C and beyond to be 

stable. In actual physical conditions, only the stable positions can be realized. The bifurcation 

point B represents the configuration common to stable and unstable configurations (sometimes 

referred to as infinitesimal stability) at which the track snaps over to the stable position at C. 

Research has also shown that, if sufficient external energy is supplied (such as by train action), 

the track can jump from a prebuckling stable configuration to a postbuckling stable configuration 

(through unstable configuration) at temperatures below ∆TBmax as shown 1 to 2 to 3 in Figure 2-
16

2(c). Hence, the range between ∆TBmin and ∆TBmax represents the buckling regime of CWR 

tracks. Whereas the track will buckle at ∆TBmax with no external energy, it can also buckle at 

∆TBmin if sufficient external energy is supplied to the track. Below ∆TBmin, the track will not 

buckle because it has only one stable equilibrium configuration.

Figure 2-2. Buckling Response Curves

The shape of the buckling response curve depends on the specific track parameters and 

conditions (i.e., track quality). Good quality track will have a large difference between the 

(a)

Deflection

Temperature increase

above neutral

δo

Pre-buckling

equilibrium

Post-buckling

equilibrium

B C

WB WC

Buckle

Deflection

Temperature increase

above neutral

δo

∆TBmax

∆TBmin

Buckle

C B

Bucking

regime

Deflection

Temperature increase

above neutral

δo

∆TBmax

∆TBmin

Buckle

1 2 3

(b)

(c)
17

maximum buckling temperature (∆TBmax), and the minimum buckling temperature (∆TBmin) as 

shown in Figure 2-3 (a). As track quality decreases, both the ∆TBmax and the ∆TBmin value 

decrease as does the ∆ between them. In fact, for low or inferior quality track, the difference 

between these temperatures can go to zero (when ∆TBmax and the ∆TBmin coalesce into one value 

∆TP), as shown in Figure 2-3(c). Such a track will buckle out slowly or progressively with the 

rail temperature rise (i.e., no sudden snap through).

Lateral deflection, δ

∆T

∆TBmin

∆TBmax

∆

Track quality: HIGH

∆: Larger 10° F

Lateral deflection, δ

∆T

∆TBmin

∆TBmax

∆

Track quality: AVERAGE

∆: Between 0°- 10° F

Lateral deflection, δ

∆T

∆TP

Track quality: LOW

∆: 0° F (a) (b) (c)

Figure 2-3. Track Quality Influence on Buckling Response Characteristics

2.2 Force Drop/Buckle Influence Zone

A feature of the sudden explosive buckles is the accompanying rail force drop (energy release) in 

the buckled zone compared with that of the prebuckling force value. This is due to the large 

lateral displacement contributing to the rail extension that releases some of the compressive load. 

The lateral displacement in the buckling zone is accompanied by the longitudinal motion in the 

outside zones, which feeds the rail into the buckling zone. The longitudinal motion will be felt 

through a substantially long section of CWR track. Thus, the rail force distribution in the 

buckled and adjoining zones is significantly altered, as indicated in Figure 2-4. This means that 

the CWR neutral temperature will be significantly altered after the buckling incident, and long 

sections of track have to be repaired and restressed. It also means that the theory of track 

buckling has to appropriately predict this energy release and force drop to correctly model 

buckling mechanism.
18

2Lo

2L

Rail force

Initial imperfection

Buckling configuration

Before buckling

After buckling

Adjoining zone Buckled zone Adjoining zone

Figure 2-4. Rail Force Distribution after Buckling

2.3 Buckling Mode Shapes

Figure 2-1 depicts the mode shape as a symmetric half sine wave called Shape I. The mode 

shape can also be a complete sine wave reflecting an asymmetric wave called Shape II, whereas

higher modes, such as symmetric Shape III, can also occur. Figure 2-5 shows these 

schematically. The actual buckled mode shape occurring in track is largely influenced by the 

shape of the initial misalignment. Tangent track typically buckles out in Shape III, whereas

curved track buckles in Shape I, and the theory has to properly account for them in terms of 

appropriate boundary conditions, as discussed later.

Z

X

L Buckled

region Adjoining

Z region

X

Z

X

L

L

L1

Symmetric mode (Shape I)

1 half wave

Anti-symmetric mode (Shape II)

Symmetric mode (Shape III)

3 half waves

Figure 2-5. CWR Track Lateral Buckling Modes
19

2.4 Track Buckling Theories

Buckling theories are based on mechanistic track model shown in Figure 2-6. The buckling 

force is the combined compressive load in the two rails, which depends on the rail crosssectional area and the temperature rise. The lateral resistance generated between the ties and the 

ballast, as well as the longitudinal and torsional resistances generated in the rail fasteners, offers 

the resistive forces to the buckling force.

z, w

x, u

Rails Ties

Torsional

spring

Lateral spring

Longitudinal spring

P P = AEαT

y, v

Figure 2-6. Track Model

The lateral resistance depends on nonlinear tie-ballast spring characteristic because the tie 

displaces laterally through the ballast. The longitudinal resistance depends on longitudinal 

spring characteristic of the rail/tie/fastener/ballast. In a poorly ballasted condition, such as with 

low ballast level in the cribs, the rail-tie structure could move longitudinally. In other cases, the 

rail creeps through the fasteners with no or negligible longitudinal movement of the ties. The 

rail-to-tie fastenings also offer rotational rigidity (modeled by torsional springs), which reacts 

against the rail’s tendency to rotate during the buckling deformations.

A number of theories have been postulated in the literature on track buckling. A review of all 

the theories to date is available in reference [17]. The theories developed under FRA research 

are divided into two basic categories, deterministic and probabilistic.

The deterministic method requires parameters to be specified with certainty as in classical 

mechanics. The buckling strength of the CWR track is evaluated using classical formulations 

and expressed in terms of the temperature increase over the neutral temperature (i.e., the 

buckling response curves of Figure 2-3). The method indicates that either track will buckle or 

not at a given temperature for a set of input parameters. A later subsection presents the required 

parameters and their characterization.

The probabilistic method provides a percent probability of occurrence of buckling at a given rail 

temperature but requires a statistical distribution of input parameters. The probabilistic theory 

can thus facilitate a risk-based approach to vehicle operations and maintenance procedures to 

manage the buckling potential. The deterministic and probabilistic methods can be used for 

static and dynamic buckling analyses and for tangent and curved tracks, as discussed in

subsequent sections.

2.5 Static versus Dynamic Buckling Model

The static buckling model ignores the effects of vehicle loads and considers the buckling of track 

resulting from longitudinal compressive loads only. In the buckling region, the resistance to 

lateral buckling is offered by a tie-ballast structure with no vehicle vertical load influence. 

Although this mode of buckling has been extensively treated in the literature and in test studies, 

it is incorrect to use these results to explain buckling occurring in conjunction with train
20

movements. The correct modeling requires the inclusion of the vehicle vertical loads and their 

influence on lateral resistance. The vertical loads are distributed along the track depending on 

the axle and truck center spacing of the vehicle. Although the lateral resistance directly under 

the wheels increases, the resistance of the track segment between the trucks can decrease because 

of the uplift wave of the section. This uplift wave is referred to as the central wave and shown 

schematically in Figure 2-7. Recession and precession waves exist (or occur) behind and in front 

of a train (as shown for two cars in the figure) where the lateral resistance can reduce. The 

central wave generally has maximum reduction in the lateral resistance and is therefore critical 

for buckling. This suggests that the track section under the vehicle can be most vulnerable to 

buckling due to loss of lateral resistance; in fact, buckling under the train is a frequent 

occurrence.

Recession

wave

Precession

wave

Increased lateral resistance

Central

Wave

(Reduced resistance)

Central

Wave

(Reduced resistance)

S

Ymax

Ymax

TCS

Y V V Y V V

Direction of travel

V V V V

Y = vertical deflection

S = axle spacing

TCS = truck center spacing

V = axle load

Figure 2-7. Definition of Uplift Waves

Buckling under the central wave uplift should be distinguished from the potential track shift that 

could occur if the wheels carry high lateral loads during negotiation of curves and lateral 

misalignments. The force that tends to move the curve laterally is the net axle force (the sum of 

the lateral forces at the two wheels of an axle). The net axle force is expressed in terms of the 

lateral to vertical force ratio (L/V). Unless this L/V ratio exceeds a threshold limit, no track shift 

or lateral movement will occur under the wheels [18]. Track shift occurs gradually under a 

wheel passage, whereas lateral buckling between the two trucks of a car can occur rather 

suddenly. Track buckling due to central wave uplift is generally not dependent on the NAL

force because the lateral deflection, if any, from the NAL load is confined to a small region 

under the wheel where the lateral resistance is the highest.

In summary, dynamic buckling theory is required for the more accurate buckling predictions. 

This means that the influence of the vehicle loads in producing the dynamic uplift is essential in 

the model, specifically, to account for the reduced resistance and the subsequent reduction in 

buckling strength.
21

2.6 Tangent versus Curved Tracks

Field tests and observations of actual buckles show that the buckling behavior of curved tracks 

can be different from that of tangent tracks. Tangent track is generally a sudden explosive type 

of buckle and can displace to either side depending on the direction of the initial misalignments 

or the weaker side of lateral resistance. Track buckles with Shape III on tangent track typically

have the amplitude of the middle wave relatively large compared with the two end waves.

Curved tracks generally buckle outward in Shape I. Because of the initial curvature, it would 

require significant energy to bend the rail in the opposite direction of its curvature (i.e., inward to 

produce the tail ends of Shape III). Another important feature of curved track buckling is the 

tendency toward progressive buckling, especially for curves with weak lateral resistance and 

with high curvatures.

Curved tracks also exhibit radial movement (breathing), especially under weak lateral resistance 

condition when in the presence of large diurnal and seasonal rail temperature changes. 

Temperature increase over the neutral can produce radially outward movement; temperature 

decrease from the neutral can produce radially inward movement. This radial breathing can be 

detrimental to the track because it reduces the lateral resistance further and may generate local 

lateral misalignments, which can precipitate buckling. Radial movement can be on the order of a 

few inches or more and can create buckling prone conditions by reducing neutral temperature, 

weakening lateral resistance, and producing local alignment defects.

2.7 Differential Equation Formulations

Buckling theories are based on the differential equations of equilibrium presented in Appendix A 

for tangent and curved tracks. These equations are derived using the variational principles of 

minimizing the potential energy of the track system. One differential equation is for the 

transverse or lateral displacement in the buckled zone. The second one is for the longitudinal or 

tangential displacement in the adjoining zone. These equations are then coupled through another 

equation for the temperature derived by using continuity requirements between the buckled and 

adjoining zones. These equations are based on the classical beam theory allowing for large 

lateral buckling displacements (requiring nonlinear strain-displacement relationship). The 

following key assumptions are made in the buckling theory:

• The two rails are represented as a beam with a combined moment of inertia and a combined 

cross-sectional area. The longitudinal force in the beam is the sum of the forces in the two 

rails. Rail cross sections before buckling remain plane after buckling.

• Lateral resistance offered by the tie is represented by a nonlinear (spring) idealization closely 

approximating test measurements.

• Longitudinal resistance offered by the fasteners and ballast to the rail can be represented by a 

linear idealization for the range of longitudinal displacements (0.5 in) typically found in 

buckling tests.

• Ties do not deform or rotate, and the connection formed by the fasteners to the rails is 

represented by a torsional spring. The torsional resistance in the tie-rail fastener is 

proportional to the rail rotation for small angles (<5°) observed in the buckling tests and in 

tests conducted on fasteners in the laboratory.

• Rail force is constant over the buckling zone length as compared with the adjoining zone, 

where the force varies in a manner depending on the longitudinal resistance of the fasteners.

• Rail temperature is uniform (longitudinally and across the rails) throughout.
22

• Adjoining zones in the tangent track experiences longitudinal displacement only with no 

lateral movement. For curved track, the adjoining zone experiences uniform radial 

movement in addition to the tangential displacement.

• The track has self-weight and no significant vertical profile irregularities.

• Lateral track misalignment has a known wavelength and amplitude, and the shape is 

approximated mathematically by a polynomial function.

2.8 Buckling Response Computation

To determine the buckling response, the relationship between the temperature rise and lateral 

displacement (Figure 2-2b) must be evaluated. For this purpose, different equilibrium 

configurations with different wavelengths are considered. Each buckling configuration gives a 

point on Figure 2-2b.

The differential equation for the buckling zone connects the rail force and the lateral 

displacement. The solution, w = w(x), of the differential equation represents buckling wave 

shapes. For a specific wave shape such as Shape I, the solution gives a relationship between the 

wavelength, L, buckling amplitude, w, and the rail force, P , in the buckled zone. Using 

continuity conditions on longitudinal displacement (and its gradient) between the buckling zone 

and the adjoining zones, a temperature equation relating the rail force, P , in the buckled zone, rail 

temperature, T, and the wavelength, L, is derived. By varying L over a range, the corresponding 

values of the buckling force, P , the amplitude, w, and the rail temperature, T, can be determined. 

From this data, the buckling response curve connecting the rail temperature and the buckling 

amplitude can be constructed.

The complexity of determining the response curve depends on the numerical scheme adopted. 

The most convenient approach is the application of Fourier series for the solution of the 

differential equations, as shown in Appendix A. The advantages of the series approach including 

the following:

• It can accommodate any nonlinearity in the lateral resistance and variations of the resistance 

under the wheel loads.

• The method can predict explosive or progressive response for straight and curved tracks with 

misalignments.

• The method works for perfectly straight tracks with no initial misalignments.

• The computational time is very small (under a few minutes) and the method facilitates easy 

programming on a PC.

An alternate approach to the foregoing method is the finite element (FE) method. The nonlinear 

FE method uses an incremental approach (i.e., the equations are formulated in terms of 

increment in temperature rise and lateral deflection). Temperature or load increments are 

selected, and a relationship is determined starting from the initial condition of zero temperature. 

This approach requires some misalignment to be prescribed a priori and may face numerical 

difficulties because of the instability at ∆TBmax. The FE method requires significant 

computational time, and its accuracy is not generally as good as the Fourier series approach used 

here.
23

2.9 Parameters Influencing Buckling Response

Foundation Modulus (Vertical)

The track foundation modulus is a measure of the vertical stiffness of the track foundation. This 

is used to determine the vertical load distribution on the ties, from which the dynamic lateral 

resistance is evaluated.

Lateral Resistance (Peak and Limiting)

The tendency of the track to buckle laterally is resisted by the reaction forces exerted by the 

ballast on the ties. The ballast lateral resistance typically exhibits characteristics such as those 

shown in Figure 2-8. The consolidated track exhibits a distinct droop in resistance after the peak

value, whereas freshly tamped track does not, it continues at the peak value. Figure 2-8a also 

shows the subparameters used to define the lateral resistance characteristic of consolidated track, 

including the following:

• Peak Lateral Resistance, FP, and corresponding displacement, wP

• Limiting Lateral Resistance, FL, and corresponding displacement, wL

For weak or recently maintained tracks, tests have shown that FP = FL and wP = wL.

In the measurement of lateral resistance, as discussed later, it is generally easy to determine FP

and wP by loading the tie laterally by no more than 0.5 in. To determine FL and wL, 

3–5 in of tie movement may be required, which is considered more destructive to the track. 

Hence, through extensive test work, correlations were developed to relate FP, wL, and FL (see 

Chapter 3), so in many cases, FP is adequate to define the resistance characteristic. Chapter 4 

further discusses the importance of FL.

FL

FP

Lateral deflection, w

Lateral resistance

Lateral resistance

FP, FL

Lateral deflection, w

WP WL

a) Consolidation track b) Freshly tamped track

Figure 2-8. Typical Lateral Resistance Characteristics

Longitudinal Resistance

Track longitudinal resistance is the resistance offered by ties and ballast to the rails because they 

tend to move in the longitudinal direction in the event of buckling, when thermal forces cause 

longitudinal movements, or in response to braking and accelerating train action. The resistance 

characteristic is generally bilinear, and in most cases, only the initial linear part will be required 

because the rail longitudinal displacement in the buckling event is small.
24

Misalignment Amplitude

The misalignment amplitude (δ0) is the size of the track misalignment prior to the occurrence of 

buckling, as shown in Figure 2-9.

Misalignment Wavelength

The misalignment wavelength (2L0) is the total length of the track misalignment before the 

occurrence of buckling, as shown in Figure 2-9.

The misalignment shape can be mathematically approximated by a simple polynomial, as shown 

in Appendix A, giving zero values of amplitude and slope at the ends of the misalignment wave.

2Lo

Track misalignment amplitude, δo

Lo

Misalignment wavelength

Misalignment

half-wavelength

Figure 2-9. Track Misalignment

Neutral Temperature (TN)

The neutral temperature is the rail temperature at which zero longitudinal force exists in the rail. 

The temperature increases in the buckling analysis are measured with respect to the neutral 

temperature. Although this parameter is required only in the safety analysis part of the 

deterministic model, a statistical distribution of this parameter is a key input in the probabilistic 

buckling strength evaluations.

Rail Size

Figure 2-10 shows the coordinate axes for a single rail beam and some rail properties (area and 

moments of inertia).

Y (vertical)

X (longitudinal)

Z (horizontal)

Rail Area

(in2)

Moment of Inertia (in4) 

(about the horizontal 

axis, Iyy)

Moment of Inertia (in4) 

(about the vertical

axis, Izz)

115 RE 11.25 65.6 10.8

119 RE 11.65 71.4 10.9

132 RE 12.95 88.2 14.6

136 RE 13.35 94.9 14.7

UIC60 11.91 73.4 12.3

Figure 2-10. Rail Beam Sectional Properties
25

Rail Temperature (TR)

The rail temperature is the maximum anticipated temperature of the rail found in the segment of 

track being analyzed. This is required together with the rail neutral temperature in the safety 

assessment parts of the analysis.

Tie-Ballast Friction Coefficient

The tie-ballast friction coefficient is a measure of the tie bottom surface influence in lateral 

resistance when the track is vertically loaded. It is required in the evaluation of the track 

dynamic lateral resistance and the uplift computation.

Torsional Resistance

The rail fasteners (which can include cut spikes/anchors or elastic fasteners) also provide 

rotational restraint against the lateral bending of the rails. The resistance is linearized with 

respect to the rail rotation, and the torsional stiffness is specified per fastener. The required input 

torsional resistance is pounds per unit length of track and is calculated from

tie spacing

torsionalstiffness torsional resistance = 2

where the factor 2 accounts for both rails.

Track Self-Weight

The track self-weight is the weight per unit length (pounds per inch) of the track structure, 

including the rails, fasteners, and ties. The following formula is used for its determination

( )

36

2 W

S

W self weight TIE RAIL

= +

where

WTIE = weight of one tie including fasteners and tie plates

WRAIL = rail weight (lb/yard)

S = tie spacing (inches)

Vehicle Characteristics

The required vehicle characteristics for dynamic buckling theory are the truck center spacing, the 

axle spacing, and the axle loads for each car. These determine the uplift characteristic in the 

central zone where buckling is likely to occur.

2.10 Buckling Safety Criteria

2.10.1 Background

Referring to Figure 2-2b, if the rails are heated to the upper critical temperature, ∆TBmax, 

corresponding to the point B, the CWR track will be in a state of unstable equilibrium, meaning 

that the track will jump to the configuration corresponding to the point C with no external energy 

supplied to it. If it is heated to a lower point in the buckling regime, as indicated in Figure 2-2c, 

some finite external energy must be supplied in the lateral plane to buckle the track laterally (to a 

point on the stable post buckling branch).
26

Clearly, trains operating at their line speeds supply energy in the lateral plane because of wheel 

set lateral movement. This vehicle energy can be accentuated if traversing misalignments or 

when negotiating curves. The energy imparted to the track is generally proportional to the 

square of the train speed. At higher speeds, higher levels of energy may be imparted to the track 

producing buckling at temperatures lower than ∆TBmax, typically under trains. Using this ∆TBmax

temperature as an allowable or safe temperature is inadmissible because of this variability (as 

well as to other sensitivities as shown later).

The energy required to buckle the track laterally as the rail temperature increases within the 

buckling regime (∆TBmax ≥ ∆Tr ≥ ∆TBmin) can be computed by using the expressions developed in 

Appendix A. Figure 2-11 shows the result of such energy computation, showing the energy 

required to buckle the track for certain assumed parameters as a function of the temperature 

increase above neutral temperature. This shows zero energy at ∆TBmax as expected at the point of 

unstable equilibrium, the maximum energy at ∆TBmin, and a very sharp decrease in energy 

required for buckling just above ∆TBmin.

Buckling energy computations have indicated that the magnitude of the maximum energy 

(referred to as the energy barrier) at ∆TBmin increases as the buckling regime (∆TBmax – ∆TBmin) 

increases. This leads to the expectation (partially verified by analysis) that, for track conditions 

exhibiting large buckling regimes, the external (train) energy will not be larger than this 

maximum, hence not sufficient to buckle that track at ∆TBmin. However, above ∆TBmin buckling 

will be possible due to the rapidly decreasing energy levels that are possible because of train 

passage.

0 5 10 15 20 25 30

Lateral deflection (in)

0

40

80

120

160

200

0 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6

Buckling energy (in-kip)

0

40

80

120

160

200

Buckling energy

∆TBmax Emin = 0

∆TBmin

∆T (°F) ∆T (°F)

Emax

10.8

Figure 2-11. Energy Required to Buckle

This aspect is incorporated in the development of the safety criterion presented below, which is 

predicated on the fact that the safe temperature level for buckling prevention can be based on

∆TBmin.

2.10.2 Buckling Safety Criteria

A buckling safety criterion is postulated based on the following stipulations:

• The permissible rail temperature rise cannot be greater than the lower buckling temperature 

∆TBmin because of the possibility of buckling at this or higher rail temperature.
27

• An adequate energy barrier should exist at ∆TBmin if ∆TBmin is to be used as a permissible 

value for the rail temperature. A 10 °F difference between ∆TBmax and ∆TBmin would 

generally assure such a barrier.

• If the difference between ∆TBmax and ∆TBmin is less than 10 °F (i.e., when buckling energies 

are small), a safety factor (SF) must be used on ∆TBmin to provide no buckling potential 

(since only the prebuckled, stable equilibrium configurations can exist). The case of 

progressive buckling requires a special treatment as indicated below.

• The determination of ∆TBmax and ∆TBmin must be based on a theoretically sound and 

validated dynamic buckling theory properly accounting for all the key parameters referred to 

earlier.

In line with these requirements, the safety criterion for the allowable temperature increase is 

determined as:

∆Tall = ∆TBmin; for (∆TBmax – ∆TBmin) > 10 °F

∆Tall = ∆TBmax – SF; for (∆TBmax – ∆TBmin) < 10 °F, and SF is between 0 and 10 °F

The ∆Tall for the special case of progressive buckling (Figure 2-3) when ∆TBmax = ∆TBmin = ∆TP

is: 

∆Tall = ∆TP – 10 °F

The inherent assumption for the progressive buckling case (typical to weak, high degree curve 

tracks) is that, at the corresponding deflections, ∆TP are not large enough to produce train 

derailments at low speeds. In the CWR-SAFE program as discussed later, these calculations are 

performed automatically, and the results are displayed to the user during the safety analysis 

phase of the program. Figure 2-12 illustrates the safety criterion. On the basis of this work, the 

UIC has adopted a similar safety criterion.

Safe temperature increase

(Tall = ∆TBmin)

∆TBmax

∆TBmin ∆ >10°F

∆T

Lateral deflection

Tall = ∆TBmin - SF

∆TBmax

∆TBmin

0 ≤ ∆ ≤ 10°F

Lateral deflection

∆T ∆T

Lateral deflection

∆TP

Tall = ∆Tp - 10°F

∆ = 0°F

10°F

Tall = ∆TBmin; ∆ > 10° F

Tall = ∆TBmin – SF; ∆ ≤ 10° F

Tall = ∆TP – 10° F; ∆ = 0

SF

Safe temperature increase

(Tall = ∆TBmin)

∆TBmax

∆TBmin ∆ >10°F

∆T

Lateral deflection

Tall = ∆TBmin - SF

∆TBmax

∆TBmin

0 ≤ ∆ ≤ 10°F

Lateral deflection

∆T ∆T

Lateral deflection

∆TP

Tall = ∆Tp - 10°F

∆ = 0°F

10°F

Tall = ∆TBmin; ∆ > 10° F

Tall = ∆TBmin – SF; ∆ ≤ 10° F

Tall = ∆TP – 10° F; ∆ = 0

SF

Figure 2-12. Illustration of Safety Criterion
28

A useful safety concept for buckling safety evaluations is the buckling safety margin (BSM), 

which is a measure of the reserve buckling strength for a given set of track and vehicle 

parameters. BSM is based on how close the track is to the safety criterion’s ∆Tall for its specific 

neutral temperature and maximum rail temperature condition [i.e., the difference between the 

allowable temperature increase (∆Tall), and the rail temperature (TR) and the neutral temperature 

(TN) difference]

BSM = ∆Tall – (TR – TN)

Four ranges of the BSM identified for practical evaluation of reserve buckling strength exist, and 

ideally, the BSM should be as large as possible to provide the greatest margin of buckling safety. 

BSM is automatically calculated in the safety analysis phase of the CWR-BUCKLE program;

evaluating the four ranges for the BSM include the following:

• No Margin – In this case, BSM as calculated by the formula above is less than zero. This 

implies that no margin of safety exists for the given set of parameters, rail temperature, and 

neutral temperature, and buckling potential is high.

• Minimum Required Range – In this case, BSM is between 0 °F and 20 °F. This implies that 

only a small margin of safety exists for the track in question.

• Adequate Range – In this case, BSM is between 20 °F and 40 °F, implying that an adequate 

margin of safety exists for the track in question.

• Desired Range – In this case, BSM is greater than 40 °F. This is considered the desired high 

range because it provides the largest margin of safety for the given track.

As an example, refer to the buckling response shown in Figure 2-11, the allowable temperature 

increase, ∆Tall = 82 °F. If the maximum rail temperature in that region is 140 °F and if the CWR 

neutral temperature is 75 °F, BSM is 17 °F, satisfying the minimum required range. This also 

means that, should the neutral temperature drop down to 58 °F, the track will have a zero BSM 

and will be highly buckling prone.

2.10.3 Buckling Safety Evaluation Methodology

A methodology for the buckling safety evaluation of CWR track segment consists of a five-step

process illustrated in Figure 2-13.
29

Figure 2-13. CWR Track Buckling Safety Evaluation Methodology

• For the specific track segment under consideration, determine the applicable track input 

parameters described in Section 2.2.5.

• Conduct buckling analysis to determine the critical temperatures ∆TBmax, ∆TBmin, or ∆TP.

• Apply the safety criterion as described in Section 2.3.2 to determine ∆Tall.

• Perform the safety analysis to determine BSM using inputs of rail neutral temperature 

and maximum rail temperature.

• Determine the range of safety (i.e., whether it is in the no margin, minimum, adequate, or 

desired range.

• Depending on the specific application or risks involved, decide if the buckling margin of 

safety is adequate or not.

• If BSM is found to be zero or less, adjustments to the track must be made to reduce the 

buckling risk.

To increase BSM to a desired level, some of the track parameters/characteristics may need to be 

changed. The following subsection describes the key parameters which provide the most 

effective way for increasing BSM. Chapter 4, which deals with parametric studies, will provide 

additional information for increasing ∆Tall and BSM.

2.10.4 Key Controlling Parameters

Track Lateral Resistance

Lateral resistance can be increased by traffic or dynamic consolidation or other compaction 

means. Ballast section can improve lateral resistance through a wider ballast shoulder and fuller

crib level. Alternative tie designs (different tie material, larger tie end cross sections, heavier 

mass, and better tie bottom and side friction resistance) could be considered for new or rebuilt 

tracks.
30

Lateral Alignment

A degraded lateral alignment condition can reduce BSM because ∆Tall is reduced. An improved 

lateral alignment will increase the ∆Tall. Hence, keeping track to good alignment geometry,

through appropriate realignment, and monitoring curves for lateral movement will increase 

buckling safety.

CWR Neutral Temperature

The railroads strive to maintain a high neutral temperature, typically between 90 °F and 115 °F, 

depending on regional temperature variations. CWR is known to move longitudinally through 

fasteners and also laterally in the case of curved tracks resulting from vehicle lateral loads and 

rail temperature fluctuations. The neutral temperature can fall to 60 °F or lower because of rail 

and track movement and maintenance actions (curve realignment or rail repair) as found in many 

tests [15, 19]. Hence, the neutral temperature drop alone can reduce BSM by as much as 30 °F 

or more. Although at present no convenient way of measuring the rail neutral temperature exists, 

the changes in neutral temperature can be monitored with basic force measuring strain gages. 

Timely destressing of CWR to recover the loss in the initial neutral temperature will be helpful in 

maintaining a high BSM, as well as making effective broken rail repairs, and exercising quality 

control on the CWR installation and relay processes to ensure a highly desired target neutral

temperature.

2.11 Summary

This chapter presented the theory of track buckling. The theory considers the vehicle and 

thermal loads on tangent or curved CWR track with lateral misalignments and nonlinear lateral 

resistance characteristics. Appendix A presents the actual mathematical formulations. This 

chapter also presented the buckling mechanism, assumptions in the development of the theory, 

and procedures used to determine the buckling strength (in terms of critical forces and 

temperatures), along with a fundamental safety criterion for buckling prevention and the key 

parameters influencing it.
31

3. Track Buckling Model: CWR-SAFE

Chapter 3 presents the fundamental theory of lateral track buckling and the parameters involved. 

The theory is computerized in a program called CWR-BUCKLE. This program accepts the 

parameters as they are measured with no simplifying assumptions. The program provides a 

scientific tool for the evaluation of buckling strength.

The need to specify all the required parameters in CWR-BUCKLE can at times be difficult, 

especially in direct application by the industry; therefore, a simpler program called CWR-INDY 

has also been developed. This program requires very simple basic inputs, such as track type, 

ballast type, crib and shoulder levels, and million gross tons (MGT). From these simplified

inputs, the true scientific parameters required for CWR-BUCKLE are estimated through 

empirical relationships built into the program, and the buckling response is determined using 

CWR-BUCKLE’s computational engine.

CWR-BUCKLE and CWR-INDY represent deterministic methods of predicting the CWR 

buckling, as opposed to the probabilistic approach in which the buckling is expressed in 

probabilistic terms, based on statistical variations of key parameter’s. Track parameters, such as 

the lateral resistance, can vary from tie to tie in field conditions. Variations and uncertainty are 

also associated with other parameters, such as misalignments and neutral temperature.

The probabilistic method overcomes the problem of specifying the parameters precisely and 

sometimes conservatively, and lends itself to a risk-based analysis and mitigation approach to 

track buckling. The probabilistic approach is available in the form of a computer program called 

CWR-RISK. This program takes the statistical parameters as inputs, and the outputs are given in 

probabilistic terms. The computational engine used for the buckling calculation is again CWRBUCKLE.

The three modules, CWR-BUCKLE, CWR-INDY, and CWR-RISK, are combined into a single 

computer program, CWR-SAFE, which can be exercised by different users for both deterministic 

and probabilistic assessment of CWR track buckling potential.

3.1 Basis of CWR-SAFE Modules

The following sections describe the basis of the three modules in CWR-SAFE.

3.1.1 CWR-BUCKLE

Because CWR-BUCKLE is a deterministic method, it is based on mechanistic buckling theory as 

explained in Chapter 2 and Appendix A. It was validated by direct test evaluations in field 

conditions on several tracks on revenue lines and also at the U.S. Department of Transportation’s 

TTC in Pueblo, CO, where static and dynamic buckling tests were carried out during 1983–1984 

and 1986–1987 on tangent and curved CWR tracks; test results are available in references [5–8]. 

To briefly highlight some of validation aspects, the following examples illustrate some key 

features and findings.

Uplift Wave Influence: Comparison of Buckling Strength under a Hopper Car and a 

Locomotive

To compare the relative influence of the central bending wave as produced by a loaded 100-ton 

hopper car and a locomotive, equal levels of misalignment were set under each of the vehicles. 
32

Vertical and lateral displacements under the stationary cars were measured as the rails were 

heated. Figure 3-1 shows a comparison of lateral displacements under each vehicle as a 

function of temperature. The misalignment growth under the hopper car was found to be more 

severe, indicating the influence of the longer uplift wave under the hopper car. The tests also 

showed that the uplift wave is a contributing factor in the misalignment growth mechanism, 

hence an important component of the dynamic buckling analysis. This test helped better 

understand the importance of the vehicle characteristics, such as the truck center spacing and 

vertical axle loads, their influence on track uplift, and the need to incorporate this influence in 

the buckling model.

Lateral deflection (in)

0.0 0.2 0.4 0.6 0.8 1.0

Temperature increase above neutral (°F)

0

50

100

150

200

250

Hopper car

Locomotive

Figure 3-1. Response of Track under Vehicles

In another test, the measured response of the track with a large misalignment under a stationary 

hopper car favorably compared with the theoretical prediction, as shown in Figure 3-2, giving 

validation of model’s predictions when the buckling response is progressive.
33

Lateral deflection (in)

0 2 4 6 8 10 12

Temperature increase above neutral (°F)

0

10

20

30

40

50

60

70

80

90

136 lbs rail

Tangent track

Peak resistance: 54.5 lb/in

Line defect: 5 inches

Experiment

Theory

Figure 3-2. Progressive Buckling Test (Tangent)

Dynamic Buckling Response Validation

A weak 5-degree curved track was tested dynamically using a locomotive and hopper car consist 

making several passes at slow speeds. After reaching 40 °F with five incremental train passes, 

no growth of initial misalignment resulted. Train passes made above 40 °F increased the initial 

misalignment to 1.1 in; at 62 °F, the curve buckled to a deflection of approximately 8 in after the 

10th pass, as shown in Figure 3-3. The measured dynamic buckling response was in good 

agreement with the theoretical predictions of CWR-BUCKLE.

Lateral deflection (in)

0 1 2 3 4 5 6 7 8 9

Temperature increase above neutral (°F)

0

10

20

30

40

50

60

70

Buckle

Theory

136 lbs rail

5 deg curve

Slag ballast

Line defect: 0.32 in/28 ft

Peak resistance: 34 lb/in

Test train: loco + 100 ton hopper car

Train passes 1-10

Final

misalignment

Figure 3-3. Dynamic Buckling of Curved Track
34

Effect of Car Length/Uplift Wave on Misalignment Growth

To further study the uplift wave influence on the lateral response, the growth of imperfections 

under the passage of different cars was monitored using strip chart recorders. Figure 3-4 is a 

snapshot from one of the charts. The intent was to evaluate the hopper car’s larger uplift wave 

influence on helping the buckling mechanism by producing larger misalignments versus a 

locomotive. An incremental growth under the hopper car versus a nonincreased deflection under 

the locomotive further proved the need for uplift based vehicle buckling theory used in the 

CWR-BUCKLE program. Vertical Force (kips) Lateral Force (kips) Lateral Deflection (kips)

0

2

4

6

0

20

40

60

Locomotive Hopper car

Negative deflection

due to loco L/V

Increment under

hopper car

0.15 in.

0.04 in.

0.15 in.

0.05 in.

0.19 in.

Initial deflection

Increment before pass

Residual deflection

Increment after pass

Figure 3-4. Strip Chart Record for 5-Degree Curve Test

The foregoing test examples illustrate some of the test validation aspects and, coupled with other 

tests and studies, provide adequate validation of the uplift based buckling theory of CWRBUCKLE.

CWR-BUCKLE Features

CWR-BUCKLE is a scientific computer program, utilizing certain input parameters of track and 

vehicle, to predict the following:

• The buckling strength, expressible in terms of the two critical temperatures, TBmax and 

TBmin, or the TP as defined previously in Chapter 2. It can predict the entire pre- and 

postbuckling response curve of the temperature versus lateral deflection.

• It can compute the buckling energy of the track at a given rail temperature.

• It predicts the rail forces before and after buckling.
35

• It predicts the allowable rail temperature increase for buckling safety and the track’s 

BMS. The BMS is indicative of the reserve buckling strength in terms of additional 

acceptable temperature increase before buckling. If the margin of safety is not adequate 

(Chapter 2), the program provides recommendations to the user on how to improve BMS.

The following example illustrates a typical CWR-BUCKLE run with the input parameters shown 

in Table 3-1.

Table 3-1. Input Parameters for CWR-BUCKLE Illustrative Example

CWR-BUCKLE Illustrative Example Inputs 

Rail size (lb/yd): AREA 136

Tie type: Concrete

Tie weight (lb): 750

Tie spacing (in): 24

Track curvature (deg): Tangent

Ballast type: Granite

Tie ballast friction coefficient: 0.86

Torsional resistance (in-kips/rad/in): 25

Longitudinal stiffness (psi): 200

Foundation modulus (psi): 10000

Peak lateral resistance (lb/in): 150

Misalignment amplitude (in): 1.5

Misalignment half-wavelength (in): 180

Rail neutral temperature (°F): 50

Maximum rail temperature (°F): 135

Vehicle type: Hopper

With the above inputs, CWR-BUCKLE provides a graph displaying buckling temperature and 

deflection as shown in Figure 3-5. The figure shows the two salient temperatures, namely the 

∆TBmax = 172 °F and ∆TBmin = 90 °F. In addition to the graph, the output includes a buckling 

results summary and a safety analysis, as shown in Table 3-2 and Table 3-3.
36

Lateral deflection (in)

0 5 10 15 20 25

Temperature increase above neutral (°F)

0

20

40

60

80

100

120

140

160

180

200

∆TBmax= 172 °F

∆TBmin = 90 °F

Figure 3-5. Buckling Deflection versus Temperature Increase above Neutral Temperature

Table 3-2 shows that if the track were to buckle out at the upper critical temperature of 171.66

°F, the corresponding amplitude would be 22.97 in with the half wavelength of 235.26 in. The 

force in the rail to cause the buckle is 439.96 kips, and at the ∆TBmax of 171.66 °F, the required 

energy to buckle the track is zero (as opposed to the energy required to buckle the track at the 

lower critical temperature ∆TBmin = 89.80 °F, which is 8.47 in-kips).

Table 3-2. CWR-BUCKLE Output Buckling Results Summary

Table 3-3 shows the safety analysis that gives the safe allowable temperature increase, Tall = 

∆TBmin = 89.8 °F, and, if the neutral temperature for this case is 50 °F, then the track should not 

buckle until Tr = 89.8 + 50 = 139.8 °F. Because the maximum rail temperature in this example 

was set at 135 °F, the BSM is approximately 5 °F.
37

Table 3-3. CWR-BUCKLE Output – Buckling Safety Analysis

Summary Conclusions

• CWR-BUCKLE is a versatile analysis tool used to perform detailed buckling evaluation 

for given track and vehicle inputs. Full-scale experiments in the field validated the 

program.

• In addition to the quantitative analysis on the buckling strength, the program outputs the 

margin of safety and recommends measures to increase this margin of safety if necessary.

3.2 CWR-INDY

As indicated above, CWR-BUCKLE requires many complex parameters for buckling analysis. 

As an alternative, the Industry Version, called CWR-INDY, was developed, which requires 

simpler inputs, such as ballast type and condition, tie type, and fastener type. Similar to CWRBUCKLE, the model requires neutral and maximum rail temperatures as inputs, and it assumes a 

vertical modulus of 6,000 psi for wood tie track and 10,000 psi for concrete tie track. With these 

simpler inputs, the program internally determines the actual numeric values for the required 

parameters and will use the same core analysis as in CWR-BUCKLE.

This section describes the basis, assumptions, and relationships used in this CWR-INDY version 

to translate the simpler input characteristics into the actual required technical parameters. The 

program has been developed to require the input parameters known to the railroad user, which 

include the following:

• Rail size (AREA 100,115,119,132,133,136, and 140; UIC 54 and 60)

• Tie type (wood or concrete)

• Ballast material (granite or slag)

• Fastener type (cut spikes, or elastic fasteners)

• Anchor/fastener pattern (every tie anchored (ETA) or every other tie anchored (EOTA))
38

• Ballast crib level (full, 3⁄4, or 1⁄2)

• Ballast shoulder width (up to 18 in)

• Tamping (yes or no)

• Consolidation level (MGT)

• Dynamic track stabilization (in terms of MGT equivalent)

• FRA track class (for lateral misalignments determination)

On the basis of the foregoing information, the following engineering input parameters are 

internally determined by the program:

• Lateral resistance

 Peak value, FP

 Limiting value, FL

 Displacement wP at FP

 Displacement wL at FL

• Torsional resistance

• Longitudinal resistance

• Misalignment amplitude and wavelength

The output of the program is the ∆Tall, the margin of safety for expected rail maximum 

temperature for a given neutral temperature, and the numerical value for the internally computed 

peak lateral resistance, FP.

Following is a description of the program’s internal algorithms, assumptions, tables and 

equations to determine the engineering parameters from the design parameter type inputs.

1. Lateral Resistance

The Single Tie Push Test (STPT) (see Appendix B) typically exhibits a characteristic such as 

shown in Figure 2-8. The technical parameters are peak resistance (FP), limit resistance (FL), 

deflection at peak resistance (wP), and deflection at limit resistance (wL).

Because this information is not usually available to railroad personnel, the industry version of the 

program estimates the lateral resistance from the track characteristics. The required 

characteristics include the following:

• Tie type

• Ballast type

• Track consolidation

• Track maintenance

• Shoulder width

• Crib level

Basis of Empirical Correlation

The peak lateral resistance of the tie under full ballast conditions can consist of the two 

components listed below. Empirical formulas based on test data have been developed for these 

components.

• The resistance component in the tamped condition
39

• The increment in the resistance resulting from traffic and the Dynamic Track Stabilizer

(DTS)

The resistance reduction due to a substandard ballast section (not having full crib or full 

shoulder) is obtained by using a reduction factor (see Equation 3-4) on the fully ballasted 

condition resistance; this factor is less than 1 and depends on the reduced shoulder and crib 

levels measured in tests [10,11].

Resistance under Full Ballast

The peak resistance for full ballast (full crib with shoulder between 12 and 18 in or more) can be 

expressed as:

FP = FP0 + ∆FP (Equation 3-1)

where FP0 is the tamped resistance, and ∆FP is an increment attributable to traffic consolidation 

and DTS.

Tamped Resistance FP0

The resistance depends on the type of tie, ballast, and type of tamping. For tamped track, 

Table 3-4 provides typical FP0 values. A range exists for these values, but the values shown here 

are on the conservative side, according to the test data from many sites [8, 9]. If lift accompanies 

tamping, subtract 300 from the FP0 values on Table 3-4.

Table 3-4. Resistances Tamped Conditions

Ballast Type Tie Type FP0 (lb)

Granite Wood 1,900

Slag Wood 1,650

Granite Concrete 2,300

Increment because of Consolidation, ∆FP

From tests [10], Figure 3-6 shows the increment caused by traffic consolidation, expressed as 

MGT, for wood ties.
40

Consolidation (MGT)

0 2 4 6 8 10

Peak lateral resistance increase (lbs/tie)

0

200

400

600

800

1000

1200

1400

Fitted equation

Data

Figure 3-6. Consolidation Influence - Wood Ties for Equation 3-2

The equation best fitting the curve for the wood tie track is the following:

F 60 MGT 944 1 e lbs

0.2 MGT

P 





 





 ∆ = • + • − − ; MGT<10 (Equation 3-2)

Tests have shown that the difference in the resistance increment for slag and granite ballast is not 

very significant [10]. Hence, Equation 3-2 is valid for slag and granite cases of the wood tie 

track.

In a similar manner, the increment in concrete tie resistance attributable to consolidation is 

provided by Equation 3-3 as a good fit to the test data on concrete ties shown in Figure 3-7:

MGT

P

0.4

F - 4.7 MGT 1360 1 lbs e

  − ∆= • + •−     ; MGT<10 (Equation 3-3)

(For MGT > 10, the model assumes asymptotic values just exceeding the MGT = 10 values.)
41

Consolidation (MGT)

0 2 4 6 8 10

Peak lateral resistance increase (lbs/tie)

0

200

400

600

800

1000

1200

1400

Fitted equation

Data

Figure 3-7. Consolidation Influence – Concrete Ties for Equation 3-3

In the case of dynamically stabilized track, the MGT term in the above equation is replaced by 

MGT + equivalent MGT in dynamic track stabilization (DTE), where DTE is an equivalent 

MGT for the given type of dynamic track stabilization operation. This equivalent MGT depends 

on the setting of the track stabilizer (frequency and head pressure), its speed, and the number of 

passes made. The DTE equivalent suggested for CWR-INDY use is 0.1 MGT based on 

European tests and UIC Code 720 [20]. This must be confirmed by tests on U.S. tracks for wood 

and concrete ties on tangent and curved tracks.

Reduction due to Reduced Ballast Levels

Optimum ballast section is one with full cribs and shoulder widths in excess of 12 in. If ballast 

component levels are reduced, the resistance will decrease, and a multiplication factor is used to 

adjust for the reduced resistance in CWR-INDY. Research has shown that there is a limiting 

shoulder width beyond which the resistance is not affected. This is between 18 and 

24 in according to reference [21], with the lower value of 18 in being assumed in CWR-INDY. 

If the ballast shoulder is less than 18 in and the crib level is not full, the lateral resistance will be 

reduced proportionately using the formula:





 



 = + +

18

Reduction Factor 1 2 3

w

d

S β β C β (Equation 3-4)

In the foregoing equation, Cd represents the ratio of existing crib depth to the full depth. Sw

represents the shoulder width 18 in (maximum) or less. Table 3-5 shows values of β1, β2, and β3. 

For full crib and 18 in or more shoulder, the factor is one, as seen in Equation 3-4.

Table 3-5. β Values for Equation 3-4

βi Wood Tie Concrete Tie

1 0.30 0.40

2 0.50 0.45

3 0.20 0.15
42

Lateral Resistance Equation

The general equation for FP becomes:

[ ] 



 



 = + + + − + + +

18 ( ) (1 ) 1 2 3

( )

0

_ w

d

MGT DTE

P P

S F F α MGT DTE γ e β β C β λ

(Equation 3-5)

where,

FP is in lb

FP0 is from Table 3-4

α is 60 for wood tie track; -4.7 for concrete tie track

γ is 944 for wood tie track; 1,360 for concrete tie track

λ is 0.2 for wood tie track; 0.4 for concrete tie track

MGT is tonnage expressed in millions of gross tons (limited to <100)

DTE is an MGT equivalent for DTS

β is from Table 3-5

Limiting Resistance

The limiting resistances are evaluated using the following empirical formulas based on test data 

in accordance to reference [9]:

• Wood tie track

Granite ballast

P P

L

P P

0.3F 500 lbs F 715 lbs

F

F F 715 lbs

if

if

 + ≥ = 

 <

(Equation 3-6)

Slag ballast

P P

L

P P

0.06F 600 lbs F 638 lbs

F

F F 638 lbs

if

if

 + ≥ = 

 <

(Equation 3-7)

• Concrete tie track

P P

L

P P

0.38F 950 lbs F 1532 lbs

F

F F 1532 lbs

if

if

 + ≥ = 

 <

(Equation 3-8)

Deflections

The deflections corresponding to FP and FL (see Figure 2-8) are also set in the CWR-INDY with 

the following values based on data from references [10, 11]:

• Wood tie track

wP = 0.3 inches

wL = 0.025 • FP + 2.6 inches (granite ballast)

wL = 0.009 • FP + 3.5 inches (slag ballast)

• Concrete tie track

wP = 0.25 inches

wL = 2.25 inches
43

2. Torsional Resistance

The torsional resistance versus the angle of rotation behavior between the ties and the rail is 

linearized based on the test data [10]. Table 3-6 shows the conservative torsional stiffness values 

for some frequently used fasteners.

Table 3-6. Torsional Stiffness, τo (for two rails)

Tie Spacing 

(inch) Fastener Type

τo

(in-kips/rad/in)

20 Cut Spikes on Wood 80

20 Pandrol on Wood 370

24 Pandrol on Concrete 10

24 McKay on Concrete 25

3. Longitudinal Resistance

Track longitudinal resistance is the resistance offered by anchors, fasteners, ties, and ballast to 

the rails as they tend to move in the longitudinal direction as a result of thermal force gradients, 

train braking, and traction forces and during buckling. This resistance is assumed to be 

proportional to the rail longitudinal displacement, hence defined through a stiffness parameter.

The longitudinal stiffness depends on the fastener type and ballast condition. Table 3-7 presents 

the values assumed in the program for wood tie anchors, Pandrol fasteners on wood or concrete 

ties, and McKay fasteners on concrete ties. The values shown are for tight anchors. For 

degraded or loose anchors, the assumed values are half of those shown in Table 3-7. The wood 

tie data presented are from the tests conducted at TTC, as reported in reference [9]. The concrete 

tie fastener resistance data comes from unpublished test data from TTC. These values are 

considered to be conservative. A loose anchor resistance program assumes half the value for

anchors.

Table 3-7. Longitudinal Stiffness (for two rails)

kf (lb/in/in)

Anchoring or Fastener Tamped Consolidated

ETA on Wood (tight anchors) 200 400

EOTA on Wood (tight anchors) 100 200

Pandrol on Wood 200 400

Pandrol on Concrete 200 400

McKay on Concrete 200 400

4. Misalignment Parameters

Misalignment amplitudes are internally set in accordance with the FRA class of track specified 

by the user, in accordance with FRA’s Track Safety Standards Compliance Manual limits. The 

program internally calculates the corresponding wavelength through a track lateral deformation 

algorithm based on pushing the track laterally to a deflection level equal to the FRA class 

misalignment amplitude specified and evaluating its corresponding wavelength.
44

CWR-INDY Features

• The inputs to CWR-INDY program for CWR buckling analysis are simpler than those for 

CWR-BUCKLE. The parameters are internally estimated from simple inputs such as the 

ballast crib and shoulder levels, types of track, and MGT.

• With the simple inputs converted into their equivalent scientific parameters, the CWR-INDY 

program uses CWR-BUCKLE as the computational engine to perform buckling analysis.

• The output of the program also includes the calculated peak lateral resistance value since this 

was not directly required as an input. In this sense, CWR-INDY can also function as a lateral 

resistance predictor. The output gives similar information as CWR-BUCKLE discussed in 

the previous subsection; however, provided as part of the output is graphical display of 

temperature versus lateral deflection.

Numerical Example

To illustrate how CWR-INDY works, this section presents a numerical example with input 

parameters similar to those used in the CWR-BUCKLE example, shown in Table 3-8.

Table 3-8. Input Parameters for INDY Illustrated Example

CWR-INDY Illustrative Example Inputs

Rail size (lb/yd): AREA 136

Tie type: Concrete

Tie weight (lb): 750

Tie spacing (in): 24

Track curvature (deg): Tangent

Ballast type: Granite

Fastener type: Pandrol

Crib level: Full

Tamped: NO

Stabilized: NO

Track consolidation (MGT): 1

Shoulder width (in): 12

Track class: 4

Rail neutral temperature (°F): 50

Maximum rail temperature (°F): 135

First, the program calculates a value for Tall, which in this example is 92 °F. The neutral 

temperature is given as 50 °F, meaning that the risk of buckling occurs at 142 °F. As in CWRBUCKLE, the input maximum temperature is 135 °F. CWR-INDY calculates the BSM as 7 °F, 

which is the difference between the maximum rail temperature and Tall. Table 3-9 displays the 

actual output shown by CWR-INDY. CWR-INDY also displays the calculated peak lateral 

resistance of 3,133 lb, which was obtained for the input parameters of concrete tie, granite 

ballast, 12-inch shoulder, full cribs, and 1 MGT of consolidation.
45

Table 3-9. CWR-Indy Output – Buckling Safety Analysis

Summary Conclusions

• The inputs for the CWR-INDY version are simpler than those for CWR-BUCKLE. The 

inputs are qualitative (tie type, ballast type, and track maintenance) and quantitative 

(consolidation level, ballast crib level, and shoulder width).

• Empirical formulas based on the available test data and engineering judgment have been 

developed to determine the scientific parameters required for CWR-BUCKLE from the 

CWR-INDY program inputs.

• The empirical formulas account for the effects of tamping, dynamic track stabilization, 

and consolidation by traffic on the track lateral resistance. Simple conservative values 

are proposed for the torsional and longitudinal resistance of commonly used fasteners 

based on test data. A detailed user’s guide for CWR-INDY as a part of CWR-SAFE is 

available in the self-help menu of the software.

3.3 CWR-RISK

The buckling safety analyses performed in CWR-BUCKLE and CWR-INDY can be considered 

as deterministic analyses because all the input parameters have definite values so that the track 

either buckles or does not. A track safety/maintenance strategy based on such deterministic 

analyses can be conservative and therefore expensive because it must be based on the worst-case

scenario for all parameters. It is expedient, therefore, to use a probabilistic approach, which can 

account for the inevitable statistical variations in input parameters. Such methodology will also 

provide improved flexibility in determining maintenance options and performing safety 

evaluations. As an example, with a risk-based approach, one can vary and select a range of 

values for the ballast condition and change the various CWR neutral temperature options to 

achieve similar levels of buckling safety. Such a choice permits more optimum allocation of 

maintenance resources.

The use of probabilistic approaches is the current trend in other technology areas requiring 

structural failure evaluations and safety assessments. The nuclear, aircraft, and naval industries 

have long benefited from such methods, which are extendable to railroad applications, 
46

particularly to developing probabilistic estimates of buckling failures. The risk methodology for 

this purpose requires not only the failure probability but also the severity or the consequence of 

the failure. For example, if buckling is predicted, does it cause a derailment and with what 

damage level? 

As applied to track buckling-induced derailments, a slow speed coal freight train operating in a 

high-degree curve may not result in the same level of damage as a high-speed corridor passenger 

train on a tangent track. Tangent track tends to buckle explosively with a large deflection, 

whereas the curved track may buckle progressively with comparatively smaller buckle 

amplitudes. Even if the parameters of the tangent and the curved track are such that they give 

equal probability of buckling, the severity of the passenger vehicle on the buckled tangent can be 

catastrophic compared with that of the freight car on the buckled curve. Therefore, the overall 

risk of buckling must be measured by the probability of the event occurring, weighted by the 

severity of the consequence or damage caused by that event.

In the present version of CWR-RISK, attention is focused on buckling probability. The severity 

aspects of the risk methodology are not treated in detail, but an assumption is made that the 

severity is proportional to the square of the train speed, which provides a rationale for railroads 

to use a slow-order policy when the potential exists for rails to buckle at high temperatures.

Buckling Probability Basics

The buckling load will be expressed in terms of the rail temperature increase over the neutral

temperature, and the strength is expressed in terms of the allowable temperature increase, ∆Tall.

Thus,

(∆T)Load = Tr – Tn (Equation 3-9)

(∆T)Strength = ∆Tall (Equation 3-10)

The fundamental parameters in the evaluation of failure probability of a structure are load and 

strength, both of which vary probabilistically in service life of the structure. The intersecting or 

interference zone in this type of graph represents the temperature regime in which the load 

equals or exceeds the strength. The probability of this load exceeding the strength is the failure 

probability of the structure (see hatched area in Figure 3-8). It can be evaluated on the basis of 

the convolution integral, as discussed in Appendix C.
47

Figure 3-8. Schematic of Interference Principle

The allowable temperature increase ∆Tall is determined for the following statistically varying 

primary track parameters:

• Lateral resistance

• Lateral misalignment

Other parameters, such as the torsional and longitudinal resistance and vehicle loads, are also 

important in the buckling strength assessment. Their influence is generally small when 

compared with that of the two primary parameters (see Chapter 4); hence, their variations in the 

field conditions are not accounted for in CWR-RISK. Figure 3-9 shows these and other 

secondary parameters, which will be treated as having prescribed deterministic values.

The load probability is obtained from the neutral temperature probability and the rail temperature 

under consideration. The load is expressed as (Tr - Tn), and only the positive part representing 

the compressive load is important since tensile load cannot cause buckling. Figure 3-9 shows the 

overall approach. The strength probability is calculated for the statistical variations of the two 

primary parameters, namely the lateral resistance and misalignments. In CWR-RISK, CWRBUCKLE is also used as the computational engine, as discussed in Chapter 5.
48

Figure 3-9. Buckling Risk Evaluation Methodology

The rail temperature is an important parameter in the load quantification. From a practical point 

of view, this represents a critical factor for the railroads on slot timing decisions for CWR track 

maintenance, and slow orders. With the probabilistic descriptions of lateral resistance, 

misalignment, and rail neutral temperature, CWR-RISK calculates the probability of buckling at 

a given rail temperature. The following will first describe the statistical inputs required in CWRRISK.

Statistical Input Parameters for CWR-RISK

In the following paragraphs, the three important input parameters, (i) rail neutral temperature, (ii) 

lateral resistance, and (iii) lateral misalignment amplitude, will be described in statistical terms. 

There is some uncertainty associated with the parameters that vary along the track. It is assumed

that a sufficient number of measurements are made over an appropriate length of track segment 

to characterize the parameters. The number of measurements and segment length are to be 

established through properly designed trial studies and experiments. Appendix D gives 

recommendations as to the minimum number of measurements required.

Rail Neutral Temperature Frequency

CWR installation temperature is typically set within a desired range established in the railroad’s 

procedures and is typically between 90 °F and 110 °F, depending on the geographic location. 

This installation temperature is the rail initial neutral temperature. However, as research has 

shown [12, 14], the neutral temperature does not stay at the rail installation temperature and can 

decrease to lower values, as low as 50 °F in some cases. These reductions generally result from 

rail and track movements (creep, curve breathing, track settlement) and track maintenance 

activities. The frequency distribution in this range is not expected to be a normal distribution, 

although this has not been evaluated. For the purpose of numerical illustration, Figure 3-10 

shows the assumed distribution based on limited U.S. testing and field experience.
49

The neutral temperature considered here is the average of the two rails. The distribution in 

Figure 3-10 is spatial. Although the neutral temperature can vary with time after CWR 

installation or rewelding and trafficking, it is assumed that a steady-state value exists at a given 

location. Figure 3-10 can be constructed on the basis of one-time (preferably spring) testing at a 

number of locations spread over the territory. Methods are available to determine the neutral 

temperature, such as through pre- and postcutting strain gage measurements or rail uplift type

methods. It is, therefore, possible to develop a database on the neutral temperatures at different 

locations to determine the frequency as shown in Figure 3-10. The CWR-SAFE User’s Guide 

provides additional information on the binning and construction of frequency distributions such 

as in Figure 3-10, which is based on measurements.

Rail neutral temperature (°F)

20 30 40 50 60 70 80 90 100 110

Frequency

0.0

0.1

0.2

0.3

0.4

0.5

75 0.43

85 0.21

95 0.00

65 0.20

55 0.12

45 0.04

35 0.00

Frequency Rail neutral 

temperature (°F)

75 0.43

85 0.21

95 0.00

65 0.20

55 0.12

45 0.04

35 0.00

Frequency Rail neutral 

temperature (°F)

Figure 3-10. Assumed Rail Neutral Temperature Distribution

Lateral Resistance Frequency

Lateral resistance varies along the track for a given track type and condition. Scatter in lateral 

resistance values along the track is inherent in the nature of the railroad environment because of 

varying cribs and shoulders, recent maintenance, tie lateral and longitudinal movements, local

disturbances, such as pumping ballast, and wet and dry condition influences. Extensive testing 

presented in reference [10] has shown that it is possible to describe a probability density of the 

resistance for a given type of track and level of consolidation. Some of the distributions found in 

the field tests approximate a normal distribution, but it is not necessary to assume such normalcy 

in the model.

Figure 3-11 presents an assumed resistance distribution. The resistance can be determined at a 

number of locations using test fixtures and measurements, such as STPT, as discussed in 

Appendix B.
50

60 70 80 90 100 110 120 130 140

0.0

0.1

0.2

0.3

0.4

0.5

110 0.2

120 0.1

128 0.0

100 0.4

90 0.2

80 0.1

72 0.0

Frequency

Lateral 

resistance 

(lb/in)

110 0.2

120 0.1

128 0.0

100 0.4

90 0.2

80 0.1

72 0.0

Frequency

Lateral 

resistance 

(lb/in)

Lateral resistance (lbs/tie)

Frequency

Figure 3-11. Assumed Lateral Resistance Distribution

Lateral Misalignment Distribution

The allowable lateral misalignment depends on the classification of track in accordance with 

FRA Track Safety Standards definitions. The current allowable misalignment amplitudes for 

U.S. track Classes 4–8 are usually given as maximum deviations from the ideal shape over a 

given chord length. For tangent track, based on a 62-chord length, the permissible deviations are 

1.5 for Class 4, 0.75 in for Classes 5 and 6, and 0.5 in for high-speed track Classes 7, and 8.

Track geometry records can be used to evaluate the probability distributions of the misalignment 

amplitudes. Although the respective wavelengths are also important, their independent 

distributions are not required as direct inputs because they are automatically generated within the 

program. Figure 3-12 shows a typical Class 4 alignment distribution. In the figure, the negative 

values are not present because the example is for a curved track. 
51

Misalignment amplitude (in)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Frequency

0.0

0.1

0.2

0.3

0.4

0.5

1.50 0.00

1.00 0.05

0.40 0.30

0.25 0.40

0.00 0.25

Misalignment (in) Frequency

1.50 0.00

1.00 0.05

0.40 0.30

0.25 0.40

0.00 0.25

Misalignment (in) Frequency

Figure 3-12. Assumed Misalignment Amplitude Distribution for Class 4 Track

For tangent tracks, the negative values are typically folded over to the positive sense so that the 

largest amplitudes dominate regardless of direction. Table 3-10 shows the deterministic input

parameters that are required in the analysis.

Table 3-10. Deterministic Input Parameters Required for CWR-RISK Analysis

Data Inputs for CWR-RISK Illustrative Example

Rail size (lb/yd): AREA 136

Tie type: Wood

Tie weight (lb): 200

Tie spacing (in): 20

Track curvature (deg): 5

Ballast type: Granite

Tie ballast friction coefficient: 1.2

Torsional resistance (in-kips/rad/in): 120

Longitudinal stiffness (psi): 200

Foundation modulus (psi): 6,000

Vehicle type: Hopper

Illustrative Example Results

For the illustrative example, the report used the distributions in Figures 3-10 to 3-12 and the 

parameters in Table 3-10. The output of CWR-RISK is a graph displaying the probability of 

track buckling as a function of rail temperature as shown Figure 3-13. The figure shows that the 

probability of buckling is zero (or very close to zero) until the rail temperature reaches a critical 

temperature denoted by Tc equaling 104 °F. The critical temperature, Tc, is based on a 1 in 1,000
52

occurring event, taken as 10-3 probability value. Beyond this point, the probability increases at a 

slow rate until the knee in the curve, after which, a steep increase with rail temperature occurs.

Rail temperature (°F)

100 110 120 130 140 150 160 170

Probability of buckling

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

TC=104 °F

Figure 3-13. Probability of Buckling versus Rail Temperature

As shown in this example, the probability of buckling for a given set of input parameters can be 

determined as a function of the rail temperature. At low rail temperatures, the probability is zero 

or very low, and at high temperatures, it can reach 100 percent. With this type of information,

several practical issues can be addressed including the following:

• Determination of critical temperature at which slow orders should be imposed

• Estimation of annual number of expected buckles over a given territory for a given annual 

rail temperature distribution

• Risk information to rail operators on train speeds at temperatures above the critical, 

depending on the level risk accepted

Chapter 5 will discuss these aspects further, and Appendix C provides the probabilistic theory for 

CWR-RISK.

3.4 Summary

CWR-SAFE a comprehensive CWR buckling program includes three basic modules for safety 

analyses.

The CWR-BUCKLE module is a fundamental buckling analysis program using a deterministic 

approach. The program has been validated by several full-scale tests and requires scientific input 

parameters. The output includes the buckling response curve, critical temperatures, safe 

allowable temperatures, energy required to buckle the track, margin of safety, and methods to 

increase the margin of safety if it is inadequate.
53

The CWR-INDY module is specially tailored for use by the rail industry. This program uses 

simple track inputs, such as ballast type, shoulder and crib level, and tie type. Like CWRBUCKLE, this program provides deterministic values for the buckling temperature, safe 

allowable temperature, and margin of safety.

CWR-RISK is the third module in CWR-SAFE for which three key inputs, lateral resistance, 

misalignment, and neutral temperature, require statistical descriptors. All other parameters 

remain deterministic. The output of the CWR-RISK program is the probability of buckling 

versus rail temperature.
54

4. Track Buckling Parameters and Sensitivity Study

This chapter presents a detailed parametric and sensitivity study on the basis of the CWRBUCKLE program in accordance with the theory presented in Chapter 2 and the program 

description in Chapter 3. This chapter describes the effects of track resistance, misalignments, 

curvature, and vehicle parameters on track buckling strength, specifically in terms of influences 

on ∆TBmax/min. Chapter 4 will also address special cases representing important aspects and 

conditions, such as static versus dynamic buckling, weak versus strong lateral strength tracks, 

and wood versus concrete tie track buckling. Appendix B presents the techniques to measure the 

key parameters.

4.1 Parametric Study Basis

In most of the parametric sensitivity studies presented below, each parameter of interest is varied 

through a reasonable range while the remaining parameters are kept at a nominal value in 

accordance with Table 4-1, unless otherwise specified.

Table 4-1. Ranges and Nominal Values of Buckling Parameters

Parameter Variable Range Nominal Value

Rail Size Area and moments of inertia 100#RE–141#RE 136#RE Track resistance Lateral 

Lateral resistance peak (FP) 50–300 lb/in 100 lb/in

Tie/ballast friction

coefficient (μf) 0.65–2.0 1.2 (wood)

0.86 (concrete)

Torsional Torsional stiffness (τo)/fastener 100–5,000 in-kips/rad 1,200 (wood)

500 (concrete)

Longitudinal Longitudinal stiffness (kf) 25–500 lb/in/in 200 lb/in/in

Curvature Degree of curvature Tangent - 10 deg 5 deg

Misalignments Misalignment amplitude (δo) 0.5–3.0 in 1.5 in

Misalignment wavelength (2L0) L0=100–500 in L0 = 180 in

Track foundation 

modulus Vertical stiffness (kV /track) 2,000–10,000 psi 6,000 psi

Vehicle 

parameters

Axle load 15,000–75,000 lb 66,000 lb

Truck center spacing (TCS) 350–700 in 506 in

4.2 Static versus Dynamic Buckling

Chapters 2 and 3 describe the important aspects of static versus dynamic buckling and this 

section will provide a numerical example to illustrate the key differences. Recalling that static 

buckling refers to a purely thermal load induced buckling (i.e., without the influence of vehicle 

loads, dynamic uplift, and train energy inputs), Figure 4-1 shows the buckling response curve for 

the input parameters noted in the figure.
55

Lateral deflection (in)

0 1 2 3 4 5 6 7 8 9 10

Temperature increase above neutral (°F)

0

10

20

30

40

50

60

70

80

90

100

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp =100 lb/in.

μ=1.20, Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 in

Dynamic (∆T Input parameters Bmax = 78° F, ∆TBmin = 75° F)

Static (∆TBmax = 91° F, ∆TBmin = 77° F)

Figure 4-1. Static versus Dynamic Buckling Behavior

The track is expected to buckle out statically at its ∆TBmax= 91 °F since the track is in a state of 

unstable equilibriums. The corresponding dynamic case is also shown in the figure, indicating a 

small buckling regime between ∆TBmax and ∆TBmin of 78 °F and 75 °F. From a safety point of 

view (and in accordance with the safety criterion in Chapter 2), the permissible temperature 

increase value should be below ∆TBmin value of 75 °F. As this example illustrates, static 

considerations provide unconservative numbers hence inadmissible for buckling safety 

evaluations, and therefore dynamic theory applications are required. Consequently, the use of 

the dynamic theory is implicit in all the subsequent parametric studies presented here.

Influence of Rail Size

The pertinent rail properties include rail cross-sectional area; cross-sectional moments of inertia 

for bending in both the vertical (Iyy) and horizontal (Izz) planes; and the track self-weight, which 

includes the weight of rails, ties, and fasteners. Table 4-2 lists these parameters according to the 

rail size (lb/yd). Figure 4-2 shows the results for the effects of rail size on dynamic buckling for 

rail sizes ranging from 100 to 140 lb/yd.
56

Table 4-2. Typical Rail Properties

Rail

(lb/yd)

Area

(in2)

Track weight

(lb/in)

Iyy

(in4)

Izz

(in4)

100 9.95 18.14 49.0 9.4

115 11.25 18.87 65.6 10.8

132 12.95 19.84 88.2 14.6

136 13.35 20.06 94.9 14.7

140 13.80 20.31 96.8 14.8

Properties shown are for a single rail, except for track weight, which includes two rails, 

wood ties, and tie plate/cut spike fasteners.

The results show that the ∆TBmax and ∆TBmin critical temperatures decrease with increasing rail 

size, with ∆TBmax showing the larger decrease. Although the rail bending moment of inertia 

increases with increasing size, the rail cross-sectional area also increases. The increase in area 

increases the thermal force, which offsets the effect of corresponding increase in bending 

stiffness, thus reducing the overall buckling strength.

Rail Size (lb/yd)

100 115 119 132 133 136 140

Temperature increase above neutral (°F)

60

70

80

90

100

110

120

60

70

80

90

100

110

120

∆TBmax

∆TBmin

Wood tie track

5 deg curve

20 in. tie spacing

Granite, Fp =100 lb/in.

μ = 1.20

Kv= 6000 psi

Kf = 200 psi

τo=120 in-kips/rad/in

δo= 1.5 in. Lo=180 in

Input parameters

Figure 4-2. Influence of Rail Size

Although a smaller rail size improves buckling strength, it would not be a preferred way to 

enhance buckling safety since rail size requirements are usually dictated by wheel loads, rail 

maximum stress/deflection, and fatigue life considerations. Additionally, within the typical 

heavy-haul rail size range of 132 to 141 lb, the realized buckling strength benefits of rail size are 

not significant.
57

Influence of Lateral Resistance

The tendency of the track to buckle laterally due to thermal loads is resisted by the ballast lateral 

reaction forces exerted through the ties. Extensive experimental studies have been performed 

under FRA sponsorship to characterize this lateral resistance through the use of an STPT method 

[10, 11, 22], in which a lateral load is applied to an unfastened tie, and the resulting force versus 

tie displacements are measured, as discussed in Appendix B. Initial force characteristic, Figure 

2-8, is approximately linear up to a maximum or peak value (FP) at a small displacement (usually

on the order of 0.25–0.5 in), which is followed by a spring softening effect (tie/ballast friction 

breaking away) down to a limiting resistance value (FL), occurring at a limiting or constant 

lateral deflection denoted by wL.

Several idealizations of this lateral resistance characteristic have been investigated for computer 

simulations, such as constant, softening, and full nonlinear (see Figure 4-3), and their influence 

on ∆TBmax/min are available in [4], where it is also shown that the differences between the 

softening and full nonlinear representations are generally negligible. Although the full nonlinear 

function is a more complete description of typical lateral response characteristics, it is also 

mathematically complex, requiring large numbers of iterations for convergence in the computer 

program. The softening lateral resistance produces results quite close to that of the full nonlinear 

characteristic; hence, for calculation of buckling response, the idealized softening lateral 

resistance characteristic is chosen, Figure 4-3 B.

Figure 4-3. Lateral Resistance Idealizations

FL

FP

Deflection, w

Lateral resistance FL

FP

Deflection, w

Lateral resistance

FP

Deflection, w

Lateral resistance

A) Constant lateral resistance

B) Softening or drooping 

lateral resistance

C) Full nonlinear lateral resistance

FL

FP

Deflection, w

Lateral resistance
58

The softening resistance function is defined by two variables: the ratio of the limiting and peak 

resistances and the deflection at the limiting resistance. These parameters depend on the type of 

ballast and level of ballast consolidation. To simplify and reduce the number of parameters 

required for definition of the softening resistance, a series of tests were conducted [9, 10] to 

correlate FL and wL to FP, which is a parameter that can be directly measured using single tie 

push tests. On the basis of this experimental work, approximate linear relationships were 

developed between FP and FL and between FP and wL as shown in [9]. Table 4-3 lists these 

correlations for granite and slag ballast. Thus, the peak lateral resistance, FP, is the primary 

parameter that is used to study the effects of lateral resistance on buckling strength. Should the 

requirement exist to use FP , FL, and wL in a specific analysis, CWR-BUCKLE has the optional 

feature to handle that case.

Table 4-3. Correlations for “Softening” Lateral Resistance

Ballast type FP/FL correlation FP/wL correlation

Granite FL = 0.3 FP + 25 wL = 0.025 FP + 2.6

Slag FL = 0.06 FP + 30 wL = 0.009 FP + 3.5

Wood tie track; units for FP and FL are lb/in; and units for wL are inches.

Influence of Peak Resistance (FP ) on Buckling

To determine the effects of lateral resistance on buckling, the parameter FP is varied over a 

practical range representing very weak or recently maintained track (FP = 50 lb/in) up to very 

strong, well consolidated track (FP = 300 lb/in). Figure 4-4 shows the buckling temperature 

results. The results show that for very weak tracks, progressive buckling can occur for 

temperature increases of 50 °F or less for the parameters considered. Such tracks are vulnerable 

to buckling in summer, even with a rail neutral temperature of 80 °F.
59

∆TBmax

∆TBmin

wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, μ=1.20

Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters

∆TProg

Peak Lateral resistance (lb/in)

50 100 150 200 250 300

Temperature increase above neutral (°F)

0

50

100

150

200

250

Figure 4-4. Influence of Peak Lateral Resistance, FP

As FP increases, the upper and lower buckling temperatures increase. For very strong tracks (FP

>200 lb/in), the minimum buckling temperature increase value is quite high (100 °F) even for the 

5° curve used in the example.

Clearly, it is important to maintain a high lateral resistance to reduce the track buckling potential. 

The railroads can use the lateral resistance condition influencing parameters (i.e., consolidation 

level, shoulder width, cribs content, tie type) to help keep FP high, thereby better ensuring CWR 

buckling safety.

Influence of Limiting Resistance (FL )

The nonlinear lateral resistance characteristic (i.e., FP versus FL behavior) has a very important 

influence on the buckling response, as shown in Figure 4-5, with the key feature of its impact on 

∆TBmin (and very little on ∆TBmax). As can be seen, as FL decreases from a value of 3600 lb/tie 

to1400 lb/tie (while keeping FP constant at 3600 lb/tie), the ∆TBmin value decreases from 121 °F 

to 89 °F. This is very important because the ∆TBmin value mostly controls the Tall determination 

in the buckling safety criterion as discussed in Section 2.3.2.
60

Lateral displacement (in)

0 2 4 6 8 10 12 14

Temperature increase above neutral (°F)

0

20

40

60

80

100

120

140

160

FL = 3600 lbs/tie, ∆TBmax/min = 143/121° F

FL = 2000 lbs/tie, ∆TBmax/min = 139/98° F

FL = 1400 lbs/tie, ∆TBmax/min = 139/89° F

FL = 1600 lbs/tie, ∆TBmax/min = 139/93° F wood tie track

136RE rail, 5° curve

20 in. tie spacing

Granite, Fp=180 lb/in.

Kf

=200 psi

Kv=6000 psi

τo=120 in-kips/rad/in

Lo=180 in., δo=1.5 in.

Input parameters

Figure 4-5. Influence of Limiting Resistance

Strong versus Weak Resistance Tracks

To provide an overall influence assessment of lateral resistance on the track buckling response, a 

comparison is made between lateral resistance conditions that would represent weak, average 

and strong lateral resistances in accordance with the representations shown in the right hand side 

of Figure 4-6.

For the baseline parameters considered, the strong track exhibits high and distinct ∆TBmax and 

∆TBmin values of 140 °F and 100 °F, respectively. The average track shows a lower and almost 

identical ∆TBmax and ∆TBmin values of 96 °F and 93 °F, while the weak case exhibits a highly 

progressive behavior. From the buckling safety considerations of Chapter 2, the strong track has 

a buckling regime of between 100 °F and 140 °F, with the safe temperature being 100 °F above 

neutral. Similarly, for the average condition, the safe temperature must be below the 93 °F 

value. For the weak track’s progressive case, beyond 70 °F large displacements occur, so the 

safe temperature should be below this number.
61

0.25

2000

3600 Strong

1600

0.35

2400

Average Average

0.5

1400

Weak

Lateral displacement (in)

Temperature increase above neutral (°F)

Strong

Average

Weak

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, μ=1.20

Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters

0 2 4 6 8 10 12 14 16

0

20

40

60

80

100

120

140

160

Lateral resistance

Lbs/tie

Lbs/tie

Lbs/tie

Deflection (in)

Deflection (in)

Deflection (in)

Figure 4-6. Buckling Response Behavior for Weak, Average, and Strong Tracks

Influence of Tie/Ballast Friction

The tie bottom surface roughness is an important parameter as it determines the component of 

the base resistance. If all other conditions remain unchanged (i.e., rail size, tie weight, ballast 

type), the lateral resistance should increase with tie bottom roughness; hence the buckling 

strength should also increase. In the case of wood ties, the tie bottom roughness controls the 

bottom surface friction resistance as the ballast tends to lock itself into the tie bottom. The 

concrete ties are relatively smooth surfaced, and even when the tie bottoms and sides are 

artificially roughened at the time of manufacture (as in some scalloped tie designs), the surfaces 

tend to become smooth in course of service life due to the pounding and grinding action between 

the tie and the ballast particles.

The tie bottom roughness factor is artificially expressed as a friction coefficient which is defined 

as the ratio of the measured base resistance to tie self-weight. Although in principle it is similar, 

this coefficient should not be confused with the Coulomb friction between two surfaces, whose 

value never exceeds unity. The lateral resistance can be expressed as the sum of the base, side, 

and end shoulder resistance components of the tie:

F = Fb+Fs+Fe Equation (4 – 1)
62

Defining the coefficient μf in terms of weight of tie, Q (including the weight of rail and 

fasteners):

Fb = μf Q (Equation 4 – 2)

μf can be considered an index of tie bottom roughness. If the vehicle contributes an additional 

load of Rv (lb/in.) on the tie, the dynamic resistance can be calculated from Equation 19 in 

Appendix A. The coefficient μf in Equation 19 is strictly a function of total tie reaction. The 

average value of μf typically varies in the range from 0.65 to 2, and Appendix B discusses its 

determination. For this parametric exercise, Table 4-4 shows assumed values of the three 

resistance components for wood tie track with a typical weight Q = 20 lb/in.

Table 4-4. Three Components of Resistance and Their Assumed Variations with μf

μf Fs (lb/in) Fe (lb/in) Fb (lb/in) Fp (lb/in)

2.0 48 24 48 120

1.0 48 24 24 96

0.65 48 24 16 88

Figure 4-7 shows the effects of the tie-ballast friction coefficient. As expected, an increase in tie 

bottom surface roughness (increasing μf) increases both buckling temperatures. Over the range 

studied here, the increase of the buckling temperatures is less than 10 °F.

Tie ballast friction coefficient

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Temperature increase above neutral (°F)

70

75

80

85

90

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite

Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 

in.

Input parameters

∆TBmax

∆TBmin

Figure 4-7. Influence of Tie-Ballast “Friction” Coefficient
63

Care must be exercised in attempting to evaluate μf influence without making the direct 

adjustments in FP along the lines of Table 4-4 (i.e., if one assumes a fixed FP and runs the 

variations on μf, results opposite in trend to Figure 4-7 will be obtained).

Influence of Torsional Resistance

As noted in Chapter 2, torsional resistance is exerted on the rails by the fasteners, and the 

resistance characteristics vary with tie and fastener types and influences the rigidity of the track 

structure in the lateral plane, hence its buckling strength. Based on torsional resistance 

measurements from tests conducted for various tie and fastener types, the response can be 

idealized as a linear response as a function of rotation:

τ = τoθ (Equation 4-3)

where

τ = applied torque per fastener

τo = torsional stiffness per rail seat

θ = rotation angle

Table 4-5 gives typical ranges of stiffness values for single rail/fastener obtained from test data 

presented in [9]. Consistent with the model’s track-beam assumption the values of τo in Table 4-

5 are to be multiplied by 2 for the two rails and then divided by the tie spacing to obtain the 

correct track/unit length dimension. The effects of fastener torsional resistance on buckling are 

examined using the linear torsional stiffness values ranging from 0 to 3500 in-kips/rad. Figure 4-

8 and Figure 4-9 show the results.

Table 4-5. Torsional Stiffness Values

Type of Tie Fastener τo (in-kips/rad)

Hardwood Pandrol 3,700–7,400

Hardwood Cut Spikes (4) 800–1,400

Concrete Pandrol 120–520

Concrete McKay 300–440
64

Figure 4-8. Torsional Stiffness Influence on Buckling Response

Torsional stiffness per fastener (in-kips/rad)

0 500 1000 1500 2000 2500 3000 3500

Temperature increase above neutral (°F)

50

60

70

80

90

100

∆TBmax

∆TBmin

∆TProg

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=100 lb/in.

Kv=6000 psi

μ=1.20

Kf=200 psi

δo=1.5 in., Lo=180 in.

Input parameters

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=100 lb/in.

Kv=6000 psi

μ=1.20

Kf

=200 psi

δo=1.5 in., Lo=180 in.

Input parameters

Figure 4-9. Influence of Torsional Stiffness on Buckling Temperatures

As can be seen from Figure 4-9, both ∆TBmax/min values increase with increasing torsional 

stiffness, and in fact, the response curves change from nonprogressive to a progressive (which is 

contrary to other parametric influences where typically a weakening of a parameter results in a 

Lateral displacement (in)

0 1 2 3 4 5 6 7 8 9 10

Temperature increase above neutral (°F)

0

20

40

60

80

100

120

140

τo = 0 in-kip/rad

τo = 1000 in-kip/rad

τo = 2000 in-kip/rad

τo = 3000 in-kip/rad

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=100 lb/in.

Kv=6000 psi

μ=1.20

Kf

=200 psi

δo=1.5 in., Lo=180 in.

Input parameters
65

progressive buckling response). The fact that the lower buckling temperature, ∆TBmin , is the 

more sensitive to torsional resistance changes (as illustrated in Figure 4-9) becomes important in 

safety considerations since it is the ∆TBmin value that controls the buckling safety criterion. The 

results further indicate that the better the rail/tie fastener connections are, (i.e., the higher the 

fastener’s torsional resistance) the higher is the buckling strength.

Effects of Longitudinal Resistance

Track longitudinal resistance is the resistance offered by the fasteners to the rails and by the 

ballast to the ties against longitudinal motion. In CWR, it becomes very important in providing 

the constraint against thermal expansion, hence in the generation of longitudinal forces and 

neutral temperature constancy. In principle, the two components of resistance can be considered 

as two springs in series, so the net resistance is controlled by the weaker of the two components. 

Therefore, in good strong ballast, the fastener resistance dominates the longitudinal restraint, so 

when ties are ineffectively anchored the longitudinal resistance can be very low. Conversely, 

with properly anchored ties, the resistance offered by the ballast becomes important. Data from 

tests (Appendix C) conducted at TTC indicate that longitudinal resistance may be idealized as in 

Figure 4-10. Due to the small longitudinal displacement that occurs during buckling (usually 

less than 0.25 in as evidenced in several buckling tests), a linear characteristic can be assumed in 

the analysis so that:

f = kfu (Equation 4-4)

where

kf = longitudinal stiffness

u = longitudinal displacement Longitudinal resistance (lbs/in.)

Longitudinal displacement (in.), u

f

kf

uo

f

o

=

f

o

uo

Figure 4-10. Typical Longitudinal Resistance Characteristic

Table 4-6 lists typical measured values of kf from tests [9] for various tie and ballast conditions 

for wood ties. For this parametric study, a typical range of stiffness values (25 to 500 psi) is 

used. Figure 4-11 shows the buckling results. As can be seen, the lower critical temperature is 

more sensitive to the changes in longitudinal stiffness, and the upper critical temperature is 

essentially independent of the changing stiffness. Since kf influence on TBmin is relatively small, 

the need to determine or know kf exactly is not as important as some of the other parameters. 
66

However, track longitudinal stiffness is very important in controlling neutral temperature 

variations and, as such, plays an important role in buckling safety management.

Table 4-6. Typical Measured Longitudinal Stiffness Values

Tie condition Ballast condition kf (psi)

ETA Consolidated 324

EOTA Consolidated 254

ETA Tamped 213

EOTA Consolidated 1⁄2 crib 126

E3TA Tamped 178

ETA = every tie anchored

EOTA = every other tie anchored

E3TA = every third tie anchored

Longitudinal stiffness (psi)

0 100 200 300 400 500

Temperature increase above neutral (°F)

70

72

74

76

78

80

∆TBmax

∆TBmin

Wood tie track

136RE rail 

5 deg curve

20 in. tie spacing

Granite, Fp=100 lb/in.

Kv=6000 psi

μ=1.20

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters

Figure 4-11. Influence of Longitudinal Resistance

Influence of Track Curvature

The effects of track curvature are investigated using curves ranging from tangent to 9°. To 

illustrate the influence of the lateral resistance on curved track buckling, the buckling 

temperatures are calculated for the three cases of strong, average, weak resistances of Section 

4.3, representing tracks of varying peak lateral resistances of FP = 70, 120, and 180 lb/in. Figure 

4-12 shows the results of this study. In each case, the results show that increasing curvature 

reduces the upper and lower buckling temperatures. For high curvature tracks or tracks with a 
67

weak lateral resistance, the buckling temperatures are drastically reduced in comparison to 

tangent track. It is important that, for weak tracks (such as with FP = 70 lb/in.), progressive 

buckling can occur in curves of about 7° and higher. The safe temperature increase for such 

tracks thus falls below 55 °F and can be vulnerable to buckling in summer, even with a rail 

neutral temperature of 80 °F. Figure 4-13 portrays the curvature influence on ∆TBmax/min 

specifically, showing the decreasing buckling strength influence with increasing track curvature.

Figure 4-12. Effect of Curvature on the Buckling Response Behavior
68

Curvature (deg)

0 1 2 3 4 5 6 7 8 9 10

Temperature increase above neutral (°F)

50

60

70

80

90

100

110

120

∆TBmax

∆TBmin

∆TProg

Wood tie track

136RE rail

20 in. tie spacing

Granite, Fp=100 lb/in.

μ=1.20

Kv

=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in.

δo=1.5 in., Lo=180 in.

Input parameters

Figure 4-13. Influence of Curvature on Buckling Temperatures

Effects of Initial Misalignments

Misalignment parameters include the track misalignment amplitude (δ0) and misalignment 

wavelength (2Lo). Several functional forms are available for idealization of the misalignment 

shape, including sinusoidal, parabolic, or fourth order polynomial representations. Of these, the 

fourth order shape is employed as it is more consistent with the Shape I buckling mode shape 

assumption, which requires zero end slopes and deflection.

Influence of Misalignment Amplitude

The misalignment effects are examined first by using typical misalignment amplitudes ranging 

from 0.5 to 3 in, representing FRA Class 8 through Class 2 alignment defects. In this example, 

the corresponding misalignment half wavelength L0 is fixed at 180 in for all amplitudes. Figure 

4-14 shows the results. It should be noted, however, that in reality a specific misalignment 

amplitude has its own corresponding wavelength, consistent with the track-beam’s bending 

flexibility. In fact, the model has an internal wavelength computational algorithm which can 

automatically calculate a wavelength for specific amplitude. As Figure 4-14 indicates, both 

buckling temperatures decrease as the misalignment amplitude increases, with the upper critical 

temperature being more sensitive to these changes. The changes in the lower critical temperature 

are more modest, and progressive buckling occurs for misalignment amplitude of about 2 in.
69

Misalignment amplitude (in.)

0.5 1.0 1.5 2.0 2.5 3.0

Temperature increase above neutral (°F)

40

60

80

100

120

140

∆TBmax

∆TBmin

∆TProg

Wood tie track

136RE rail, 5 deg

20 in. tie spacing

Granite, Fp=100 lb/in.

μ=1.20

Kv

=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in.

Lo=180 in.

Input parameters

Figure 4-14. Influence of Misalignment Amplitude on Buckling Temperatures

Influence of Misalignment Wavelength

The misalignment wavelength effects are studied using fixed misalignment amplitude of 1.5 in

and varying misalignment half-wavelengths from 150 to 500 in. Figure 4-15 shows the results. 

The lower critical temperature is relatively insensitive to the effects of wavelength; however, the 

upper critical temperature decreases sharply as the wavelength is reduced. Progressive buckling 

condition is reached at a half wavelength of approximately 150 in, when the ratio of 

misalignment amplitude to misalignment wavelength is largest.
70

Misalignment half wavelength (in.)

150 200 250 300 350 400 450 500

Temperature increase above neutral (°F)

40

60

80

100

120

140

160

∆TBmax

∆TBmin

∆TProg

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=100 lb/in.

μ=1.20, Kf

=200 psi

Kv=6000 psi

τo=120 in-kips/rad/in.

δo=1.5 in. 

Input parameters

Figure 4-15. Influence of Misalignment Wavelength on Buckling Temperatures

The foregoing results indicate that control of line defects in terms of amplitudes and wavelengths 

helps in increasing the CWR track buckling strength. Sharp kinks with small wavelengths 

should be minimized in revenue service conditions.

Effects of Track Foundation Vertical Stiffness

As noted in Section 2, the presence of vehicle loads causes an uplift in the track, which is 

partially dependent upon the track foundation stiffness. The effects of foundation stiffness are 

examined using a typical range of 3,000 to 10,000 psi and are shown in Figure 4-16. ∆TBmax is 

more sensitive to stiffness variations than ∆TBmin, and for stiffness values below 4,000 psi the 

response becomes progressive.
71

Vertical stiffness (psi)

3000 4000 5000 6000 7000 8000 9000 10000

Temperature increase above neutral (°F)

70

75

80

85

90

∆TBmax

∆TBmin

∆TProg

Wood tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=100 lb/in.

μ=1.20, Kf

=200 psi

τo=120 in-kips/rad/in.

δo=1.5 in., Lo=180 in.

Input parameters

Figure 4-16. Influence of Track Foundation Vertical Stiffness

Effects of Vehicle Parameters

As discussed in Section 2 and implied by Figure 4-1, the effect of vehicle loads is to cause rail 

uplift and thus reduction of buckling strength in comparison with the static buckling case. The 

primary vehicle parameters controlling the uplift are the axle loads and the truck center spacing 

(TCS), which vary with vehicle size and type.

Influence of Axle Loads

To examine the effects of these vehicle parameters on buckling, axle load are first studied using 

a typical hopper car dimensions. The axle load is varied to reflect changes in the car’s cargo 

weight, ranging from the empty weight load of 15,575 lb to the maximum gross weight load of 

65,750 lb. The axle spacing and truck center spacing for this car are 70 and 506 in, respectively. 

All other parameters are set at the default values. Figure 4-17 shows the buckling results. The 

lower critical temperature is relatively insensitive to the effects of axle load and is essentially 

constant. However, the upper critical temperature decreases with the increasing axle load. This 

is an important aspect of dynamic buckling since the rapid decrease in the upper critical 

temperature quickly reduces the energy barrier for buckling. The practical implication is that 

lighter cars tend to have larger buckling margins of safety than heavier cars.
72

Axle Load (lbs)

10000 20000 30000 40000 50000 60000 70000 80000

Temperature increase above neutral (°F)

70

75

80

85

90

Wood tie track

136RE rail, 5° curve

20 in. tie spacing

Granite, Fp=100 lb/in.

μ=1.20

Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in.

Lo=180 in., δo=1.5 in.

Input parameters

Truck center spacing: 506 in.

Axle spacing: 70 in. 

∆TBmax

∆TBmin

Figure 4-17. Influence of Axle Load on Buckling Temperatures

Effect Truck Center Spacing

The effects of vehicle truck center spacing are examined by varying TCS from 350 to 700 in to 

reflect changes in car length, while keeping the axle spacing at 70 in and an axle load of 65,000 

lb constant. Figure 4-18 shows the results. For small values of TCS (less than 470 in for the 

example shown), the behavior is progressive and with increasing ∆TP values. The upper and 

lower buckling temperatures diverge at a TCS of about 480 in. The upper buckling temperature 

then increases while the lower buckling temperature continues to decrease with a slight upswing 

for large TCS values. At these large TCS values, the upper and lower temperatures approach the 

values predicted by static buckling theory (91 °F and 77 °F, respectively, as shown by Figure 4-

1) since the effect of the vertical load is not felt in between the trucks, thereby simulating a static 

buckling condition. It can be seen that critical truck center spacing is about 480 in corresponding 

approximately to that of hopper cars. This is considered critical because this has the lowest 

temperature at which progressive buckling can occur (i.e., no energy barrier as compared to the 

longer truck center spacing).
73

Truck center spacing (in)

Temperature increase above neutral (°F)

∆TBmax

∆TBmin

∆TProg

Wood tie track

136RE rail, 5° curve

20 in. tie spacing

Granite, Fp=100 lb/in

μ=1.20

Kv=6000 psi

Kf=200 psi

τo=120 in-kips/rad/in.

Lo=180 in., δo=1.5 in.

350 400 450 500 550 600 650 700 750

20

30

40

50

60

70

80

90

100

110

120

Input parameters

Axle spacing: 70 in.

Axle load: 66 kips

Figure 4-18. Influence of TCS on Buckling Temperatures

Effects of Track Structure: Concrete versus Wood Ties

Concrete tie CWR track is generally regarded as stronger than the wood tie track from a buckling 

strength point of view. The higher strength of concrete tie track is attributed to its larger tie

weight, size and shape, and usually better quality ballast, all contributing to increase the lateral 

resistance. However, other factors, such as the higher friction coefficient of wood ties, smaller 

tie spacing for wood ties (20 in for wood versus 24 in for concrete tie track), and typically higher 

torsional resistances for cut spike fasteners, become important in offsetting the perceived 

concrete tie weight, size, and shape advantage.

To make a reasonable comparison, in this report we will postulate an average condition for the 

wood and concrete tie track while keeping as many parameters constant as possible (i.e., rail 

size, curvature, ballast type, and track alignment condition). The average condition for wood ties 

is taken as that shown in Figure 4-6 with a peak resistance of 120 lb/in, while the average 

condition for concrete ties is taken as indicated in Figure 4-19 with a peak resistance of 142 lb/in. 

as measured in recent tests [26]. As Figure 4-19 indicates, for the average conditions stipulated, 

the concrete tie track has a higher ∆TBmax but a lower ∆TBmin than for wood tie tracks. Because of 

the larger difference between the concrete tie ∆TBmax and ∆TBmin than for the wood tie, the 

buckling energy at the respective ∆TBmin values is higher for concrete than wood, indicating that, 

while buckling safety at the concrete tie ∆TBmin may be assured, at the wood tie ∆TBmin it is not. 

According to the safety criterion of Section 2.3.2, the safe allowable temperature Tall (wood) is 

84 °F while the Tall (concrete) is 80 °F, implying slightly better buckling strength for wood tie tracks 

for the parameters chosen. Figure 4-20 shows a similar comparison for a weak or recently 

maintained track. For the weak wood tie track, the response is progressive buckling, while for 
74

the concrete tie, track although distinct ∆TBmin/max exist, the response is close to progressive. For 

this weak condition, the buckling safety criterion gives a Tall (wood) = 60 °F, and Tall (concrete) = 

56 °F, again showing a slightly higher buckling strength for wood tie tracks. On the other hand, 

the wood tie track buckles more progressively, therefore construed to be weaker in the buckling 

sense. At Tall (wood) = 60 °F the track has already incurred an additional half inch of lateral 

deflection adding to the initial misalignment of 1.5 in. It is evident that progressive buckling 

should be avoided due to rapid incurrence of lateral deformations with temperature increase.

136RE rail

5 deg curve

24 in. tie spacing

Granite, Fp=142 lb/in.

μ=0.86

Kv=10000 psi

Kf

=200 psi

τo=25 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters for

average concrete tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=120 lb/in.

μ=1.20

Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters for

average wood tie track

Concrete tie track

Wood tie track

Lateral displacement (in)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Temperature increase above neutral (°F)

0

10

20

30

40

50

60

70

80

90

100

110

120

Figure 4-19. Buckling Response for “Average” Wood versus Concrete Tie Tracks
75

Lateral displacement (in)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Temperature increase above neutral (°F)

0

10

20

30

40

50

60

70

80

90

100

110

120

136RE rail

5 deg curve

24 in. tie spacing

Granite, Fp=80 lb/in.

μ=0.86

Kv

=10000 psi

Kf

=200 psi

τo=25 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters for

weak concrete tie track

136RE rail

5 deg curve

20 in. tie spacing

Granite, Fp=70 lb/in.

μ=1.20

Kv=6000 psi

Kf

=200 psi

τo=120 in-kips/rad/in

δo=1.5 in., Lo=180 in.

Input parameters for

weak wood tie track

Concrete tie track

Wood tie track

Figure 4-20. Buckling Response for “Weak” Wood versus Concrete Tie Tracks

Summary

The results of the parametric studies show that several parameters have a significant impact on 

the buckling temperatures, ∆TBmax/min, notably track lateral and torsional resistances, 

misalignments, and track curvature. As will be discussed in Chapter 6 on the safety aspects of 

CWR, for safety considerations, the ∆TBmin values are the more important ones since they drive 

the safety criterion. Special cases addressing conditions and features, such as static versus 

dynamic buckling, weak versus strong lateral resistance tracks, and wood versus concrete tie 

track buckling, have also been presented with their implications on the track buckling response. 

It is important to note that the results presented represent trends and sensitivity influences for a 

set of fixed parameters. For a specific track type or condition, actual representative parameters 

should be used for buckling analysis. Table 4-7 presents a summary of the parametric analysis 

results in terms of desirable values for providing high buckling strength.
76

Table 4-7. Desired Parameters for High Buckling Strength

Parameter Desired Value Comments

Rail size Small

Buckling strength goes up with decreasing rail 

size; however, rail size selection is based on 

fatigue and wheel load considerations. 

Peak lateral resistance High

Buckling strength goes up with increasing lateral 

resistance. After maintenance, ensure adequate 

consolidation by traffic or mechanical 

stabilization. Also, ensure/maintain adequate 

ballast shoulders and crib content.

Tie/ballast friction 

coefficient High

Buckling strength increases with increasing 

tie/ballast friction. Consolidation tends to 

increase the coefficient. For concrete ties, rough 

bottom surfaces help.

Torsional resistance High

Buckling strength increases with fastener 

torsional resistance. Avoid spike killed ties and 

broken or missing insulators on concrete ties.

Longitudinal resistance High

The benefit of this parameter is more in 

controlling the rail neutral temperature by 

limiting longitudinal movement. Anchor 

effectiveness is very important, and ETA versus 

EOTA is better.

Curvature Low

Buckling strength decreases with curvature. To 

counteract this influence, maintain high neutral 

temperature and lateral resistance.

Misalignment amplitude

Misalignment wavelength

Small

Large

Good alignment is very important. More 

frequent track geometry inspections are useful, 

as is controlling/monitoring curve breathing.

Track foundation modulus High Vertically stiffer track is better from buckling 

point of view.

Axle load Small

Increased axle loads tend to reduce the upper 

buckling temperatures but with very little 

influence on ∆TBmin. High axle loads also require 

larger rail sections, which can further reduce 

buckling strength.

Truck center spacing Small or Large Within the range of typical truck center spacing, 

the intermediate values tend to be the worst.
77

5. Probabilistic Evaluations of Track Stability

This chapter will use the probabilistic theory presented in Chapter 3 to define probabilistic 

measures of track buckling potential, based on the input parameters (lateral resistance, neutral 

temperature, and misalignment) being expressed in statistical terms. Unlike the case of the 

deterministic approach, which gives the margin of safety that could vary from location to 

location; the probabilistic approach can give a gross index of the buckling potential on a given 

territory. The probability of buckling at a given rail temperature can be expressed by the number 

of expected buckles over a given track section. From this and knowing the annual rail 

temperature over the region, one can compute the probable number of buckles in a year. This 

type of information will be valuable to the operating agencies for the following reasons:

• The industry can compare the predicted annual number of buckles using the probabilistic 

methodology in a given territory to the actual reported incidents and accidents attributed 

to track buckling and this information for planning future preventative maintenance 

programs.

• By parametric studies, one can determine the optimum parameter levels (lateral 

resistance, neutral temperature, and misalignments) to minimize the annual number of 

buckles and thus reduce the number of buckling accidents.

• For a given level of buckling probability (risk acceptance), railroads can perform tradeoffs between maintaining high resistance, improving lateral alignment conditions or 

restressing the rails to higher neutral temperatures.

• The probabilistic theory will also be useful in the decision-making process on slow orders 

when the rail temperature is high. Speed reductions and the temperatures for their

imposition can be rationally determined using a risk-based approach and the probabilistic 

buckling theory.

Annual Number of Buckles

Chapter 3 presents the approach to determine the probability of buckling at a given rail 

temperature. Since the rail temperature changes daily and seasonally, i.e., producing its own 

temperature frequency, it becomes important to evaluate the cumulative probability of buckling 

over a year. This will be called the annual probability of buckling, Pa, from which the expected 

number of buckles over a chosen length of track section can be determined as explained in the 

following paragraphs.

To determine Pa, the annual rail temperature (TR) frequency distribution for the year is required. 

This can be derived from the past records on the rail temperature or alternately they can be 

deduced from the air temperature records available from the U.S. Meteorological Society. If the 

air temperature is used as a basis, the rail temperature must be adjusted higher than the air’s since 

the rail acts as a black body absorbing heat. The difference between the rail and the air 

temperature can be 25–35 °F based on the field data. Then the annual probability of buckling is 

the product of the probability of buckling at the rail temperature Pb and the annual frequency of 

the rail temperature, PT, or:

Pa = PbPT (Equation 5-1)

Chapter 3.3 identifies the critical temperature, Tc, above which the buckling probability becomes 

significant. Since Pb is zero or very small for TR<Tc, the rail temperature frequency for TR<Tc is 
78

not important. What is important is the number of days in a year for the rail temperature, TR, to 

be in the range of Tc and above to the maximum rail temperature, TM.

Expected Number of Buckles per Annum

To derive the expected number of buckles in a year, one must first define an appropriate track 

segment length that is vulnerable to a buckling occurrence. For this, a track segment equal to 

that of one car length is chosen since buckling usually takes place in the uplift region under a car, 

as discussed in Chapter 2. If one considers that each car is capable of inducing a buckle at the 

segment, then the annual number of potential buckling events at the segment is the total number 

of cars passing over the segment in a year. However, if one buckle occurs at this segment under 

one car in a train, then all potential buckles under all other cars in the train can be ignored since 

one buckle per train at one time is sufficient to cause a derailment. Hence, the number of 

potential buckles at a given segment can be reduced to the annual number of trains passing over 

the segment.

Alternatively, one can consider that only one buckling event is possible for a total track segment 

occupied by one train.

Defining

nc = number of cars in a typical train

lc = typical car length (expressed in miles)

nt = number of trains in a typical day

np = number of potential buckling events possible

eb = expected number of buckles per year

Then

np per mile = number of train lengths in 1 mile = 1/(nclc)

np per mile per day = nt/(nclc)

np per mile per annum = 365 nt/(nclc)

eb, expected number of buckles per year per mile = Pa365nt/(nclc)

Supposing eb is required for a 100 mile segment, then

eb, expected number of buckles per year for 100 miles = Pa365 X 100nt/(nclc)

eb = 3.65X104 Pant/(nclc)

eb = (3.65X104 PbPTnt)/(nclc) (Equation 5-2)

Equation 5.2 shows that the annual number of buckles for the 100-mile segment will increase 

with the following:

• The number of trains

• The probability of buckling at a given temperature (which depends on track parameters 

such as lateral resistance, rail neutral temperature or track geometry)

• The temperature frequency over Tc (i.e., more hot days will increase buckling)

The last term in the equation represents the length of the train. This indicates that shorter train 

increases the number of buckles versus a longer train. This is because only one buckle per train 

is considered as a potential event, and a 100 mile segment will have more train lengths for a 
79

shorter train than a longer one. As an illustration of the foregoing theory, the following data will 

be used.

nc = 100 (number of cars per train)

nt = 10 (number of trains per day)

lc = 1/100 (car length in miles)

then equation 5-2 simplifies to

eb = 0.365X106 PbPT (5-3)

Numerical Example

As an illustration, the annual rail temperature distribution is used as shown in Table 5-1. Using 

the table, the frequency distribution, PT, can be constructed as a function of rail temperature, TR. 

Figure 5-1 shows the resulting frequency distribution PT for 2 °F intervals obtained by 

interpolation and normalization.

Table 5-1. Annual Rail Temperature Data

Air Temp (oF) Rail Temp (oF) Number of Days 

per Year

30 60 15

40 70 30

50 80 50

60 90 70

70 100 70

80 110 70

90 120 30

100 130 20

110 140 10
80

Rail temperature (°F)

50 60 70 80 90 100 110 120 130 140 150

Annual rail temperature frequency, PT

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

Rail temperature (°F)

50 60 70 80 90 100 110 120 130 140 150

Annual rail temperature frequency, PT

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

Figure 5-1. Rail Temperature Distribution

Baseline Case

As an example, consider a baseline case (Case A) for which the lateral resistance, neutral 

temperature, and the misalignment distributions are shown in Table 5-2. Figure 5-2 shows these 

distributions graphically. CWR-RISK takes the input misalignment and the lateral resistance 

data and formulates a distribution representing the strength of the track. Because the input data 

is an array with probabilities built in, the strength curve is also an array that is a function of 

probability. Inside the CWR-RISK program, this array is then interpolated to find values for 

every 1-degree increment. Next, the data is normalized so that all the probabilities sum to unity. 

This array represents the strength of the track represented by the allowable temperature, ∆Tall, 

which is obtained from the program.

Table 5-2. Input Distribution for Baseline Case A

Lateral 

resistance

(lb/in)

Frequency

(%)

Rail neutral 

temp

(°F)

Frequency

(%)

Misalignment

amplitude

(in.)

Frequency

(%)

70 0.00 50 0.00 0.00 0.29

85 0.05 60 0.04 0.25 0.35

100 0.25 70 0.21 0.50 0.22

115 0.40 80 0.50 1.00 0.08

130 0.25 90 0.21 1.25 0.04

145 0.05 100 0.04 1.50 0.02

160 0.00 110 0.00 1.75 0.00
81

Lateral resistance (lb/in.)

60 80 100 120 140 160 180

Frequency

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Rail neutral temp (°F)

40 50 60 70 80 90 100110120

Rail neutral temp (°F)

40 50 60 70 80 90 100110120

Misalignment amplitude (in.)

0.0 0.5 1.0 1.5

Figure 5-2. Distributions for Baseline Case A

As discussed in Chapter 3, CWR-RISK also calculates the load density curve. This is basically 

the rail neutral temperature minus the rail temperature (i.e., the rail temperature in degrees above 

the neutral temperature). CWR-RISK then overlays the load distribution curve onto the track 

strength curve and evaluates the buckling probability through the use of the convolution integral. 

The probability of buckling at a rail temperature is then output from the program. Figure 5-3

gives an example of a 5° curve with assumed track characteristics as in Table 5-3.

Figure 5-3. Buckling Probability at a Given Temperature, Case A

Rail temperature (°F)

100 105 110 115 120 125 130 135 140 145 150

Probability of buckling,Pb

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Tc= 114° F
82

Table 5-3. Assumed Curved Track Characteristics

CWR-RISK Illustrative Example Inputs 

Rail size (lb/yd): AREA 136

Tie type: Concrete

Tie weight (lb): 750

Tie spacing (in): 24

Track curvature (deg): 5

Ballast type: Granite

Tie ballast friction coefficient: 0.86

Torsional resistance (in-kips/rad/in): 25

Longitudinal stiffness (psi): 200

Foundation modulus (psi): 10000

Vehicle type: Hopper

Figure 5-3 presents the relationship between the buckling probability and the rail temperature. 

From this graph, one can identify the critical rail temperatures, Tc, at which the buckling 

probability just begins to be significant as defined in Section 3.1.3. This is the critical 

temperature (Tc) above which there is a more finite level of buckling risk. The critical 

temperature is 114 °F for Case A with the assumed parameters given in Table 5-3.

Figure 5-4, derived from Equation 5-3, gives the expected annual number of buckles over 100 

miles of track as a function of rail temperature for the annual cycle in accordance with the PT

distribution of Figure 5-1. If vehicle operations are permitted for the full range of expected rail 

temperature up to 140 °F, the expected number of buckles per annum is 28. If the vehicle 

operations were to be stopped at a lower temperature, for example 130 °F, then the number of 

buckles will reduce to five as seen from Figure 5-4. Thus, the data in the figure can be used to 

identify the maximum permissible rail temperature for vehicle operations for a chosen maximum 

permissible annual number of buckles. This limiting temperature for train operation is defined as 

TL. The dashed line shows a TL = 130 °F if the permissible number of buckles per annum is five.
83

Rail temperature (°F)

110 115 120 125 130 135 140

0

5

10

15

20

25

30

Expected number of buckles

Per 100 miles per annum, eb

Tc = 114° F

eb = 28

TL = 130° F

Rail temperature (°F)

110 115 120 125 130 135 140

0

5

10

15

20

25

30

Expected number of buckles

Per 100 miles per annum, eb

Tc = 114° F

eb = 28

TL = 130° F

Figure 5-4. Number of Buckles versus Rail Temperature (Case A)

Slow Orders

As stated earlier, Tc represents the critical rail temperature below which the probability of 

buckling can be considered as zero; hence full normal line speed can be permitted for the 

vehicles up to this rail temperature. It has already been concluded that, for a chosen maximum 

number of permissible buckles in a year, a limiting temperature, TL exists. At temperatures 

above this TL vehicle speeds should be stopped (or slow ordered to 10 mph where practical). 

Given that the speed at Tc is the maximum permissible speed Vmax and at TL the speed is zero, 

the issue is to determine a permissible speed for the rail temperatures within this temperature 

regime of Tc≤TR≤TL. Since at present no consensus and common rationale for hot weather slow 

order policy determinations exists, a risk-based speed reduction formula is proposed here.

If the damage due to buckled track derailment can be assumed to be proportional to the vehicle 

kinetic energy and hence to the square of the speed, the suggested speed formula [13] for the 

same level of damage at all speeds is the following:

Vr/Vmax = { 1-Pb(T)/Pb(TL) }0.5 Equation 5-4

Where

Vr = Reduced speed

Vmax = Permissible maximum authorized line speed

Pb(T) = Buckling probability at rail temperature, T

Pb(TL) = Probability of buckling at the limiting temperature, TL

For Case A, using the foregoing formula, the speed ratio is calculated and shown in Figure 5-5

for an assumed five or less annual buckles. As seen from the figure, the permissible speed falls 

off rapidly from its maximum value at Tc to zero at TL. In practical applications, the graph may 
84

be idealized into a bilinear approximation by the lines GH and HI or any other linear 

combination in the gray zone.

Figure 5-5. Permissible Speeds for Case A

Parametric Studies

5° Curve

To show the benefits of probabilistic analysis, this section presents a parametric study. As 

discussed in Section 5.1, the annual number of expected buckles depends mostly on how well the 

track is maintained in regard to the lateral resistance, the rail neutral temperature, and its 

alignment. The railroads can make choices in the maintenance of these three key parameters and 

attain the same level of buckling safety assurance (i.e., the same number of annual buckling 

events over a given territory). To illustrate this point, four possible cases of improvements in 

maintenance on the baseline track are considered, Case B, Lateral resistance, Case C, Neutral

Temperature, Case D, misalignment amplitude, and case E improvement on all three parameters. 

Figure 5-6 through Figure 5-9 show schematically the potential improvements.
85

Lateral resistance (lbs/in) Rail neutral temp (°F) Misalignment amplitude (in.)

Frequency

60 80 100 120 140 160 180

0.0

0.1

0.2

0.3

0.4

0.5

0.6

40 60 80 100 120 0.0 0.5 1.0 1.5 2.0

Improved 

Lateral Resistance

Figure 5-6. Distributions for Case B

Figure 5-7. Distributions for Case C
86

Figure 5-8. Distributions for Case D

Figure 5-9. Distributions for Case E

Case B shows improvement in the lateral resistance distribution by having full cribs, maintaining 

a good shoulder geometry, and ensuring high levels of ballast consolidation. The result of such 

measures will be the better distribution of lateral resistance with an increase in the mean value 

and the range as compared to the resistance distribution shown in Figure 5-2 (baseline Case A). 

The other two distributions of the neutral temperature and misalignment are the same as in Case 

A.
87

Case C has an improved neutral temperature distribution when compared to the baseline Case A. 

This can be achieved by proper restressing of rails after rail repair and after track realignment, 

maintaining effective rail anchors and high toe loads on elastic fasteners, and minimizing curve 

breathing in wintertime, using adequate ballast inside and outside the shoulder and monitoring 

the curve position. The distributions of lateral resistance and misalignment in Case C are the 

same as in the baseline.

Case D represents improvements in the alignment over the baseline. This improvement can be 

accomplished through more frequent geometry inspection and alignment corrections. The 

generation and growth of misalignments can also be mitigated by reducing vehicle lateral forces 

and high cant deficiency curves. The distributions of the lateral resistance and the neutral 

temperature in Case D are the same as in the baseline case.

Finally, Case E, where all three improvements occurring simultaneously are incorporated; this 

can represent an ideal (strong) track as shown in Figure 5-9.

Clearly, one would expect the baseline to result in the highest number of buckles per year and the 

ideal case to result in the lowest number of buckles. The question of interest is how well the 

relative improvements will perform with respect to buckling safety and their relative cost of 

implementation. The performance benefits will be quantified using the probabilistic theory, but 

the cost benefit analysis will not be presented because it depends on individual railroad practices 

and procedures.

Performance Comparisons

Figure 5-10 through Figure 5-13 show the expected number of buckles and the annual 

probability of buckling as a function of rail temperature for the first four cases. Table 5-4

summarizes salient results for all cases. The results include the critical temperature Tc, the 

limiting temperatures for the number of buckles under 5 per annum, and the expected number of 

buckles if normal vehicle operations speeds are continued up to TR = 140o F. The dramatic 

decrease in the number of buckles at 130o F, for example, compared to 140o F operations can be 

clearly seen. An inspection of the results for all the cases at 140o F shows that the neutral 

temperature distribution improvements will also bring significant reduction in the annual number 

of buckles. At 130o F, the neutral temperature distribution improvement alone may be adequate 

to eliminate all the potential buckles under the assumed conditions. The improvements in all 

three parameters (Case E) will result in zero buckles even at 152o F. A figure for Case E is not 

shown because of the zero number of expected buckles up to 152o F.
88

Rail temperature (°F)

110 115 120 125 130 135 140

0

5

10

15

20

25

30

Expected number of bucklesPer 100 miles per annum, 

e b

Tc = 114° F

e b = 28

TL = 130° F

Rail temperature (°F)

110 115 120 125 130 135 140

0

5

10

15

20

25

30

Expected number of bucklesPer 100 miles per annum, 

e b

Tc = 114° F

e b = 28

TL = 130° F

Figure 5-10. Number of Buckles versus Rail Temperature (Case A)

Rail temperature (°F)

110 115 120 125 130 135 140

0

5

10

15

20

Expected number of bucklesPer 100 miles per annum, eb

Tc = 126° F

eb = 17

TL = 135° F

Figure 5-11. Number of Buckles versus Rail Temperature (Case B)
89

Rail temperature (°F)

110 115 120 125 130 135 140

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Expected number of buckles

per 100 miles per annum, e b

Tc = 132° F

eb = 0.6

Figure 5-12. Number of Buckles versus Rail Temperature (Case C)

Rail temperature (°F)

110 115 120 125 130 135 140

0

5

10

15

20

Expected number of bucklesPer 100 miles per annum, eb

Tc = 126° F

eb = 17

TL = 135° F

Figure 5-13. Number of Buckles versus Rail Temperature (Case D)
90

Table 5-4. Comparison of Track Cases

Case Description Tc

TL for 5 

buckles per 

annum

Number of 

buckles per 

annum at 

TR = 140 °F

Number of 

buckles per 

annum at 

TR = 130 °F

A Weak Baseline 

Track 114 °F 130 °F 28 5

B

Improved 

Lateral 

Resistance

126 °F 136 °F 12 0.66

C

Improved Rail 

Neutral 

Temperature

134 °F >140 °F 0.06 0

D Improved 

Alignment 126 °F 135 °F 17 0.56

E

Improvements In 

All Three 

Parameters

152 °F >152 °F 0 0

Tangent Track

Analysis has also been conducted in which the baseline case parameters (Case A) are applied to 

tangent track to study how track curvature affects the number of buckles per year. Figure 5-14 

shows the results for tangent track.

Rail temperature (°F)

110 115 120 125 130 135 140

0.0

0.5

1.0

1.5

2.0

Expected number of buckles

Per 100 miles per annum, eb

Tc = 130° F

eb = 1.87

Figure 5-14. Number of Buckles versus Rail Temperature for Tangent Track (Case A)
91

Applying the Case A parameters to tangent track as opposed to 5° track increases Tc from 114 °F 

to 130 °F. In addition, the number of annual buckles decreases from 28 to less than 2 for every 

100 miles per annum, showing that curved tracks are more vulnerable to buckles as compared to 

tangent tracks.

Slow Orders Parametric Study

Figure 5-15 shows the permissible speed ratios (Vr/Vmax) with respect to rail temperature for 

each of the cases of the parametric study. These graphs are derived using the equation in Section 

5.1.4. As expected, the baseline Case A indicates quick reduction in speed at lowest 

temperatures. With improvements, the curves show higher speeds at relatively high 

temperatures. Case C performs far superior than case B and case D and allowing a vehicle to 

maintain full speed up to 140 °F. This means that an effective way of maintaining high-speed 

operation at high rail temperatures is through the assurance of a better (higher) rail neutral 

temperature condition. The ideal Case E outperforms the individual cases and allows full speed 

up to 152 °F.

Rail temperature (°F)

110 115 120 125 130 135 140

0.00

0.20

0.40

0.60

0.80

1.00

A

C, E

D

B

Speed V/Vmax

Figure 5-15. Permissible Speeds for All Cases

Summary

• By determining the distributions of the key input parameters through measurements, 

railroads can optimize their maintenance strategies for minimization of buckling 

accidents and associated costs.

• The probabilistic buckling theory provided in CWR-RISK predicts the buckling 

probability of CWR track at any given rail temperature. The graph relating the buckling 
92

probability and the rail temperature depends largely on the three key parameters of lateral 

resistance, neutral temperature, and misalignment. These three parameters are expressed 

in terms of statistical distributions. The buckling probability graph identifies the critical 

temperature at which buckling probability becomes significant and can define slow order 

requirements at elevated temperatures.

• From the probability versus rail temperature curve, the expected number of buckles per 

annum over a given track length can be predicted, if the annual temperature distribution 

of the rail is known. This information permits more practical measures for buckling 

safety assurance through targeted track maintenance operations and slow orders.

• A limiting rail temperature for vehicle operations can be identified from the analysis, 

based on the accepted number of buckles per annum over a given territory.

• The influence of alternate maintenance strategies on buckling safety, such as control of 

ballast resistance, neutral temperature, and track alignment or limiting the vehicle 

operations at lower temperatures can be evaluated. This will allow for performing costbenefit analyses on maintenance procedures to reduce the buckling related damages.

• A risk-based theory is proposed for slow order management. The theory predicts the 

critical temperature to initiate slow orders and the appropriate speed reduction up to a 

limiting temperature.
93

6. Safety Aspects of CWR

This chapter presents CWR buckling safety aspects, including safety limits, test validation of the 

limits, and implementation methodologies of the safety limits for buckling prevention and 

mitigation.

Safety Limits

As discussed in the previous chapters, two approaches to the safety evaluation of CWR track 

exist. One is the deterministic method, which focuses on the safe allowable temperature for a 

given set of track conditions and vehicle parameters. When the rail temperature exceeds this 

allowable temperature, vehicle operations should be curtailed. The second method is the 

probabilistic method, which gives a finite probability of buckling at any rail temperature 

exceeding the critical temperature, Tc. This method can also calculate the expected number of 

buckles per annum over a given track section length and requires annual rail temperature 

distribution information. When the rail temperature exceeds Tc, vehicle operations can be

continued at reduced speeds as discussed in Chapter 5.

The two methods should not be considered as alternative choices in the management of CWR 

safety. Rather, they complement one another in overall safety assurance and provide reasonable 

flexibility for railroads use and implementation. The deterministic method is appropriate for 

specific track sections with a history of buckling incidents and for day-to-day buckling safety 

assurance on chosen sections of track. The probabilistic method provides a global approach for 

buckling safety improvement for extended periods of operations over a larger CWR territory.

Deterministic Method

The allowable rail temperature is the sum of the safe allowable temperature increase and the 

CWR neutral temperature. The safe limits can be presented in one of the following schemes:

1. Application of CWR-BUCKLE

2. Use of graphics/charts prepared for a range of track conditions/parameters via CWRBUCKLE

3. Uncoupled approach

The following paragraphs discuss these in detail.

1. Application of CWR-BUCKLE

The safety criteria and evaluation methodology illustrated in Chapter 2 can be applied directly by 

exercising CWR-BUCKLE. With the pertinent track and vehicle input parameters, the program 

calculates the allowable temperature increase and performs the safety analysis. The required 

controlling parameters for buckling safety, such as lateral resistance and rail neutral temperature, 

are used to develop the safety criteria.

The advantage of this method is its ability to account for all the parameters in the system and 

provide a quick calculation of allowable temperatures. The disadvantage is that it requires some 

knowledge of the required input parameters and some expertise in running the program.
94

2. Use of Graphs/Charts

In this method, charts are prepared giving the maximum permissible temperature for buckling 

safety in terms of two the two primary parameters: lateral resistance and CWR neutral 

temperature. Lateral resistance is chosen as a primary variable in the determination of the safe 

allowable temperature increase with the other (secondary) parameters set at their average or 

nominal values. For an assumed line defect and track curvature, the maximum allowable rail 

temperature can be plotted against the lateral resistance over a range of neutral temperature.

Figure 6-1 through Figure 6-4 constitute examples of safety limit charts for 136# rail CWR 

concrete tie track with Tangent, 3°, 5°, and 7° curvature for an FRA Class 6 line defect of 0.5 in, 

while Figure 6-5 through Figure 6-8 are examples for Class 4 line defect of 1.5 in. The assumed 

fixed parameters for the four track conditions and the two classes of track are as shown in the 

inset in Figure 6-1.

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

Fixed Parameters

Rail size (lb/yd): AREA136

Tie type: Concrete

Tie weight (lbs): 750

Tie spacing (in): 24

Ballast type: Granite

Tie ballast friction coefficient: 0.86

Torsional resistance (in-kips/rad/in): 25

Longitudinal stiffness (psi): 300

Foundation modulus (psi): 10000

Vehicle type: Hopper

Figure 6-1. Buckling Safety Limits Chart for Tangent with 0.5” Line Defect
95

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

80

Figure 6-2. Buckling Safety Limits Chart for 3° Curve with 0.5” Line Defect

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

90

80

Figure 6-3. Buckling Safety Limits Chart for 5° Curve with 0.5” Line Defect
96

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

90

80

100

Figure 6-4. Buckling Safety Limits Chart for 7° Curve with 0.5” Line Defect

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

80

Figure 6-5. Buckling Safety Limits Chart for Tangent with 1.5” Line Defect
97

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

90

80

100

Figure 6-6. Buckling Safety Limits Chart for 3° Curve with 1.5” Line Defect

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

90

80

100

Figure 6-7. Buckling Safety Limits Chart for 5° Curve with 1.5” Line Defect
98

Allowable rail temperature (°F)

100 110 120 130 140 150 160

Lateral resistance (lbs/tie)

1000

1500

2000

2500

3000

3500

4000

TN (°F) = 60

70

90

80

100

Figure 6-8. Buckling Safety Limits Chart for 7° Curve with 1.5” Line Defect

The advantage of these graphical representations is that the safety limits can be easily read off 

the graphs for implementation without the need to run CWR-BUCKLE. The disadvantage is that 

graphs must be fairly extensive to cover all track parameters in addition to lateral resistance, 

(e.g., different rail size, wood tie tracks, and fastener resistances). The parametric study result of 

Chapter 4 can also be invoked to help in gaining a better insight to parametric influences of the 

safety limits.

3. Uncoupled Approach

The safety limits in the charts for the maximum allowable rail temperature are derived from the 

combined effects of the allowable rail temperature increase (which is dependent on lateral 

resistance) and the neutral temperature. For a given maximum rail temperature at a given 

geographical region, one can determine the minimum required lateral resistance for a possible 

lowest neutral temperature. For example, referring to the previous figures, the required lateral 

resistance can be calculated on the assumption that the neutral temperature drops off to a value 

between 60 °F and 70 °F from its intended value of 95 °F or higher. From the safety limit charts, 

Table 6-1 shows the required values of lateral resistance in a region with 140 °F maximum rail 

temperature.

Table 6-1. Minimum Resistance Values for Neutral Temperature of 60°F and Rail 

Temperature of 140 °F

Curvature (Degrees)

Required Lateral Resistance (lb/tie)

0.5” Line Defect 1.5” Line Defect

Tangent 1400 2100

3 1800 2800

5 2100 3200

7 2500 > 3500
99

In this table, the neutral temperature is independently set at a minimum limit of 60 °F. For 

curvatures of 7° and greater, the minimum resistance values required will be over 3500 lb/tie. 

Such high values are generally difficult to ensure consistently because of maintenance activities, 

and curve movement. However, if a limit of 70 °F is stipulated for the neutral temperature, the 

resistance values fall into the range as shown in Table 6-2.

Table 6-2. Minimum Resistance Requirement for Neutral Temperature of 70 °F and Rail 

Temperature of 140 °F

Curvature (Degrees)

Required Resistance (lb/tie)

0.5” Line Defect 1.5” Line Defect

Tangent <14001 <14001

3 <14001 1950

5 1500 2300

7 1800 2700 1 Although resistances below 1400 lb appear in Table 6-2, such low values are not recommended in 

practice since they may give rise to other problems such as track shift/curve breathing and general 

instability under high vehicle loads.

These tables show that lateral resistance and neutral temperature can be independently specified 

to obtain the allowable rail temperature values.

Probabilistic Approach

This approach is useful in planning overall CWR track maintenance to control the buckling 

incident rate within a certain limit, such as less than one buckle per every 500 miles per annum 

on a given territory. Because the thousands of miles of railroad track exhibit variable conditions, 

the probabilistic approach is more appropriate for overall buckling safety evaluations. The 

probabilistic approach predicts a critical temperature, Tc, above which buckling risk exists. The 

method can also predict the number of buckles in the territory that can occur annually for a rail 

temperature in the range of Tc and TMAX. The maximum permissible temperature for vehicle 

operations, TL, which is equal or less than TMAX, can be determined for a specified limit allowed 

on the permissible number of buckles.

This method will also provide a speed reduction formula for vehicle operations when the rail 

temperature exceeds the critical temperature Tc. The speed reduction formula provides a rational 

basis for hot weather slow order practices.

In summary, the method specifies the following:

Tc = critical temperature above which buckling risk exists

NB = maximum number of permissible buckles per annum per section of track

TL = maximum permissible rail temperature to limit the number of buckles below NB

As stated in Chapter 5, the value of Tc can be derived using the frequency distributions of the 

lateral resistance, neutral temperature, and misalignment amplitude. The limit values in these 

distributions, (i.e., the lowest values of lateral resistance and neutral temperatures together with 

the highest value of the misalignment amplitude) can also be used in the deterministic approach 
100

to evaluate Tc. This is because the CWR-RISK predicted Tc value is the same as the Tall in the 

deterministic approach of CWR-BUCKLE for those specific input parameters.

The maximum permissible number of buckles, NB, is to be specified based on the railroad’s risk 

acceptance in terms of potential derailment impact, possibly in conjunction with historical data 

on buckled track derailments for the particular territory. Table 6-3 shows one hypothetical 

specification and is intended for illustrative purpose only. The Tc numbers can be determined 

from CWR-RISK application to specific territory parameters/conditions, while the maximum 

permissible buckles from historical records and railroad risk acceptance.

Table 6-3. Example Specifications on a Territory for Normal Freight (no Hazmat)

Track Class Minimum Tc °F

Permissible Number of 

Buckles per 100 Miles per 

Year

3 120 .1

4 130 .05

5 135 .01

6 140 .001

In addition, Table 6-3 shows the specifications on the critical temperature Tc, which represents 

the rail temperature at and below which the buckling probability is zero (10-6

). To preserve this 

Tc, the lowest values of the lateral resistance and neutral temperature (measured or stipulated for 

the territory and used in the analysis to determine Tc) must be maintained. Tc should increase for 

higher track classes (assuming similar values for lateral resistance and neutral temperature), for 

which the annual buckling rates should decrease. Depending on the assumptions of the 

maximum permissible buckles, operation can be performed above the stipulated Tc (i.e., to a 

level TL as inferred in Chapter 5).

Safety Limit Validations

Deterministic Limits

The buckling theory was validated with full-scale tests as discussed in Chapter 3. The limits as 

derived in the Deterministic Method were also validated by direct full-scale tests [6, 7] on the 

CWR tracks at TTC in Pueblo, CO.

In the tests designed to validate the safety limits, two tangent track tests, two 5° curved track 

tests, and one test on a 7.5° curve were conducted. In each case, the rails were heated artificially, 

and multi cars consist made several passes at rail temperatures near and above the safety limit 

values. The tangent and 5° curved tracks showed only a very small growth in misalignments and 

were considered essentially stable. The 7.5° curve withstood the longitudinal forces generated at 

the safe allowable temperature but buckled out explosively at about 10 °F above the safe 

allowable temperature. This is consistent with the minimum margin of safety of 10 °F built into 

the prescribed safe allowable temperature.

Referring to Table 6-4, tangent tracks were tested up to temperatures slightly higher than the safe 

allowable. The highest temperatures for the 5° curved track tests were limited because of 

prevailing cloudy and windy conditions on the specific days of testing.
101

Table 6-4. Buckling Safety Limit Test Results

Test #

Curve

Peak Tie Lateral 

Resistance (lb/in.)

Misalignment ∆T °F

Max. Vehicle

Speed (mph)

Number Cars

Final 

Misalignment 

Amplitude (in.)

Amplitude

Comments

(in.)

Length (in.)

Theory

Test

1. Tan I 0 68 0.88 335 84 92 20 48 0.98 No Buckling

2. Tan II 0 79 0.81 319 86 100 55 67 0.82 No Buckling

3. Curve I 5° 83 0.55 307 77 70 40 63 0.55 No Buckling

4. Curve II 5° 98 0.7 311 80 79 40 52 0.84 No Buckling

5. Curve III 7.5° 89 0.75 315 52 62 34 24 3.94 Track Buckled 

Under Train

Safety Limit Implementation

This section will describe the methods used to implement the deterministic limits. A formal 

methodology for the implementation of the probabilistic approach and its extension to risk-based 

safety performance standards is currently under development.

As stated previously in Section 6.1, the implementation is facilitated if the uncoupled approach is 

followed. For a given geographical region with known maximum rail temperature at and above 

which vehicle operations will be stopped (or are performed at a very slow speed), a set of 

required limiting values can be established:

• Lateral resistance FP ≥ FP

1

• Neutral temperature Nt ≥ Nt

1

• Misalignment amplitude δ0 ≤ δ0

1

1 Limiting value for each parameter given in Table 6-1 and 6-2

Among the three limiting parameters, the neutral temperature is the most difficult one to monitor 

and control. It is expedient, therefore, to assume a minimum value for it and attempt to control 

other parameters at the required values. Although the neutral temperature is usually set during 

CWR installation between 90 °F to 115 °F (or even higher depending on the geographic location 

of the CWR), it can drop to much lower values for the several reasons discussed in previous 

chapters. It becomes appropriate to assume as a safety factor a reduced value for neutral 

temperature such as 60 °F or 70 °F. On this basis, the corresponding minimum required value 

for the lateral resistance can be determined from the charts in Figures 6-1 to 6-8 for line 

conditions as given in the insert of Figure 6-1. Indeed, Table 6-1 and Table 6-2 values for the 

minimum required lateral resistance were deduced for assumed minimum neutral temperatures of 

60 °F and 70 °F, respectively, and serve as an example for the determination of the FP*, Nt*, and 

δ0* values.
102

The CWR safety implementation is to be carried out through monitoring and controlling of the 

foregoing parameters within the limits indicated by the starred quantities. The following 

paragraphs discuss some procedures/methods for monitoring and controlling these parameters.

Lateral Resistance

Monitoring Hardware/Software

Monitoring lateral resistance at limited critical locations can be carried out using the STPT 

method described in Appendix B in accordance with the measurement requirements in Appendix 

D. In some cases, CWR-INDY’s track lateral resistance predictor (see Equation 3.5 in Chapter 

3) can be used to estimate the lateral resistance. The current STPT fixture can be drastically 

simplified in size and operation by requiring capture of only the peak resistance values which 

occur within small (0.3”) displacements. The current method’s large displacement measurement, 

the X-Y plotter, and the electric pump are not required; a simple hand pump is adequate to 

operate the hydraulics to produce the required loads to move the tie.

Monitoring Locations

Some of the critical conditions at a given location include the following:

• Recently maintained sections (surfaced, realigned, tamped, or after other operations, such 

as ballast and tie renewal)

• Substandard ballast section identified visually to have inadequate ballast

• Sections which have undergone dynamic track stabilizer (DTS) and/or traffic 

consolidation

Monitoring Methodology

As explained in Appendix D, the resistance is identified as the average for a 50 foot cell, which 

is chosen because it is representative of a buckle length under a car. One measurement per 50 

foot concrete tie track and two measurements for wood tie track are deemed adequate to evaluate 

the resistance within a prescribed tolerable error as discussed in Appendix D.

The lateral resistance data will be examined and compared with the required values as indicated 

by the safety limits in the uncoupled approach. Those cells with resistance values less than the 

stipulated values in the safety limits must be restored to the safety limits and above using one of 

the following methods.

Measures to Modify and Control Lateral Resistance

Track lateral resistance can be controlled at and above a specified value by one or more of the 

following methods, including, full ballast-section, traffic consolidation, and mechanical 

consolidation.

Full Ballast Section: Good quality ballast, such as granite with good particle size and 

interlocking capability, full cribs and shoulder widths of at least 18 in would contribute to 

providing increased lateral resistance. Heavier concrete ties with roughened bottoms and 

scalloped designs are also useful in increasing the lateral resistance.

Traffic Consolidation: The ballast must be consolidated after any maintenance activity such as 

surfacing, tamping, and realignment. An empirical relationship exists between the lateral 
103

resistance and the traffic tonnage (MGT) as given by Equation 3-5 in Chapter 3. However, 

consolidation by traffic usually needs some time to accumulate to the level (usually around 0.1 

MGT) required due to the typical low speed application. To ameliorate this, other means of 

consolidation is performed as discussed below.

Mechanical Consolidation: The ballast can be consolidated by use of mechanical means such as 

the DTS. Care must be taken on the settings (frequency, down-pressure, stabilizer speed) since 

these can significantly affect the resulting levels of consolidation achievable by the machine. 

Typically the stabilizers do not compact the ballast shoulders and for increased resistance 

shoulder compaction may also be useful. DTS applications should be periodically calibrated 

against the STPT measurements to verify their effectiveness and ensure correct settings and 

reliability. Coupling site-specific information with these measurements, CWR-SAFE could be 

exercised to evaluate buckling vulnerability at high temperatures.

Neutral Temperature

Monitoring Hardware

Monitoring CWR neutral temperature is a difficult task since at present no practical 

measurement capabilities which can evaluate the absolute rail force in a nondestructive manner 

exist, although significant research has been carried out for its development. Research to date, 

however, has resulted in a methodology which can be implemented as an interim solution. As 

stated earlier, one identifies a threshold lower limit (such as 60 °F or 70 °F) for the neutral 

temperature for the assurance of buckling safety. The monitoring methodology will be used to 

determine locations where the neutral temperature can fall below the threshold limits. The 

following paragraphs will discuss methods to control the neutral temperature above the threshold 

limit.

The authors conceptualized the Rail Uplift Device (RUD) in 1987 to determine the absolute 

force in the rail by correlating its deflection under an applied vertical load to the tension or 

compression in the rail. The rail fasteners are removed over a given length to provide free rail 

bending. Based on the principle, the AAR’s Track Loading Vehicle was used to successfully 

map the neutral temperature over some sections of track on the BNSF railroad [15]. In general, 

it is possible to design a portable device to lift the rail with fasteners removed and measure the 

rail deflection under a given vertical force level, thus correlating rail deflection level to the rail 

longitudinal force in the rail.

Since RUD can be cumbersome for systematic applications, its use can be limited to a one time 

application to determine the absolute rail force. A strain gage can be fixed at the location and a 

reference can be provided to the strain gage using the rail force measured by the RUD. This will 

avoid rail cutting, which is the usual way of providing a zero reference to the strain gage. Thus, 

the strain gage usage in conjunction with RUD will eliminate the undesirable rail cutting. With 

the initial reference from RUD, the strain gage can provide all subsequent data at the location 

relatively easily and quickly and for a sufficiently long time if protected properly.

Strain gages can also be applied to monitor neutral temperature changes without knowing the 

absolute neutral temperature values. Strain gages can be installed and zeroed at any temperature, 

Ti. If a reading is taken at a temperature, TR, then the neutral temperature change can be 

calculated using the formula
104

∆TN = eR - α(TR -Ti) (Equation 6-1)

where eR is the strain gage reading at the rail temperature TR. If ∆TN indicates a drop of more 

than a prescribed value (30 °F, for example), then the track segment’s rail may need to be cut to

determine the correct absolute TN and restress as required. Alternatively, a RUD type device can 

be used if rail cutting must be avoided.

Monitoring Locations

The following critical locations should be identified and monitored as a priority:

• Locations with a buckling history

• Recently maintained locations (surfaced or realigned)

• Winter repaired and readjusted rail segments

• Locations consistent with heavy braking and traction forces

• Locations with excessive rail movement or bunching

• Locations with sudden changes in longitudinal resistance such as bridges and turnouts

• and other locations as described in Section 2.2 of Appendix D

Monitoring Methodology

Monitoring the exact rail force and neutral temperature is an involved process. However, the 

concept of minimum required neutral temperature is very useful for monitoring as it simplifies 

the problem to a large extent.

Appendix D shows that one measurement in a 100 foot section is adequate to determine the 

neutral temperature within ±5 °F maximum error and a confidence limit of 90 percent. Each of 

the critical locations is divided into a number of cells of 100 foot in length. One or two cells 

should be tested for the minimum neutral temperature values (say 60 °F or 70 °F) for the critical 

segment. Testing is done through either a RUD (if available) or through the strain-gage/rail cut 

method. If a minimum condition is found, the necessary restressing (or other required 

maintenance action) can be performed.

Other interim monitoring schemes include the selective monitoring of rail break repairs which 

require a rail-plug insertion. These plugs can be pre-instrumented with strain gage coupons 

(with the initial zeroes established), and once the welds are made, the new (or corrected) neutral 

temperature is automatically determined. In such fashion, more effective control on neutral 

temperature could be attained.

Measures to Control

Control of neutral temperature is important at the time of installation of new CWR, when rewelding rails after destressing and restressing, and during rail fracture repairs requiring a new 

plug insertion. Neutral temperature control may also be required in conjunction with 

maintenance operations, such as curve realignments and surfacing operations involving large 

lifts.

1. CWR Destressing

It is a practice in the railroad industry to destress the rail by cutting whenever the heat patrols 

observe the rail becoming wavy and tight in the fasteners on summer days. This operation will 

relieve the compressive load and is usually followed up by re-welding the rail after cutting some 
105

rail out and removing anchors and fasteners on either side of the cut locations. Research has 

identified the following relevant technical issues and concerns:

• The correct determination of the lengths of rail to unfasten

• The correct determination of the amount of rail to cut out so that an adequate amount of 

gap for welding (usually 1 in.) exists for the target neutral temperature readjustment

Industry practices on these issues are not well defined and need rational guidelines which can be 

based on the analytical and experimental work performed in [16]. When a rail in tension or 

compression is cut, the rail force is redistributed from zero at the cut location to a steady state 

value at some distance (referred to as the influence zone). This zone can be as long as 1000 ft on 

either side of the cut depending on the precut rail force level and the rail/tie longitudinal 

resistance. The unfastening or de-anchored length can be computed according to the formula 

developed in [16]. If the actual de-anchored length is smaller than that calculated residual

stresses would remain after re-welding resulting in a reduced neutral temperature at these 

locations.

Once the correct de-anchored length is prescribed, the required gap size (or rail plug length) can 

be made for the required target neutral temperature setting.

2. Restressing

To restress the CWR to the desired neutral temperature, the gap after destressing must be 

adjusted to 1 in before welding when the rail temperature reaches the neutral temperature. If the 

rail temperature is lower than the desired neutral temperature due to unfavorable conditions 

(such as cold ambient temperatures), the use of artificial heating or mechanical tensors are 

typically used to induce the required force level to produce the intended neutral temperature. In 

the case of solar or artificial heating, the gap adjustment can be made by additional cuts on the 

rail ends or more fasteners can be removed to obtain the correct gap size. If the initial gap is not 

optimal, the railroads may weld at a lower temperature, resulting in a lower neutral temperature, 

or they may wait for a higher temperature to obtain the 1 in. gap for welding.

The gap size is very critical when hydraulic tensors are deployed to set the neutral temperature. 

In the case of solar or artificial heating, the rail temperature can be readily measured before 

welding. In the case of tensors, the only guiding factor is the change in the gap size under the 

application of tensors, which is correlated to the desired neutral temperature by look-up tables. 

Currently, these look-up tables assume that the rails at the junction of the unfastened and 

fastened zones are fully constrained against longitudinal movement under the tensor application, 

especially at large load levels. This may be an incorrect assumption and can contribute 

significant errors on the nonconservative side in the resulting neutral temperature.

As an illustrative example, consider the situation when one applies a tensor on 136# rail at 40 °F 

with a 325 foot de-anchored section on either side of the gap. Assume that the longitudinal 

resistance, f0, in fasteners is 20 lb/in/rail, and the required neutral temperature is 100 °F. In the 

current railroad practice, the required gap is:

∆1(current practice) = 1 + 2Ldα∆T = 4′′ (Equation 6-2)

Then according to [16], if the ends are moving with the tensor-applied force, the correct gap is 

given by:
106

∆2 (analysis) = ∆1 + [AE(α∆T)2/f0] = 6.95′′ (Equation 6-3)

The gap of 4′′ results in a neutral temperature of only 70 °F, incurring an error of 30 °F when 

using existing guidelines.

Therefore, the present railroad practices need improved guidelines on gap adjustment for 

installing CWR at the correct neutral temperature using tensors. The current gap size 

specifications should be increased by the correction term of [AE(α∆T)2/f0]. Since this involves 

the longitudinal resistance f0 and the de-anchored length Ld, which must be evaluated specifically 

for each site, a black box device, such as the one discussed in Appendix E currently under 

development, will be useful to the industry. In the interim, current best estimate guidelines for 

improved destressing and rail break repairs based on recent tests and analyses is provided in [28].

3. Increased Longitudinal Resistance

The fastener longitudinal resistance is an important parameter to be controlled to minimize rail 

longitudinal movement through the fasteners and thus retain the high neutral temperature set 

when the CWR is installed or destressed. Train braking and traction forces and unequal diurnal 

rail heating (sun and shade) resulting in unequal thermal loads tend to move the rail, which may 

alter the neutral temperature.

To increase the longitudinal resistance in wood tie track with cut spikes, it is important to have 

effective (tightly applied and maintained) rail anchors. Considerations may be given to 

anchoring every tie rather than every other tie, especially in areas where excessive longitudinal 

rail motion is evident, or using elastic fasteners.

On concrete tie tracks with elastic fasteners, the toe load between the clip and the rail base is an 

important factor contributing to the longitudinal resistance. The toe load reduces in service 

conditions, resulting in decreased resistance. The reduction in toe load can result from fatigued 

clips, worn pads, abraded rail seats, missing insulators, or to being incorrectly driven on.

4. Curve Breathing

Curves are usually well ballasted on the outside to withstand the rail compressive load in 

summer and high lateral vehicle loads. In winter, the rail tensile forces and slow speed freight 

train loads on the low rail can push the curve in towards its center. The result can be curve 

movement radially inward in winter and outward in spring and summer. Curve breathing not 

only weakens the lateral resistance and initiates alignment defects but also alters rail neutral 

temperature. To control neutral temperature changes in the curved track, generous ballasting on 

the inside of the curves must also be provided. Curve staking is often recommended to monitor 

curve breathing induced alignment variations, which are also indicators of possible neutral 

temperature shifts. Correcting excessive curve movement may require realignment, possibly 

coupled with destressing for neutral temperature readjustment.

5. Maintenance Operations

Whenever the CWR track is subjected to any maintenance operation involving moving the track, 

some loss may occur in its neutral temperature. This is particularly true when the track is lifted 

during tamping or when being mechanically lined-in-or-out. Some existing curves may need to 

be realigned for larger radii to permit higher speeds and sometimes to accommodate new lines. 

The changes in neutral temperature should be monitored at a few critical locations by means of 
107

strain gages. It is not necessary to cut the rail to monitor the change unless the change during the 

particular maintenance action is large (such as in excess of 20 °F). In such cases, the track must 

be destressed and readjusted to the correct neutral temperature.

Misalignment

Monitoring

Currently the monitoring of lateral misalignment is done by the railroads through geometry cars 

at various intervals to comply with Federal specifications. It would be helpful and desirable to 

schedule additional track geometry inspections in the summer months, especially when hotweather kinks could become potential buckling prone conditions.

Control

It is important to keep the misalignments under the specified FRA limits for the class of track. 

Initial growth of misalignments can sometimes arise from track shift conditions caused by lateral 

loads acting on weak spots, such as weld misalignments and weak ballast locations. These initial 

misalignments can grow under thermal loads and train action and must be corrected through 

appropriate maintenance actions. After maintenance to correct track misalignments (especially 

after surfacing and realignment), the ballast must be consolidated to an appropriate level by 

either artificial consolidation methods or through slow speed traffic.

Track Buckling Mitigation

In addition to the safety implementation methodologies discussed in the previous section, the 

following approaches are advisable in helping to minimize the problem of track buckling.

Vehicle Dynamic Loads

Vehicle net axle dynamic loads on curves with cant deficiency may cause progressive track shift, 

leading to a buckling scenario, particularly in the presence of high compressive loads. Excessive 

unbalanced speeds must be avoided, although this is the trend in modern high-speed passenger 

trains to reduce travel time. Likewise, the net axle lateral-to-vertical force ratios must be low 

(L/V ≤ 0.4) while the vehicle negotiates any lateral misalignment on tangent or curved tracks, as 

per [27]. In the similar vain, heavy traction and braking forces should be minimized, since these 

can also trigger buckles under hot temperature conditions.

Slow Speeds

Slow speeds are recommended whenever rail temperatures are above the Tc critical values 

discussed in earlier sections. Slow speeds on curves and misaligned tracks not only reduce track 

shift potential but also reduce the damage should a buckling event occur.

Train Handling

One important aspect of train handling is the development of procedures for the engineer to 

engage brake applications or implement other evasive measures when encountering a buckle in 

front of the locomotive caused by the preceding train. Such procedures are very difficult to 

devise on the account of the many variables of each particular scenario (such as speed, consist 

length, freight commodity, physical track location/conditions and size of the buckle in front of 

the train). Therefore, in lieu of such procedures, efforts should be focused on catching the 
108

buckle by the previous train causing it. Since buckles tend to occur usually toward the rear end 

of the consist often allowing the end vehicles to negotiate the buckles. It would be useful for the 

last car to contain appropriate train end devices to identify buckling occurrences and alert the 

locomotive engineer up front to warn the dispatcher or the following train of the potential 

dangerous track condition. Such devices could be based on simple accelerometer based sensors 

which indicate unusual vehicle lateral motion or rocking condition usually associated with 

traversing a large track misalignment. It is not conceptually difficult to design and implement 

such devices.

Summary

• CWR buckling safety limits can be derived directly by exercising the CWR-BUCKLE 

program with the appropriate input parameters. Alternatively, the safety limits can be 

assembled in the form of charts which show the relationship between the track 

parameters (lateral resistance, neutral temperature) and the allowable rail temperatures 

for safe vehicle operations with regard to lateral buckling. The relationship between 

lateral resistance and neutral temperature can be uncoupled in the form of minimum 

required values. This approach has certain advantages, including simplicity and ease of 

implementation without the need to determine the exact values of the parameters.

• Safety methodology using the probabilistic method has also been developed and 

presented. This method can provide more flexibility to the industry than offered by the 

deterministic method because it provides risk-based approaches to buckling prevention 

(such as stipulating the overall permissible number of buckles per annum in a given 

railroad territory). The method provides a global approach to safety specification and is 

complementary to the deterministic method, which is more suitable for specific locations 

and for day-to-day operations.

• The deterministic theory’s safety limits were validated in full-scale dynamic buckling 

tests. The probabilistic approach needs validation, which may be accomplished through 

case studies in collaboration with the industry.

• Methods of safety limit implementation using the deterministic approach are presented. 

Monitoring and controlling the lateral resistance at or above the desired value are 

discussed. Estimating the resistance using the empirical equations inherent in CWRINDY and using STPT equipment for direct measurement are available methods for this 

purpose. Control of lateral resistance can be achieved by providing a full ballast section 

and ties with good bottom friction characteristics and effective consolidation after 

maintenance.

• Neutral temperature monitoring and control are also important in CWR safety assurance. 

At present, monitoring can be done primarily with strain gages. The reference value 

required by the strain gage can be obtained by rail cutting or alternatively by a RUD, as 

described in this chapter. Control of neutral temperature may be accomplished through 

appropriate destressing and restressing operations. This requires knowledge of the proper 

unfastening or de-anchoring length on either side of the rail cut or fracture and the proper 

gap size for adjustment. Effective anchoring and increased longitudinal resistance of 

fasteners can also help maintaining desired neutral temperatures in service. Ample 

ballast on the inside and outside of curves can control curve breathing, and then the 
109

neutral temperature changes. Control of neutral temperature will also be necessary 

during track realignment and some maintenance operations.
110

7. Conclusions and Recommendations

The following conclusions are based on the technical studies presented in this report along with 

recommendations for further research.

Conclusions

• A critical mechanism in CWR track lateral buckling under thermal loads is the rail

bending wave (uplift) between the trucks of a car. The uplift wave is controlled by 

several parameters, including, track vertical stiffness, truck central spacing, tie-ballast 

friction coefficient, and axle loads. Buckling under a car is identified as the more 

frequent occurrence as opposed to the static case (i.e., without vehicle influences).

• Buckling can be explosive or progressive. Explosive buckling (sometimes referred to as 

snap-through) is characterized by equilibrium jumps between the two critical 

temperatures (identified as TBmax and TBmin) on the buckling response curve that defines

the relationship between temperature increase and lateral displacement. At the upper 

critical temperature, TBmax, the track buckles out automatically without any external 

energy. In contrast, to precipitate buckling at the lower critical temperature, some finite 

energy supplied by moving vehicles is required. Progressive buckling, sometimes 

referred to as a slow-buckle, is manifested by increasing lateral deflections with 

increasing temperatures when no distinct TBmax and TBmin exist. This usually occurs for 

weak track conditions.

• A dynamic buckling theory was formulated on the basis of the uplift mechanism taking 

into account large deflections and nonlinearities in resistance parameters. The theory was 

validated by full-scale field tests which simulated explosive and progressive buckling 

characteristics under vehicles. The theory and tests identified critical track and vehicle 

parameters that govern track buckling, and specialized measurement techniques were 

developed for their determination.

• The principal track parameters that influence CWR buckling include track lateral and 

torsional resistances, misalignments, CWR neutral temperature, rail sectional properties,

tie-ballast friction characteristics, and track foundation stiffness. The knowledge and 

measurement of these parameters is important, thus this report presents an analysis and 

measurement techniques for them.

• The dynamic buckling theory has been divided into two groups. The first uses the theory 

in a deterministic manner, whereas the second relies on statistical principles. Computer 

codes have been developed for the deterministic and the probabilistic approaches. The 

deterministic codes are CWR-BUCKLE and CWR-INDY, which, for a given set of 

parameters, can ascertain whether or not the track will buckle out at a given rail 

temperature. The probabilistic approach, which evaluates the probability of buckling as a 

percentage between 0 (no buckling) and 100 (buckling) for a given scenario, is coded in 

CWR-RISK. All three modules are incorporated in a master code, CWR-SAFE, which 

operates in Windows environment on a PC.

• CWR-BUCKLE requires measured input parameters (such as the nonlinear lateral 

resistance characteristic defined in terms of peak and limit resistance values). The 
111

program output includes buckling response data which can be graphically represented, 

upper and lower critical temperatures (TBmax, TBmin), the safe allowable rail temperature 

increase, and the margin of safety against buckling. The program contains a track quality 

based safety criterion to determine allowable temperatures and buckling safety margin.

• CWR-INDY needs only physically describable input parameters, such as the type of 

track, ballast type, ballast crib level and shoulder width, and consolidation levels. These 

are used in the program to deduce the required scientific parameters quantitatively 

through the built-in empirical equations in the code. The output of CWR-INDY is the 

allowable temperature for buckling (or the buckling strength) and the margin of safety. 

The output also includes numerical value of the peak lateral resistance. CWR-INDY is 

intended for quick on-the-spot industry use for buckling strength evaluations.

• CWR-RISK works with inputs described through the statistical distributions of the three 

key parameters of lateral resistance, misalignment amplitude, and neutral temperature. 

The output of this code is the buckling probability expressed as a function of the rail 

temperature. From this output, critical temperatures for buckling prevention can be 

determined, and the expected annual number of buckles in a territory can be evaluated.

• The parametric study generated using the CWR-BUCKLE revealed information of 

practical significance on CWR buckling strength in terms of key variables and their 

sensitivity on the buckling response characteristics. For example, the peak value of 

lateral resistance has a significant influence on the upper temperature, TBmax, as has the 

lateral misalignment. The lower buckling temperature, TBmin, is influenced by the 

limiting value of the lateral resistance, torsional resistance, and lateral misalignment. 

Both temperatures are strongly influenced by track curvature. Vehicle truck center 

spacing and axle loads also impact buckling temperatures through their influence on the 

dynamic (uplift) lateral resistance.

• The parametric study based on the probabilistic approach identifies a critical temperature, 

Tc, below which the buckling probability is zero. Above this Tc value, the probability is 

found to increase rapidly with rail temperature, and thus, Tc is identified as the 

temperature for the deployment of preventative measures such as slow orders. The 

results of the probabilistic approach can also be used to define a limiting rail temperature, 

TL, beyond which vehicle operations should be stopped or carried out at low speeds. This 

TL can be determined by choosing a number of acceptable buckles per year; thus the 

railroads can control future annual number of buckles by adhering to Tc as the safe limit 

and curtailing operations (such as slow-orders) beyond it up to TL.

• Controlling Tc and the number of buckles by an appropriate choice of TL will also require 

controlling the low-end values of the distributions of lateral resistance, neutral 

temperature, and the high-end values of lateral misalignments. Numerical examples 

indicate that controlling the neutral temperature’s low-end values is particularly effective 

for buckling prevention.

• Based on a risk methodology, a speed reduction formula has been devised relating 

vehicle speed (as a percent of maximum line speed) to the rail temperature, TR > Tc. The 

formula produces zero speed when T = TL, the limiting rail temperature for vehicle 

operations. Applying speed restrictions between Tc and TL becomes effective in limiting 
112

the annual number of buckles to accepted values, as well as reducing the consequences of 

the buckling induced derailments.

• Safety limits are formulated on the basis of the deterministic approach. The limits were 

validated by full-scale tests with moving vehicles on heated CWR test segments. For the 

safety implementation of these limits, the approach is recommended that stipulates the 

adherence to established minima for lateral resistance and neutral temperature for a given 

FRA class of track (i.e., for a stipulated maximum allowable misalignment). Methods to 

verify these minimum values include the STPT device or CWR-INDY for lateral 

resistance prediction, and the RUD and strain gage for neutral temperature.

• Using the data collected previously, the required sample rates for the measurements of 

lateral resistance and neutral temperature have been estimated. In the case of lateral 

resistance, an error of 15 percent is tolerable for the resistance range of practical interest 

(1500 to 2000 lb). This will not contribute to an error of more than 5 °F in the allowable 

rail temperature for safe operations of vehicles on CWR tracks. To evaluate the 

resistance within this maximum permissible error and for a confidence limit of 90 

percent, one sample in a 50 ft section is adequate for concrete tie track, and two samples 

over a 50-foot section for wood tie tracks. Similarly, for the case of neutral temperature 

measurement, for a tolerable error of 5 °F, one measurement over a 100-foot section 

seems adequate for the measurement with 90 percent confidence limit.

Recommendations

• The probabilistic approach and its application to risk-based buckling prevention practices 

should be demonstrated on cooperating railroads. This would require testing for the 

statistical variations in lateral resistance, neutral temperature, and lateral alignment within 

a characteristic or representative line segment. The data would be used in CWR-RISK to 

develop safer operating temperature regimes for CWR, as well as to indicate potential 

cost-effective maintenance strategies.

• Implement the Rail Uplift technique onto a moving platform, such as on a hi-railer or a 

dedicated rail car, to facilitate an easy and quick measurement of the minimum required 

neutral temperature.

• Develop a simplified STPT and procedure for a quick peak resistance measurement (i.e., 

mobilizing the test tie only through a small 0.2–0.4 in deflection by eliminating complex 

hydraulics and simplifying data acquisition).

• A rail destressing force indicator can be developed to more effectively install and destress 

CWR by setting the neutral temperature more correctly. Such an instrument can also be 

used to calibrate the hydraulic rail tensors currently in maintenance use to minimize 

possible errors in their present application.

• Continue testing revenue service tracks for neutral temperature determination to better 

identify buckling prone conditions and to provide data toward establishing better neutral 

temperature management practices, including more effective destressing and repairing 

broken rail repair procedures.
113

• Using STPT, further quantify the sampling rates established in Appendix D. Further,

correlate with field data the empirical equations for lateral resistance in CWR-INDY. It 

is also important to continue lateral resistance mapping on revenue lines to better define 

statistical variations for CWR-RISK applications, as well as for data to further develop 

and validate CWR-INDY’s lateral resistance predictor.

• Evaluate the risk-based slow order methodology presented here against railroad 

experience to develop more extensive applications strategies, including the promulgation 

of risk-index based approaches to buckling prevention.
114

8. References

1. U.S. DOT/Federal Railroad Administrations Accident/Incident Bulletins 149-165 (1980-

1996); and Annual Reports on Safety Statistics Annual Reports (1997-2003).

2. Samavedam, G., “Buckling and Post Buckling Analyses of CWR in the Lateral Plane,” 

British Railways Board, R&D Division, Technical Note, TN TS 34, 1979.

3. Kish, A., and G. Samavedam, “Analysis of Thermal Buckling Tests on United States 

Railroads,” DOT/FRA/ORD-82/45, 1982.

4. Kish, A., G. Samavedam, and D. Jeong, “Influence of Vehicle Induced Loads on the 

Lateral Stability of CWR Track,” DOT/FRA/ORD-85/03, 1985.

5. Samavedam, G., A. Kish, and D. Jeong, “Experimental Investigation of Dynamic 

Buckling of CWR Tracks,” DOT/FRA/ORD-86/07, November 1986.

6. Kish, A., and G. Samavedam, “Analyses of Phase III Dynamic Buckling Tests,” 

DOT/FRA/ORD-89/08, February 1990.

7. Kish, A., and G. Samavedam, “Dynamic Buckling Test Analyses of a High Degree CWR 

Track,” DOT/FRA/ORD-90/13, 1991.

8. Kish, A., and G. Samavedam, “Dynamic Buckling of Continuous Welded Rail Track: 

Theory, Tests, and Safety Concepts,” Transportation Research Record, 1289, Proceedings 

of Conference on Lateral Track Stability, 1991.

9. Samavedam, G., A. Purple, A. Kish, and J. Schoengart, “Parametric Analysis and Safety 

Concepts of CWR Track Buckling,” Final Report, DOT/FRA/ORD-93/26, December 

1993.

10. Samavedam, G., A. Kanaan, J. Pietrak, A. Kish, and A. Sluz, “Wood Tie Track 

Resistance Characterization and Correlations Study,” Final Report, DOT/FRA/ORD94/07, January 1995.

11. Kish, A., D.W. Clark, and W. Thompson, “Recent Investigations on the Lateral Stability 

of Wood and Concrete Tie Tracks,” AREA Bulletin 752, October 1995, pp 248-265.

12. Samavedam, G., A. Kish, and D. Jeong, “The Neutral Temperature Variation of 

Continuous Welded Rails,” AREA Bulletin 712, 1987.

13. Kish, A., and G. Samavedam, “Risk Analysis Based CWR Track Buckling Safety 

Evaluations,” Proceedings of Conference on “Innovations in the Design and Assessment 

of Rail Track,” December 1999, Delft University of Technology, The Netherlands.

14. Kish, A., and G. Samavedam, “Longitudinal Force Measurement in Continuous Welded 

Rail from Beam Column Deflection Response,” AREA Bulletin 712, Vol. 88, 1987.

15. Kish, A., S. Kalay, A. Hazell, J. Schoengart, and G. Samavedam, “Rail Longitudinal 

Force Measurement Evaluation Studies Using the Track Loading Vehicle,” Bulletin 742, 

American Railway Engineering Association, Bulletin No. 742, 1993.

16. Samavedam, G., “Investigation on CWR Longitudinal Restraint Behavior in Winter Rail 

Break and Summer Destressing Operations,” DOT/FRA/ORD-97/01, 1997.
115

17. ERRI Report, Samavedam, G., “Theory of CWR Track Stability,” D202/RP3, Utrecht, 

February 1995.

18. Samavedam, G., et. al., “Analysis of Track Shift Under High-Speed Vehicle-Track 

Interaction,” Technical Report DOT/FRA/ORD-97/02, 1997.

19. Thomson, D., G. Samavedam, W. Mui, and A. Kish, “Field Testing of High Degree 

Revenue Service Track for Buckling Safety Assessment,” Technical Report, 

DOT/FRA/ORD-92/02, November 1993.

20. Union of International Railways (UIC) Code 720R, “Laying and Maintenance of CWR 

Track,” 2nd Edition, December 2002.

21. Klaren, J.W., and J.C. Loach, “Lateral Stability of Rails, Especially of Long Welded 

Rails,” Question D14, ORE, Utrecht, 1965.

22. Sussmann, T., A. Kish, and M. Trosino, “Influence of Track Maintenance on the Lateral 

Resistance of Concrete Tie Track,” Transportation Research Board Record, 1825, Paper 

No. 03-3694.

23. Samavedam, G., and A. Kish, “Track Lateral Shift Model Development and Test 

Validation,” DOT/VNTSC/FRA Report, 2002.

24. Harnett, D., “Statistical Methods, Third Edition,” The Philippines: Addison Wesley 

Publishing Company, Inc., 1982.

25. Kristoff, S., “Track Lateral Strength Measurements at Union Pacific Railroad Sites,” 

Foster-Miller Report prepared for Union Pacific Railroad, 2001.

26. Kish, A., T. Sussman, and M. Trosino, “Effects of Maintenance Operations on Track 

Buckling Potential,” International Heavy Haul Association Technical Conference, May 

2003, Dallas, Texas.

27. Kish, A., G. Samavedam, and D. Wormley, “New Track Shift Limits for High-Speed Rail 

Applications,” World Congress for Railway Research (WCRR 2001), November 25-29 

2001, Cologne, Germany.

28. Kish, A. and G. Samavedam, “Improvements in CWR Destressing for Better 

Management of Rail Neutral Temperature,” Transportation Research Board 2005 Annual 

Conference, January 9-12, 2005.
A-1

Appendix A .

Mathematical Formulation of CWR-Buckle

A.1 Buckling Theory

The equation to analyze the lateral stability of continuous welded rail (CWR) track can be 

derived by applying the principle of minimum potential energy under the assumptions of 

Chapter 2. Through the use of variational calculus, the equations of equilibrium are 

presented in the form of two highly nonlinear differential equations for tangent and 

curved tracks. These differential equations become solvable when the infinite track 

domain is divided into two regions: (1) a buckled zone where longitudinal displacement 

is neglected and (2) an adjoining zone that extends to infinity where lateral displacement 

is neglected (Figure A-1). The equations for the two regions are coupled through another 

equation, which yields the temperature. For an assumed buckled length, 2L, the 

temperature increase, the lateral track deflection, and the compressive force in the rails 

can be calculated through the solution of the following differential equations.

Figure A-1. Geometry and Coordinates
A-2

A.2 Differential Equations and Solution

The basic differential equations are formulated using the large deflection theory and 

principles of variational calculus as in [2].

Buckling Zone (0 ≤ x ≤ L): Tangent Track

The differential equation is

( ) [ ] 2

o

2

2

2

o 4

4

zz

dx

d w

F w(x) P

dx

d w P

dx

d w EI + − τ = − − (A-1)

where 

E = the modulus of elasticity, 

Izz, = the area moment of two rails for lateral bending (i.e., about the vertical 

axis), 

P = the longitudinal force in the rails, 

F[w(x)] = the lateral resistance distribution function, 

τo = the linear torsional stiffness of fasteners (both the lateral resistance and the 

torsional stiffness are expressed here per unit track length), 

wo = the initial imperfection distribution, and 

w = the deflection in the lateral direction.

The solution of this differential equation can be expressed in terms of an infinite 

trigonometric series, assuming the initial misalignment and the buckling mode are 

symmetric.

w(x) =

m=1,3,5,...

∞

∑ Am cos

mπx

2L

 

   

(A-2)

d2wo

dx2 =

m=1,3,5,...

∞

∑ bm cos

mπx

2L

 

   

(A-3)

F[w(x)] =

m=1,3,5,...

∞

∑ am cos

mπx

2L

 

   

(A-4)

Where

Am = −(am + Pbm)

EIzz

mπ

2L

 

   

4

− P − τ ( o) mπ

2L

 

   

2

(A-5)

The foregoing expressions satisfy the requirements of zero deflection and moment at the 

ends of the buckled zone. In addition, the zero slope condition will be satisfied by 

stipulating
A-3

0

2

m mA sin m

m 1,3,5,...

 =



 



 π ∑

∞

=

(A-6)

The foregoing equation which is solved by an iteration scheme, gives the relationship 

between the assumed buckling length 2L, and the compressive load P .

Adjoining Zone (x  ≥ L)

The longitudinal resistance is assumed proportional to longitudinal displacement. The 

governing differential equation in the adjoining zone (x>L) is derived from equilibrium 

considerations in the longitudinal direction. Thus, if proportional longitudinal resistance 

is assumed, then the differential equation in the adjoining zone is

AE d2U

dx2 = kfU

(A-7)

where 

A = the cross sectional area of two rails, 

E = the modulus of elasticity, 

kf = the slope of the longitudinal resistance versus longitudinal displacement curve 

(i.e., longitudinal resistance stiffness), 

U = the longitudinal displacement.

The solution to this equation is

U(x) = C1e

ψx + C2e−ψx (A-8)

where

However, the solution must be bounded for very large values of x. Then, U = U′ = 0 in., 

the limit as x approaches infinity, and C1 = 0. After differentiation,

U'(x) = −ψC2e−ψx

or

U' = -ψU (A-9)

Buckling Zone (0 ≤ θ ≤ φ): Curved Track

The governing differential equation in the buckled zone (0 ≤ θ ≤ φ) for curved track is

[ ] 2 d

wo

2d

2 R

P

R

P F w( ) 2 dw2d

2 R

P o

4 d

w4d

4 R

EIzz

θ

= − θ + −

θ 













 − τ

+

θ

(A-10)

ψ

2 = kf

AE
A-4

As in the tangent case, the solution to this equation can be expressed in terms of an 

infinite trigonometric series

w(θ) =

m=1,3,5,...

∞

∑ Am cos

mπθ

2φ

 

 

  

 

(A-11)







 







φ

πθ = θ ∑

∞

= 2

m b cos

d

d w

R

P m

m 1,3,5,...

2

o

2

2 (A-12)

[ ( )] 





 







φ

πθ = ∑

∞

= 2

m F w x a cos m

m 1,3,5,...

(A-13)

P 

R =

m=1,3,5,...

∞

∑ P 

R

cm cos

mπθ

2φ

 

 

  

 

(A-14)

where

2

2

m

2 R

P o

4

2

m

4 R

EIzz

bm 2 R

P

cm R

P

am

Am





 





θ

π















 − τ  − 

 





θ

π

















− − +

= (A-15)

The foregoing equations satisfy zero deflection and curvature requirements at the ends of 

the buckled zone. An equation similar to that of A-6 satisfying the zero slope condition 

relates the compressive force, P , to the assumed buckling length, 2L.

Adjoining Zone: Curved Track

The differential equation of longitudinal equilibrium that applies to the adjoining zone (θ

>φ) for the curved track case, again assuming proportional longitudinal resistance, is

AE

R2

d2U

dθ2 = kfU

(A-16)

Recall that L = Rφ and x = Rθ. Thus, the general solution to this equation is

U(θ) = C3e

Rψθ + C4e−Rψθ

(A-17)

where

ψ

2 = kf

AE

The function F in A-1 and A-10 represents the lateral resistance. F can be expressed as:

F = FP F(w) (A-18)

where FP is the peak value, which can be a function of x, and F(w) is the functional 

dependency on the lateral displacement w.
A-5

For the case where vehicle loading is present, the peak resistance is a function of the 

longitudinal distance along the track

(A-19)

where 

FP = the peak value of static lateral resistance, 

μf = the tie-ballast coefficient of friction, 

Q = the self weight of the track, 

Rv(x) = the ballast vertical reaction to the vehicle wheel loads on the track.

The vertical reaction can be calculated from classical beam on elastic foundation theory 

as shown later. Uplift occurs when the sum of the vertical deflection and the self weight 

of the track is less than zero or mathematically when [Q+Rv(x)]<0.

The Fourier coefficient that accounts for the effect of lateral resistance on the track, am, is 

derived from the following integral

For tangent track

[ ] dx

2L

m x F w(x) cos

L

2

a

L

o

m 



 



 π = ∫ (A-20)

For curved track

[ ] θ 





 







φ

πθ = θ ∫

φ

d

2

m F w( ) cos

L

2

a

o

m (A-21)

The integrals are evaluated using Filon’s integration scheme. If the lateral resistance 

function F[w(x)] is a constant value, Fo, then this integral can be evaluated in closed form

am = 4Fo

mπ

sin mπ

2

 

   (A-22)

The Fourier coefficient that accounts for the effect of initial imperfection in the track, bm, 

is derived from the following integral

For tangent track

dx

2L

m x

cos

dx

d w

L

2 b

L'

o

2

o

2

m 



 



 π = ∫ (A-23)

where













>

≤ =

o

o

o L L

L L

if

if

L

L

L'
A-6

and

L0 = misalignment half-wavelength

For curved track

θ 





 







φ

πθ

φ θ = ∫

φ

d

2

m

cos

d

2 d w b

'

o

2

o

2

m (A-24)

where













φ > φ

φ ≤ φ

φ

φ φ =

o

o

o if

if '

Misalignment

The initial imperfection shape is assumed as a fourth degree polynomial

wo(x) = δo 1 − 2 x

Lo

 

 

  

  

2

+

x

Lo

 

 

  

  

4  

 

 

 

 

 

(A-25)

where 

δo = the offset or the misalignment amplitude, 

L0 = half the wave length over which the misalignment occurs.

For the imperfection shape shown above, evaluation of A-23 results in the following

For L ≤ L0

bm = − 16δo

mπLo

2 1 − 3 L

Lo

 

 

  

  

2

1 − 2 2

mπ

 

   

2  

  

 

  

 

 

 

 

 

 

sin mπ

2

 

   

(A26)

For L > Lo

bm = − 16δo

mπLo

2 −6

L

Lo

 

 

  

  2mπ

 

   cos

mπLo

2L

 

   + 2 −1+ 3 2L

mπLo

 

 

  

  

2  

 

 

 

 

 sin mπLo

2L

 

   

 

 

 

 

 

 (A27)

For φ ≤φo

bm

R2 = − 16δo

mπLo

2 1 − 3 L

Lo

 

 

  

  

2

1 − 2 2

mπ

 

   

2  

  

 

  

 

 

 

 

 

 

sin mπ

2

 

   

(A-28)

For φ >φo
A-7

bm

R2 = − 16δo

mπLo

2 −6

L

Lo

 

 

  

  2mπ

 

   cos

mπLo

2L

 

   + 2 −1+ 3 2L

mπLo

 

 

  

  

2  

 

 

 

 

 sin mπLo

2L

 

   

 

 

 

 

 

 (A-29)

Also,





 



 = 





 





 = ∫ 2

sin 4

2

cos

2 π

π

θ

φ

πθ

φ

φ

m

m

d mc

o

m (A30)

Temperature Calculations

The temperature equation is derived by using continuity requirements on the longitudinal 

displacement between the buckled and adjoining zones. It can be shown that

U(L) = − P L

AE − Z + α∆TL (A-31)

U' (L) = − P L

AE

+ α∆T (A-32)

where L is the buckled length and Z is defined below. Using A-10

T = P

AEα

+

Zψ

α(1 + ψL) (A-33)

where

dx

L

o

o w'w' 2

2 w' Z ∫ 















= + (A-34)

The equation for Z can be rewritten, after an integration by parts

ww" dx

2

w' Z

L

o

o

2

∫ 













 = − (A-35)

The expression for Z can be expressed as an infinite series by applying Fourier analysis

Z = L

4

m=1,3,5,...

∞

∑ Am

2 mπ

2L

 

   

2

− 2Ambm

 

  

 

  

(A-36)

In a similar fashion to the tangent analysis, the temperature equation for curved track is

T = P

AEα

+

ZRψ

α(1 + ψL) (A-37)
A-8

where the equation for Z can be written as (after integration by parts and application of a 

Fourier series)

ZR =

m=1,3,5,...

∞

∑ 2L

mπR2 Am sin mπ

2

 

   +

mπ

2

 

   

2 Am

2

4L − AmbmL

2R2

 

  

 

  

(A-38)

In the limit as R approaches infinity or as the track curvature becomes tangent, the 

expression for ZR approaches the expression for Z in the tangent case since bm for the 

tangent case is identical to bm/R2 for curved. Thus, the two temperature equations also 

reduce to the same expression in the limiting process.

Vertical Deflection and Reaction Calculation

Quasi-static load idealization is assumed to be adequate in determining loss of resistance 

in the uplift region, which occurs due to the vertical track deformation under wheel loads. 

The differential equation for the vertical deflection v is

EIyyv' ' '' +Kvv = δi(x − xi ∑ )Vi + Q (A39)

where

EIyy = combined flexural rigidity for the two rails in the vertical plane

Kv = track foundation stiffness (assumed constant)

Vi = vertical wheel loads

δi = Dirac delta functions

Q = track weight/unit length

In this equation, the effects of vertical imperfections and the compressive load in the rails 

are excluded for the sake of simplicity.

After solving equation A-39 under the boundary conditions at infinity, v = v ́ = 0, one can 

compute the distributed foundation (tie ballast) reaction Rv(x) given by

Rv = Kv • v(x) (A-40)

A.3 Energy Required for Buckling

As stated earlier, the upper buckling temperature represents stability in the infinitesimal 

sense requiring no external energy for snap-through explosive buckling. At temperatures 

lower than this (but higher than the lower buckling temperature), the track can buckle out 

upon the application of a finite external energy. Thus the energy needed to cause 

buckling can be used as a measure of the degree of stability. This measure will be useful 

in the development of CWR safety limits.

Referring to Figure 2-3c in Chapter 2, the pre-buckling state is represented by position 

(1) while the post-buckling unstable branch is represented by position (2). It is assumed 

that, if the track can be brought into position (2), it will automatically move to position 

(3).
A-9

The following quantities are defined.

V1 = strain energy in the rails at stable equilibrium position (1)

V2 = strain energy in the rails at unstable equilibrium position (2)

W = work done against resistances by moving track from position (1) to position (2)

Ω = energy required to move track from position (1) to position (2)

By an energy balance

Ω = (V2 - V1) + W (A-41)

The strain energy components are given by the following integrals

dx

AE

P

V ∫

∞

∞ =

0

2

1 2

1 (A-42)

where

P∞ = -AEαT

Here, for simplicity, neglect the energy due to bending in the pre-buckling state

dx

dx

d w

2

EI dx

AE

P

2

1 V

2

o

2

2 zz

o

2

2 ∫ ∫

∞ ∞















 = + (A-43)

In the curved track case, the longitudinal force distribution becomes

P =

P for 0 ≤ θ ≤ φ

AE 1

R

du

dθ − αT  

   for θ > φ

 

 

 

 

 

 (A-44)

The work done against the lateral and longitudinal resistances are given by the following 

integrals

W F[w(x)]dw dx

0

w(x)

0

1 ∫ ∫

∞

= (A-45)

W f[u(x)]du dx

0

u(x)

0

2 ∫ ∫

∞

= (A-46)

Here f is the longitudinal resistance.

Thus, the total work done against ballast resistance (lateral and longitudinal) is

W = W1 + W2 (A-47)

The difference in strain energy is calculated from the following equation

dx

dx

d w

2

EI dx

AE

P P

2

1 V V

2

0

2

2

zz

0

2 2

2 1 ∫ ∫

∞ ∞

∞

















+ − − = (A-48)
A-10

This equation shows that the total strain energy is the sum of two components: one due 

to compressive axial force and the other due to beam bending. The evaluation of these 

integrals is performed with the aid of the Fourier analysis.

(A-49)

The work done against lateral resistance can be evaluated from A-45 once the lateral 

resistance is expressed in terms of a mathematical function.

The work done against a linear longitudinal resistance f = kfu is given by

W2 = kf

4ψ

3

P

AE − αT

 

 

  

 

2

(A-50)

V2 − V1 = AE

2

P

AE

P

AE

L +

1

2ψ

 

 

  

 +

αT

ψ

 

  

 

  − (αT)

2 L +

3

2ψ

 

 

  

 

 

  

 

  

+

EIzz

64L3

m=1,3,5,...

∞

∑ (mπ)

4 Am

2
B-1

Appendix B.

Techniques for Parameter Measurements

The following paragraphs describe the hardware and measurement techniques for the 

critical parameters developed as part of this track stability research program. Some of the 

measurements/technique can be greatly simplified for the purpose of routine safety 

assurance and implementation by the rail industry, as discussed in Chapter 6.

The hardware and measurement techniques for the following parameters are discussed.

1. Lateral Resistance

2. Torsional Stiffness

3. Longitudinal Resistance

4. Tie-Ballast Friction Coefficient

5. Rail Force and Neutral Temperature

B.1 Lateral Resistance Measurement

A number of research organizations have measured track lateral resistance in the United 

States and abroad. The recommended measurement scheme mobilizes only a single tie. 

Some of the previous techniques require lateral movement of a cut panel or the entire 

track section by a concentrated lateral load. In the case where only a single tie is 

mobilized, the resistance is directly represented by the load-deflection response of the tie 

and provides the spring type resistance required by analysis.

The advantages of the Single Tie Push Test (STPT) over the panel test include the 

following:

• STPTs yield a more fundamental and correct characteristic of the ballast 

resistance.

• The test is easy to set up and perform.

• The hardware is man-portable and can be used by track crew with minimal 

training.

• The obtained test data also provides the parameter required for other track 

analyses, such as lateral panel shift

The disadvantage of the STPT is the variation of the results from tie to tie. However, an 

arithmetic average of the individual test results is adequate for buckling analysis. 

Appendix D discusses the sampling rate for a statistically significant value.

B.1.1 Test Hardware

Lightweight portable devices were developed for the wood and concrete tie tracks. The 

devices are shown in Figure B-1 for a wood tie and Figure B-2 for a concrete tie. The 

device consists of a hydraulic control unit with a pump and a fixture with a hydraulic 

cylinder. With the spikes, rail anchors, and tie plates removed from both rails, the 

assembly is set to grab the test tie which is now free to move laterally under the rails. 

The hydraulic piston mounted on the fixture reacts with the force required to move the tie 
B-2

against one of the rails. Hydraulic pressure can be provided by the hand pump or by an 

electric pump to speed up the operation.

A pressure transducer or load cell in line with the piston and pressure gage in the control 

unit (as a backup) indicates the load applied. A rotary potentiometer mounted on the tie 

measures the displacement with respect to the stationary second rail. The loaddisplacement relationship is plotted using an X-Y plotter. Alternately, a modern Data 

Acquisition System can display this relationship on a monitor and store the data on a 

computer.

Figure B-1. STPT Equipment for Wood Ties

Figure B-2. STPT Equipment for Concrete Ties

B.1.2 Typical Results

A large number of track resistance tests were conducted using the STPT device at the 

Transportation Test Center’s (TTC) test tracks in Pueblo, CO, and on a number of 

railroads. Figure B-3 shows typical results for relatively strong, medium, and weak wood 

and concrete tie tracks. In general, two salient points on the load versus deflection 

characteristics exist: the peak FP, occurring at displacements on the order of 0.25-0.5 in

and the limiting value, FL, at about 3-5 in of tie displacement. The softening or drooping 

behavior becomes more pronounced for high FP values; whereas for low FP’s, the 

resistance is practically constant with FP being close or equal to FL.
B-3

Figure B-3. Typical STPT Results for Strong, Average, and

Weak Track for Wood and Concrete Ties

The most recent application of the STPT for track lateral resistance evaluation was at an 

Amtrak concrete tie track segment at New Carrollton, MD. The tests were to evaluate 

track lateral strength and stability after track maintenance and subsequent stabilization 

operations on Class 4 mainline track segment. The tests consisted of performing track 

lateral resistance measurements using the STPT, and comparing the resulting 

measurements after maintenance and different consolidation operations for subsequent 

buckling safety evaluations. Figure B-4 provides a data summary, and additional test 

results are available in [22]. The data was also used to conduct buckling safety analyses 

using CWR-RISK to evaluate buckling risk implications resulting from the maintenance 

actions [26].
B-4

Figure B-4. STPT Concrete Tie Measurements on Amtrak

B.1.3 Torsional Stiffness Measurement

Torsional resistance is offered in the fasteners when the rail tends to rotate with respect to 

the ties i.e. resisting the in-plane bending, such as during buckling. Many rail fastener 

systems were tested, including two elastic fastener designs, as well as conventional wood 

tie plate/cut-spike systems. Laboratory and field tests were designed to evaluate pure rail 

rotation in fasteners while and to account for the tie rotation in the ballast (such as in the 

case of very torsionally rigid fasteners), respectively.

In the laboratory tests, the tie was held rigidly in a fixture, and equal and opposite loads 

were applied to the rail as shown schematically in Figure B-5 via hydraulic cylinders. 

The rail displacement was measured, and rotation was computed with the moment arm. 

All tests were conducted to at least a 5-degree rotation or until the onset of tie material 

crushing or failure. In cases of concrete tie fasteners, testing was also suspended if any 

insulators broke.
B-5

Hydraulic cylinder

and load cell

Hydraulic cylinder

and load cell Tie plate

and rail spikes

Rail section

String pot

Tie

Figure B-5. Methodology for Torsional Stiffness Measurement

In the field tests, wood ties with cut-spike construction under existing service conditions 

were considered. The rails were cut into 40-inch sections. Two configurations of loading 

were initially devised, as shown in Figure B-6. Configuration A, was eliminated from 

further consideration because it applied a reactive lateral force on the fasteners. In 

Configuration B, the setup produces pure torque without lateral load on the fasteners. 

The test section chosen was 136 RE jointed rail with rail anchors on every other tie 

permitting an assessment of anchor influence.

Tie/plate/spikes Hydraulic cylinder and load cell

FR

Fa

Tr

Hydraulic cylinder

and load cell

Hydraulic cylinder

and load cell

Fa

Fa

Tr

Tie/plate/spikes

Configuration A Configuration B

Tr = torque

Fa = applied force

FR = resultant force

Figure B-6. Torsional Test Loading Configurations

B.1.4 Concrete Tie Test Results

Two elastic fastener systems were tested on concrete ties. Several test runs were 

performed to determine the effect of fully driven clips versus partially driven. No 

apparent difference in torsional resistance response was observed. Further comparisons 

were made between polyurethane pads versus rubber pads—again no significant 
B-6

differences were observed in the response curves. Overall, both systems exhibit a soft 

torsional resistance response in comparison with wood tie systems, as seen in the 

following section. The reason for the soft response is primarily due to the elasticity 

offered by the resilient pads and the softness of the insulators. This elastic attribute of the 

pads allowed the applied torque to be absorbed through insulator crushing rather than by 

the clips themselves, a fact observed through the insulators examination upon completion 

of each test run. The response for the two elastic fasteners was in the same range, 

suggesting fastening types on concrete ties play little role in determining the torsional 

resistance. Rather, the existence of insulators and pads appears to be the controlling 

factor. Typical results for the two systems are available in [9].

B.1.5 Wood Tie Results

Wood tie fastening systems tested produced a much stiffer resistance than the two elastic 

fattener systems on concrete ties. The specific fasteners tested on wood ties were a 

typical cut-spike construction, and elastic fastener with lock and screw spikes. Torsional 

response was compared for a variety of wood tie types and condition. Tie type and 

condition had little or no influence on the response characteristic, whereas the fastener 

type itself was a significant parameter. As expected, elastic fastener on wood ties were 

the stiffest, regardless of the use of lock versus screw spikes, followed by eight spikes per 

plate, four spikes per plate, and finally the softest wood tie characteristic was offered by 

two spikes per plate. Typical wood tie test results are available in [9].

B.2 Longitudinal Resistance

The longitudinal resistance of the track is defined as the resistance offered to the rail as it 

moves or tends to move in the longitudinal direction. In the case of curved track, this 

direction of movement can be considered to be tangential to the curve.

The resistance to the rail movement in the longitudinal direction is offered by the 

fasteners that hold the rail to the tie. The rail may move through the fasteners with the 

ties stationary, or the rail and the ties may move through the ballast when the fasteners 

provide a rigid connection. Figure B-7 shows the two scenarios schematically. To 

distinguish the two cases, one could define Figure B-7b to represent the fastener 

longitudinal resistance, as opposed to Figure B-7c, which represents the ballast 

longitudinal resistance. The net resistance to the rail can be a combination of these two 

resistances. That is, at small loads in the rail, initial rail movement can occur in the 

fasteners with respect to the tie. With increasing load, the rail may move along with the 

tie in the ballast. The behavior of the system is also dependent on the vehicle load.
B-7

A'

A''

u

u

f

f

Pad

Tie

a) Initial position

b) Loaded rail moves with

respect to tie stationary in 

ballast

c) Loaded rail moves along

with tie in ballast

Figure B-7. Longitudinal Resistance Mechanism

The longitudinal resistance is an important contributor to the stability of CWR in the 

following manner:

• It controls the distribution of the rail longitudinal force and movements, thus 

affecting the rail neutral temperature.

• It provides the necessary reaction to train braking and acceleration forces.

• It controls the rail break gaps when pull-apart failures occur.

The factors that may significantly influence the longitudinal resistance include the 

following:

• Anchor effectiveness

• Toe load retention of fastener

• Pad shear stiffness

• Tie type and size

• Ballast type/condition

• Vertical vehicle load

The resistance can be expressed per fastener (per rail) or per tie (two fasteners per tie). 

The track longitudinal resistance is obtained by dividing the individual tie longitudinal 

resistance by the tie spacing. 

Track longitudinal resistance =

Individual tie longitudinal resistance

Tie spacing

The tie spacing in the above equation is the spacing of anchored ties (i.e., ties with 

fasteners attached to the rail). Wood ties with cut spike construction are generally 

anchored on every other tie. The spacing to be used in the foregoing equation should be 

twice the actual physical spacing of ties.
B-8

Figure 4-10 in Chapter 4 shows a typical response from a longitudinal resistance test. 

The resistance is idealized as linear-constant. The slope of the linear part is called

longitudinal stiffness, kf, whereas the constant fo is called steady state value of the 

resistance.

B.2.1 Measurement of Longitudinal Resistance

The fastener longitudinal resistance can be measured in a laboratory using the pull-test, in 

which the rail is pulled through its fastener with the tie held rigidly by an external 

support. For the measurement of the ballast longitudinal resistance, a field setup is 

necessary. Figure B-8 shows a schematic of the arrangement used at TTC. A panel is cut 

and pulled longitudinally, reacted by adjacent portions of the track. The panel must have 

a minimum number of ties to prevent rotation over their long axes when subjected to 

loads via the rail.

Figure B-8. Longitudinal Resistance Test Hardware

B.3 Tie-Ballast Friction Coefficient

The tie-ballast friction coefficient is determined using the STPT fixture for small vertical 

loads (≈ 1 kip) on the tie and the Track Loading Vehicle (TLV) for larger vertical loads

(> 5 kips). The case with small vertical load on the tie is appropriate for calculating the 

loss of resistance in the central region between the trucks, which is the scenario for track 

buckling.
B-9

The friction coefficient is defined as the friction force at the bottom surface of tie 

interfacing the ballast divided by the weight of the tie. This interface is not like that 

between two bodies with plane surfaces exhibiting the typical Coulomb friction property. 

The ballast particles can physically penetrate the wood tie surface, thereby giving rise to 

an apparent friction coefficient larger than one in some cases where the tie has been in 

service over a long time. Under passing vehicles causing vibrations and vertical 

movement of ties, the friction coefficient falls off to a lower value compared to the value 

in a statically embedded state of the tie. This value of the friction coefficient during the 

tie movement is more appropriate for use in the dynamic buckling theory than the static 

value.

In the application of TLV to measure the friction coefficient, the TLV hydraulics applied 

vertical load on the tie and pushed the tie laterally (Figure B-9). Several ties with similar 

ballast shoulder and crib were tested with and without the vertical load. The difference 

between the peak values of the lateral resistance with and without vertical load on the tie 

divided by the vertical load on the tie was taken as the tie-ballast friction coefficient. The 

friction coefficient was also evaluated for a range of vertical loads as well. The friction 

coefficient at no applied vertical load on the tie, other than its self-weight, is also 

evaluated by testing a number of ties with and without ballast in shoulders and cribs. The 

difference in the peak values of the resistance divided by the tie weight was taken as the 

tie-ballast base friction coefficient. The values of the friction coefficient were found to 

be in the range of 0.9 to 1.2 for the wood tie and about 0.85 for the concrete tie, the 

ballast in both cases being granite.

Figure B-9. TLV Based Friction Coefficient Measurement

B.4 Rail Force and Neutral Temperature

A critical review of the techniques to measure rail force and neutral temperature revealed 

that only two techniques are available to measure rail force with acceptable accuracy and 

reliability. These are the strain gage and the Rail Uplift Technique (RUD). The strain 

gage is well known in the literature and has been used extensively in a number of 

measurements for the FRA/Volpe projects on CWR. The authors [14] originally 

conceived the Rail Uplift Technique and extensive tests were carried out at TTC to 

validate the technique.

Strain gage technique needs rail cutting to provide a “reference” for the absolute force. 

The RUD can yield absolute force after calibration which is not site dependent. The 
B-10

device has an analytic basis as explained in the following paragraph. The disadvantage 

of the device is that is requires freeing the rail from anchors or fasteners, which makes it 

somewhat cumbersome in application.

B.5 Rail Uplift Device

Vertical deflection depends on the magnitude of the rail longitudinal force when rail-ties 

fasteners are removed over some length, restrained vertically at the ends of the freed 

portion, and subjected to a concentrated uplift load at the center. Clearly, longitudinal 

compressive load will increase the deflection of the beam-column, and tensile force will 

reduce it. For a given length of rail, the vertical force required to produce a specified 

deflection is a measure of these rail forces. The concept implementation is based on the 

fact that the rail can be held pinned at the two end points by the wheels of a rail car. This 

automatically fixes the length of the rail and boundary conditions at the ends of the rail 

beam. The spikes and anchors between the inner wheels of the two trucks of the car must 

be removed. Figure B-10 shows schematically the rail uplift method.

Figure B-10. Rail Uplift Device Principle

An analytic model was developed to relate the applied vertical force and deflection to 

longitudinal rail force. The model showed that the relationship between the applied 

vertical force and the rail longitudinal force is linear. A device can be operated without 

rail buckling for small deflections (< 2”) over the practical range of the rail longitudinal 

force ±120 kips and the rail temperature range 40 °F to 100 °F. Instead of correlating the 

deflection with the rail force for a constant applied vertical load, one can correlate the 

required uplift force for a fixed vertical deflection with the rail longitudinal force.

B.6 Validations

Tests were conducted at TTC, Pueblo, CO, on a tangent and a 5-degree curved track to 

validate the technique. A special instrumentation car with inner wheel spacing of 340 in

was adapted to provide a maximum vertical force of 30 kips. The test sections were 

instrumented with strain gages to measure the rail force. The variation in the rail force 
B-11

was achieved through destressing at reasonable high neutral temperatures for tensile 

loads and by means of artificial rail heating for compressive force levels. Figure B-11 

shows the mean regression line for all the test data and the theoretical prediction, which 

are in good agreement. Test validations were also performed on a 5-degree curve.

-100 -75 -50 -25 0 25 50 75 100

14

15

16

17

18

19

Tension Compression

Tangent track

Experiment

Theory

V (kips)

P (kips)

Figure B-11. Rail Uplift Principle Validation

B.7 Field Application

The RUD has been applied in the field to map the neutral temperature on BNSF railroads 

[15]. This was performed using AAR’s TLV, which was modified to apply the uplift 

force and measure the uplift deflection (Figure B-12). Several thousand feet of the track 

was mapped, including a tangent and 5-degree to 6-degree curved section. Figure B-13 

shows an example of the measured rail neutral temperature variation.

Figure B-12. TLV Based Rail Uplift Measurements
B-12

Figure B-13. Examples of Neutral Temperature Measurements Using the Rail 

Uplift Device (Locations Spaced 300 to 500 ft Apart)
C-1

Appendix C.

Theoretical Basis of CWR-Risk

This appendix will present analytic details of evaluating the probability of buckling on 

the basis of the convolution integral used in Chapter 3. Figure C-1 shows the 

distributions of load and strength. These are represented by ψ and φ respectively. If the 

strength curve (φ) is totally separated to the right of the load curve, ψ, no interference 

will exist between the curves. This means that the strength is always greater than the 

load, hence be no buckling will occurred. In cases where there is some interference, as 

seen in Figure C-1, a range of temperatures (A to D) exists in which load can exceed 

strength, thereby giving rise to buckling situations. To evaluate the buckling probability 

at a temperature in the interference range, the convolution integral is used. To facilitate 

an understanding of this integral and its evaluation, it is convenient to represent the load 

by x and the strength by y.

Figure C-1. Frequency Distributions of Load and Strength, and 

the Interference Zone

x = load (T-TN), 

where 

T = rail temperature, 

TN = neutral temperature

ψ(x) = load frequency (see Figure C-2), represents the probability of x being equal to a 

specified value X by 

P(x = X) = ψ(X)
C-2

Figure C-2. Frequency Distribution Function of Load

The probability load function is defined as 

Ψ(x) = dx

x

0

x ∫ψ( ) (see Figure C-3)

and represents the probability of x being equal to or less than a specified value X by 

P(x ≤ X) = Ψ(X)

Figure C-3. Probability Function of Load

y = strength, ΔTall

φ(y) = strength frequency (see Figure C-4)

and represents the probability of y being equal to a specified value Y by

P(y = Y) = φ(Y)
C-3

Figure C-4. Frequency Distribution Function of Strength

The probability strength function is defined as 

Φ(y) = ∫ φ

y

0

(y) dy (see Figure C-5)

and represents the probability of y being equal to or less than a specified value Y by 

P(y ≤ Y) = Φ(Y)

Figure C-5. Probability Function of Strength

Referring to Figure C-1, consider point C representing the load x and point B 

representing the strength y, which is less than x. The probability of x ≥ y is, by the 

product law, φ(y) * ψ(x). Now let C vary from B to D. The total probability of C 

assuming greater or equal values of B, between B and D, equals
C-4

∫ψ φ

D

B

(x) (y) dx

The above integral y and φ(Y) are fixed; only x and hence ψ (x) are varied.

Hence in the interval BD, the P(C ≥ B) = ∫ψ φ

D

B

(x) (y) dx

To obtain the sum total probability of load exceeding strength, let B vary from A to D

P (C ≥ B) for interval AD = dy

D

A

D

B

(x) (y) dx ∫ ∫ 



















ψ φ

= dy

D

A

y

0

x y dx

D

0

x y dx ∫ ∫ ∫ 



















ψ( ) φ( ) − ψ( ) φ( )

= dy

D

A

y

0

y y x dx ∫ ∫ 



















φ( ) − φ( ) ψ( )

Since [ ] 1

1

0

D

0

ψ x dx = x = ∫ ( ) Ψ( )

Also ( ) [ ( )] (y)

y

0

y

0

∫ψ x dx = Ψ x = Ψ

( ) [ ] dy

D

A

y 1 y ∫ P Buckling = φ( ) − Ψ( )

The integral involves only one variable, y, and is confined to the interference zone.

Numerical Work

For numerical work, points A and D (A being the starting point of the strength curve and 

D being the end point of the load curve which define the range of temperature in the 

interference zone) are identified in the CWR-SAFE program. The interval A to D is 

divided into an even number of intervals (n). Define the intervals between A and D: 

y(1), y(2), ....... y(n+1).
C-5

The program identifies the non-zero values shown as √ in the following matrix. The last 

value ψ and the first value of φ are zero as they represent points A and D respectively.

ψ(1) = √ φ(1) = 0

ψ(2) = √ φ(2) = √

............................

ψ(n+1) = 0 φ(n+1) = √

The program computes

Ψ(1) = 0

Ψ(2) = [ ψ(1) + ψ(2) ] / 2

Ψ(3) = { Ψ(2) + [ ψ(2) + ψ(3) ] / 2 }

Ψ(n+1) = { Ψ(n) + [ ψ(n) + ψ(n+1) ] / 2 }

The program defines Z(i) = φ(i) [ 1 - ψ(i) ] for i = 1, 2, ...n+1

which is the integrand in P(Buckling) shown above.

The program computes by a trapezoidal rule ∫

n+1

1

Z(i)dy

i.e., ∫ ∑ +

+

= + +

n

2

Zn 1

n 1

1

Z1

2

Z(i) 2Z(i)dy
D-1

Appendix D.

Measurement Requirements and Sample Rates for Lateral Resistance 

and Neutral Temperature

Appendix B describes the methods and hardware to measure the two key parameters of 

lateral resistance and neutral temperature. The measurements are needed to provide input 

to CWR-SAFE for monitoring and controlling the parameters for safety assurance in 

revenue lines. The measurements are typically required for two scenarios. One is over a 

defined small segment of track for use in the deterministic approach for buckling safety 

evaluation. The second one is for the determination of distribution of the parameter over 

a territory for use in the probabilistic approach.

D.1 Lateral Resistance

D.1.1 Sampling Rate Basis

A number of lateral resistance measurements were made under the FRA research 

program on different track territories, including those of Amtrak, Union Pacific and 

Transportation Technology Center using the Single Tie Push Test (STPT) device. In 

many of these measurements, a 50-foot long test zone was considered as a characteristic 

length (approximately the truck center spacing for vehicles where buckles typically 

occur). A number of test samples were taken over each of the 50-foot sections, and a 

mean value of resistance was identified for each test segment. For a wood tie track with 

about 20” tie spacing, there are 15 testable ties since only alternate ties can be tested. For 

the concrete tie with 24” spacing, the test fixture tends to react on two adjacent ties on 

each side of the test tie, thereby limiting the maximum number of testable ties to about 9 

per a 50-foot long test zone. The mean value of 9 ties in the 50-foot test zone is 

considered as a representative number for the resistance in the zone.

Whereas for research purposes, one can test 9 ties in concrete and 15 ties in wood in a 50 

foot test zone; this is not considered to be practical for rapid safety evaluations by the 

industry. It is also necessary to minimize excessive track disturbance to the track. 

Therefore, the problem of minimum required number of tie samples in the zone should be 

addressed. The mean, Fn, of the selected number of samples, n, must be reasonably close 

to the average resistance, F0, obtained by testing the maximum possible number of ties in 

the zone. A tolerance for the difference e = F0 - Fn must be established first.

Tolerance

Table D-1 gives a sample of allowable temperature increase for lateral resistance values 

in the range of 1500 to 3000 lb/tie. This table also shows the allowable temperature 

values for a resistance drop of 15 percent from the nominal values in the range. Table D2 shows other parameters assumed in this analysis.
D-2

Table D-1. Sensitivity of Allowable Rail Temperature with Respect to Lateral 

Resistance and Peak Value

Tangent Track 5-Degree Curve

Nominal 

Lateral 

Resistance 

(lb)

∆T1 ∆T2 δT ∆T1 ∆T2 δT

1500 94.63 92.21 2.42 82.56 78.98 3.58

2000 99.49 96.64 2.85 88.74 85.3 3.44

2500 104.01 100.64 3.37 93.98 90.1 3.88

3000 108.42 104.45 3.97 98.86 94.48 4.38

∆T1 = Allowable Temperature for Nominal Value of Resistance (°F)

∆T2 = Allowable Temperature for 85% of Nominal Resistance (°F)

δT = Change in Allowable Temperature (°F)

Table D-2. Assumed Parameters

Data Inputs 

Rail Size: Area 136

Tie Type: Wood

Tie Weight (lb): 200

Tie Spacing (in): 20

Ballast Type: Granite

Tie-Ballast Friction Coefficient 1.2

Torsional Resistance (kips/in./rad): 100

Longitudinal Stiffness (lb/in./in.): 200

Foundation Modulus (psi): 6000

Peak Resistance (lb/in.): 100

Misalignment (in.): 0.625

Misalignment Half-Wavelength (in.): 180.6

Rail Neutral Temperature (°F): 80

Maximum Rail Temperature (°F): 140

Vehicle Type: Hopper

It is seen that the corresponding change in the allowable temperature does not exceed 5

°F when the change in lateral resistance is under 15 percent. It will be assumed that the 

buckling temperature error of ±5 °F is tolerable in practical applications. The change in 

allowable temperature tends to reach or exceed 5 °F only at high resistance values (>3000 

lb/tie), but such high resistance situations generally do not create buckling problems. 

Therefore, it will be concluded that a tolerance of ±15 percent in the lateral resistance 

measurement is permissible for the wood tie track. The same conclusion can also be 

reached for the concrete tie track.
D-3

D.1.1.1 Sampling Rate

The required sampling rate for a permissible tolerance of ±15 percent resistance will be 

determined using the statistical equations of confidence levels and intervals. As stated, a 

large number of resistance measurements were made, and it was found that these can be 

reasonably represented by normal distributions, as shown in [10]. A normal distribution 

is represented by the two parameters, namely the mean and the standard deviation.

• Mean value, F0

F F / N

N

i 1

0 ∑ i

=

= (D-1)

Fi represents the peak resistance value of the ith tie, and N is the number of ties used in 

the 50 ft zone, which is at least 9.

• Standard deviation, σ

(F F ) (N 1)

1/ 2 N

i 1

2

i 0 − 















σ = ∑

=

− (D-2)

The determination of the sampling rate is as follows. Suppose that instead of testing N = 

9 or greater number of ties in the 50 foot zone to determine the average resistance, F0, 

assigned to the zone, one opts to test a much smaller number of ties, n < < N. The mean 

of the n number of test tie data, Fn, will be different from F0. For a maximum permissible 

difference of ±e ( e = F0 − Fn ) or for a

Confidence Interval = (Fn – e ≤ F0 ≤ Fn + e), (D-3)

one can establish a relationship between the number of samples and the probability of 

finding the mean value within the confidence interval. The sampling rate, n, is given by 

the well-known formula [24] in statistics

n = (Cσ/e)2 (D-4)

Here C is a constant which depends on the required Confidence Level. If the Confidence 

Level is designated by Y, the relationship between Y and C is shown in Table D-3. This 

table is valid for normal distributions and has been well documented in [24].

It is usual practice in sample sizing to examine the three Confidence Levels presented in 

Table D-3. The higher the stipulated Confidence Level, the larger the value of C and the 

sampling rate increases with the square of C. A 90 percent Confidence Level is adequate 

in many of the engineering applications. This will be recommended for CWR track safety 

monitoring. In view of the built-in margin of safety in the proposed safety limits, a 

higher Confidence Level may not be justified as it increases the monitoring requirements 

for the lateral resistance.
D-4

Table D-3. Relationship between Confidence Level and

Parameter C (for Normal Distribution)

Y

(Probability or Confidence Level)

C

(Parameter)

90% 1.645

95% 1.960

99% 2.576

For the assumed tolerance e = 15 percent and 90 percent confidence level, the sampling 

rate n is calculated from equation D-4 for a number sites with historical data collected 

using the traditional method of testing 9 or more ties in 50 foot zones.

Table D-4 provides the required sampling rate for tracks tested at TTC, UP, and Amtrak. 

Wood and concrete ties and slag and granite ballast materials were considered. Tamped, 

stabilized under revenue traffic, and artificially stabilized conditions were also variables 

in the data.

Table D-4. Examples of Sampling Rate Calculations

Site 

Location

Track 

Type

Zone 

# Initial Condition MGT Measured Mean (lb)

1

Std 

Dev

Required 

Sampling 

Rate2

TTC

Ref [10]

Wood 1 Slag Ballast .1 1469 258 4

2 Granite Ballast, 

Trafficked 25 1993 397 5

3 Slag Ballast, 

Trafficked 25 2374 357 3

4 Slag Ballast 0 1038 148 2

5 Slag Trafficked 100 3176 560 4

6 Granite Trafficked 25 2206 411 4

UP

Ref [25]

Wood 1 Granite Tamped 0 3300 460 2

2 Granite Trafficked .4 3371 420 2

3 Granite Trafficked .6 3430 510 3

Amtrak

Ref [22]

Concrete 1-aGranite 

Consolidated - 3483 275 1

1-b Granite Tamped 0 2000 128 1

1-c Granite Stabilized 0 2611 231 1

2-a

Granite 

Consolidated - 3030 197 1

2-b Granite Tamped 0 1900 145 1

2-c Granite Stabilized 0 2355 217 1

3-a

Granite 

Consolidated - 3610 239 1

3-b Granite Tamped 0 1885 100 1

3-c Granite Stabilized 0 2575 198 1

4 Granite Tamped 0 1905 152 1
D-5

1 Measured Mean was for 9 or more ties in the case of 50 foot wood tie sections and 5 for 50 foot concrete 

tie sections. 2 Required Sampling Rate is for an error within 15 percent of measured mean with 90 percent confidence 

value.

Although some variability occurs in the results for the sampling rate, it appears that a 

sampling rate of one tie for a 50 foot concrete section and 2 ties for a 50 foot wood tie 

section can be recommended to determine the resistance with an error of 15 percent with 

a Confidence Level of 90 percent.

D.1.2 Evaluation of Lateral Resistance Statistical Distribution on a Territory

To determine the distribution of lateral resistance that is applicable over a large territory 

of a railroad division, it is not necessary to measure the resistance on every 50 foot track 

segment. Adequate data can be collected by first identifying weak locations for selective 

measurements and super imposing data from the remaining normal track 

conditions/locations.

The representative weak locations include the following:

A. New or recently maintained segments

B. Stabilized segments before full speed restoration

C. Degraded track (poor ballast and tie and settling subgrade conditions)

D. High curvature CWR track with excessive radial movement

E. Locations with previous buckling incidents

A and B type track segment resistances can be estimated using CWR-INDY’s lateral 

resistance estimator (see Equation 3.5 in Chapter 3). This is because, for new and

recently maintained tracks, the lateral resistance is fairly uniform, and CWR-INDY’s 

predictor is fairly good for such conditions. Conversely, where larger variations are 

expected, such as in C, D, and E type segments, the STPT method will be required. The 

sampling rates for those measurements shall be as per the recommendations made in 

Subsection D.1.1.

To generate data for normal or typical track segments, isolate representative sections of 

the following four categories: wood and concrete, each with curvatures from 0° to 4° and 

from 5° to 9°. A representative section is one that exhibits similar characteristics as the 

majority of the territory (i.e., one which has similar ballast, tie type, ballast section, and 

MGT). The length of the representative section can be as large as a quarter mile. Using 

the recommended sampling rates (i.e., one tie/50 foot of concrete tie and two ties/50 foot 

of wood tie, the lateral resistance measurement should be carried out).

Combining the measurements for the weak locations with those for the normal section, 

one can construct a database for the territory and deduce the statistical distribution 

(Figure D-1). The database can be updated at selected locations on an annual basis, 

particularly for the weak locations during early summer conditions. For normal 

segments, the measurement of resistance must be made only once.
D-6

Figure D-1. Statistical Distribution of Resistance

D.2 Neutral Temperature

D.2.1 Sampling Rate Basis

Several tests were carried out on the evaluation of neutral temperature on revenue lines 

and TTC tracks under the sponsorship of the FRA. The tests were performed for 

different purposes, including the following:

1. Mapping of neutral temperature on long stretches of track (5° and 6° curves and 

tangent on BNSF, [15])

2. Determination of daily and seasonal neutral temperature changes in revenue lines 

and at TTC

3. Validation of longitudinal restraint model for winter rail break

4. Restressing CWR at the required neutral temperature (on TTC track, [16])

The tests for item 1 revealed that over several thousand feet of track at BNSF test sites 

the neutral temperature varied in the range of 110 °F to 60 °F. The measurement spacing 

varied from 300 to 500 ft. Hence, it is inferred that a 60 °F minimum is achievable on 

revenue lines. This number is used in the safety implementation presented in Chapter 6.

The tests for item 2 revealed that daily temperature fluctuation caused by rail longitudinal 

movement through fasteners and anchors could result in a neutral temperature change of 

several degrees (< 10 °F). These tests also revealed the influence of curve breathing on 

neutral temperature.

The tests carried out at TTC with the objective stated in item 3 showed that the 

longitudinal resistance in a typical wood tie track varies in the range of 15 to 30 lb/in, 

depending on the anchoring pattern and other factors.

To estimate the potential variations in the neutral temperature along a rail, the following 

analysis has been used. Consider the equilibrium of a single rail of length l (Figure D2).
D-7

Figure D-2. Evaluation of Change in Rail Longitudinal Force

From equilibrium considerations

f0l = P1 – P2 (D-5)

where f0 is the maximum longitudinal resistance (per rail per unit length). By definition 

of rail neutral temperature at locations 1 and 2.

P1 = AEα (TR-TN1) (D-6)

P2 = AEα (TR-TN2) (D-7)

where 

A = the cross sectional area of a single rail, 

E = Young’s Modulus, 

α = the coefficient of thermal expansion, and 

TR = rail temperature.

TN1, 2 = rail neutral temperature at locations 1 and 2.

From equations D-5, D-6, and D-7, it is seen that

(T N1-TN2) = ∆TN = f0l/AEα. (D-8)

Here ∆TN = change in neutral temperature between locations 1 and 2.

This equation represents the possible variation of neutral temperature between two 

locations on one rail, which are a distance, l, apart. If an error of 5 °F is permitted in the 

neutral temperature, then the maximum permissible distance between two measurement 

points is

lmax = (5AEα/f0)

The error of 5 °F in neutral temperature is the same value used for the error in the 

allowable temperature, Tall , due to the 15 percent permissible error in the lateral 

resistance. The longitudinal resistance, f0, is typically 25 lb/in. For typical rail cross-
D-8

sections, it can be shown that lmax ≈ 50 ft. In practice, this will be larger than 50 ft since 

the longitudinal resistance is determined by a point below A on the initial linear line, 

particularly if the rail is uncut (Figure D-2).

If the neutral temperature measurement point is chosen at P1, then the permissible lmax on 

the left side of location 1 in Figure D-2 will be the same as 50 ft. It can therefore be 

recommended that for every 100 ft rail section, one measurement point be used for the 

stated maximum permissible error of 5 °F. This is further confirmed from the data 

analysis carried out in the following paragraph.

Table D-5 shows an example analysis of data collected on TTC track during the winter 

rail break tests. The sample number of strain gage locations used in the tests to measure 

the neutral temperature was 8 for every 100 foot. The mean of the 8 strain gage values is 

considered as the value against which the required sample rate for a maximum tolerable 

error of 5 °F is evaluated. It can be recommended that one sample over a 100 foot rail 

section is adequate to define the neutral temperature with 90 percent confidence limit for 

the maximum tolerable error of 5 °F.

Table D-5. Required Sampling Rate for 100 ft Long Tangent Track (90% Confidence 

Limit, 5°F Error on the Mean Value).

Test Anchoring

TN

1

Measured 

Mean (°F)

Std Dev 

(°F)

Required 

Sampling 

Rate

WRB1 EOTA 81.2 2.5 1

WRB2 EOTA 78.8 2.1 1

WRB3 EOTA 102.2 2.6 1

WRB4 EOTA 118.7 2.4 1

WRB5a ETA 98.7 0.8 1

1 Number of data points = 8

ETA = Every Tie Anchored

EOTA = Every Other Tie Anchored 

WRB = Winter Rail Break Test

D.2.2 Evaluation of Neutral Temperature Statistical Distribution on a Territory

To determine the distribution of the neutral temperature on a territory, the following 

potential weak spots should be identified:

A. High degree curves exhibiting large radial (spring time movement)

B. Rail repaired/neutral temperature readjusted locations

C. Recently maintained locations (lined or lifted)

D. Bottom of steep grades, and locations with high traction and braking

E. Locations experiencing rail movement through the fasteners

F. Locations with a buckling history

G. Locations with rail fracture/pull apart history
D-9

For the weak locations (A to G), a one time monitoring with strain gages or a rail uplift 

device (RDU) will be required at the recommended minimum sampling rate. The 

monitoring is aimed to determine the lowest TN value and to assure 60 °F minimum 

neutral temperature. Neutral temperature data should also be gathered on normal or 

typical track segments representative of a territory. This can be done on a quarter mile 

tangent wood and concrete segment and a quarter mile of 5° wood and concrete tie 

segment, which are good representations of the territory in terms of CWR parameters, 

such as age of rail, fastener type and condition. Perform measurements on these 

segments on both rails either via strain gauging or via RUD in accordance to the 

sampling methodology of Section 2.1. Thus, the statistical distribution on CWR neutral 

temperature can be assembled by combining the A-G data and the normal data for 

probabilistic evaluations.
E-1

Appendix E.

Rail Destress Force Indicator (RDFI)

Rail destressing and restressing are routine operations for the railroads, required 

whenever rail is intentionally cut in summer or fractured due to fatigue cracks and 

excessive longitudinal tensile forces developed in winter. Intentional cuts in summer will 

relieve excessive compressive forces and avoid potential buckles under high longitudinal 

compressive loads. Rail restressing is required to install or reweld the rails at a desired 

neutral temperature.

To install and reweld the rail at the desired neutral temperature, railroads depend on rail 

heating or hydraulic tensors. Rail heating can be artificial using propane gas, although 

this practice is being discontinued because of safety reasons. Solar heating is resorted to 

in many cases.

One specific aspect of destressing is de-anchoring or unfastening the rail from the ties 

after cutting the rail. This will ensure that the rail compressive force is eliminated, not 

only at the cut location, but also through a substantial portion of the rail on either side. 

Some of the railroad practices recommend a de-anchoring length from 5 rail lengths or 

195 ft to longer lengths on either side of the cut location, but the guidelines are vague and 

not rationally formulated. After de-anchoring over whatever length the track crew 

decides on, a gap adjustment is performed at the prevailing rail temperature that will 

yield the desired neutral temperature under tensor application. The gap adjustment 

includes a 1” welding allowance.

Figure E-1 shows the definition of rail gap and the sequence of events in summer rail 

cutting and rewelding at the desired neutral temperature. Figure E-2 shows a similar 

situation when winter rail break occurs, and tensors are applied to restress the rail to the 

desired neutral temperature.
E-2

Figure E-1. Definition and Measurement of Cutout Length and Rail 

Gap in a Summer Rail Cut Scenario

In the summer scenario, an optimum amount of steel must be removed and the rail deanchored over a required length so that a gap of about 1′′ is achieved at the desired rail 

welding (neutral) temperature. Likewise, in winter after rail fracture or cuts, the correct 

amount of steel must be added, and/or an optimum tensor force must be applied to reduce 

the gap to about 1” for rewelding after the anchor removal over an optimum length.
E-3

Figure E-2. Definition and Measurement of Rail Gap (Tensor Application)

A need to develop an instrument to facilitate the destressing and restressing operations in 

an optimal manner exists. Large errors can occur in the resulting neutral temperature if 

these operations are not performed properly. (For example, see Section 6.3.2). A 

breadboard device has been previously developed and tested, which consists of a strain 

gage and a temperature probe and uses a calculation program to output the optimum deanchoring or unfastening length and the gap for a prescribed target neutral temperature. 

Initial rail movement due to cutting is a manual measurement which is also required as 

input into the bread-board device.

This measurement is made using two scribe marks on the rail head on either side of the 

cut and can be automated for easy use by the crew in future devices. The computer 

program back-calculates the longitudinal resistance at the site using the rail force from 

the strain gage and the resulting rail displacement from the rail cutting. On the basis of 

the analytic model presented in [16], the device outputs the required de-anchored length 

and the final adjusted gap length for a desired neutral temperature which is also an input 

entry to the device. The strain gage can be used to monitor the correctness of the newly 

set neutral temperature, as well as for subsequent monitoring to track how neutral 

temperature changes. Current research is addressing the development of such a prototype 

device in a field-functional version known as Rail Destress Force Indicator.
F-1

List of Abbreviations and Acronyms

AAR Association of American Railroads

CWR continuous welded rail

DTE Equivalent MGT in Dynamic Track Stabilization

DTS Dynamic Track Stabilization

EOTA every other tie anchored

ERRI European Rail Research Institute

ETA every tie anchored

FRA Federal Railroad Administration

MGT million gross tons (traffic)

NAL net axle lateral load

RDFI Rail Destress Force Indicator

RUD Rail Uplift Device

SF safety factor

STPT Single Tie Push Test

TCS truck center spacing

TTC Transportation Technology Center

UIC Union Internationale des Chemins de Fer (Union of International 

Railways)
G-1

List of Symbols

Ω Energy required to buckle the track

ψ Load frequency

β Constant values for adjust the lateral resistance

α Rail steel coefficient of thermal expansion

θ Rotation angle

σ Standard deviation

φ Strength frequency

μf Tie-ballast friction coefficient

δo Initial misalignment amplitude

τo Torsional stiffness per rail fasteners

τ Applied torque per fastener

∆TBmax Upper buckling temperature increase

∆TBmin Lower buckling temperature increase

∆Tr Rail temperature increase

2L Buckling wavelength

2Lo Misalignment wavelength

A Rail cross sectional area

Cd the ratio of existing crib depth to the full depth

C Confidence level constant

E Rail steel modulus

e Tolerance (permissible error)

eb Expected number of buckles per year

eR Strain gage reading at the rail temperature

f Longitudinal resistance

Fb Resistance from tie bottom

Fe Resistance from shoulder

FL Limiting lateral resistance

Fn Mean value of n samples

FP Peak lateral resistance

Fs Resistance from tie sides

FP0 Tamped resistance

∆FP Increment due to consolidation

Iyy Rail section moment of inertia about the lateral axis through centroid

Izz Rail section moment of inertia about the vertical axis through centroid

kf Longitudinal stiffness

kip Unit of force equaling 1,000 pounds-force

Kv Track vertical stiffness

L/V Lateral to vertical force ratio

Ld Deanchoring length on one side of rail cut when destressing

lmax Maximum permissible distance between two measurement points

lc Typical car length (expressed in miles)

N Number of ties used in the 50 foot zone

NB Maximum number of permissible buckles per annum per section of track
G-2

n Sampling rate

nc Number of cars in a typical train

nt Number of trains per day

np Number of potential buckling events possible

P Rail force in the buckled zone

P Longitudinal rail force

Pa Annual probability of buckling

Pb Probability of buckling

PT Relative frequency distribution of rail temperature

Q Track weight per unit length

R Radius of curved track

Rv Vehicle contributes an additional load

S Tie spacing

Sw Shoulder width

Tall Allowable rail temperature increase

Tc Critical temperature for slow orders

TBmax Upper critical temperature

TBmin Lower critical temperature

TL Limiting temperature for train operations

TMAX or TM Maximum rail temperature

TN or N Rail neutral temperature

TM Maximum rail temperature

TP Temperature at which increased lateral displacements begin to occur with 

small increases in rail temperature

TR Annual rail temperature frequency

U Rail longitudinal displacement in adjoining section

u Rail longitudinal displacement in buckled zone

V Axle vertical load

Vmax maximum permissible speed

Vr Reduced speed

v Rail vertical displacement

w Lateral deflection

wB Pre-buckling deflection

wC Post buckling displacement

wo Initial misalignment function

wP Peak lateral displacement

Wtie Weight of one tie including fasteners and plates

Wrail Rail weight

Y Confidence level