US Department of Transportation Federal Railroad Administration Estimation of Rail Wear Limits Based on Rail Strength Investigations Office of Research and Development Washington, DC 20590 D.Y. Jeong Y.H. Tang O. Orringer U.S. Department of Transportation Research and Special Programs Administration Volpe National Transportation Systems Center Cambridge, Massachusetts 02142 DOT/FRA/ORD-98/07 DOT-VNTSC-FRA-98-13 December 1998 Final Report This document is available to the public through the National Technical Information Service, Springfield, Virginia 22161 This document is also available on the FRA web site at www.fra.dot.gov 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 content or use thereof. 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 object of this report. REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information.Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE December 1998 3. REPORT TYPE & DATES COVERED Final Report - January 1997 4. TITLE AND SUBTITLE Estimation of Rail Wear Limits Based on Rail Strength Investigations 6. AUTHOR(S) David Y. Jeong,1 Yim H. Tang,1 and O. Orringer 1,2 5. FUNDING NUMBERS R9002/RR928 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 1 U.S. Department of Transportation Research and Special Programs Administration Volpe National Transportation Systems Center Cambridge, MA 02142-1093 2 Tufts University Mechanical Engineering Department Medford, MA 02155 8. PERFORMING ORGANIZATION REPORT DOT-VNTSC-FRA-98-13 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Department of Transportation Federal Railroad Administration Office of Research and Development Washington, DC 20590 10. SPONSORING OR MONITORING AGENCY REPORT NUMBER DOT/FRA/ORD-98/07 11. SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY 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. 12b. DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) This report describes analyses performed to estimate limits on rail wear based on strength investigations.Two different failure modes are considered in this report:(1) permanent plastic bending, and (2) rail fracture.Rail bending stresses are calculated using the classical theory of beams on elastic foundation.The effect of wear is modeled as a geometric change of the rail section due to loss of material from wear.Two different wear patterns are examined:(1) vertical rail-head height loss, and (2) gage-face wear from the side of the rail (referred to as gage-face side wear).An elementary plastic-collapse criterion is used to estimate wear limits based on failure by means of rail bending.An approximate method that was previously developed to analyze the growth of internal transverse defects is also applied to estimate wear limits on the basis of fracture strength.These analyses reveal that rail-wear limits estimated with the fracture-mechanics approach are more restrictive (i.e., conservative) than those based on the plastic-bending approach. Therefore, for safe operations on railroad tracks, allowable rail-wear limits should be estimated on the basis of fracture strength. 15. NUMBER OF PAGES 44 14. SUBJECT TERMS detail fracture, dynamic load factor, fracture mechanics, fracture strength, gage-face wear, head-height loss, plastic collapse, yield strength, wear 16. PRICE CODE 17.SECURITY CLASSIFICATION OF REPORT Unclassified 18. SECURITY CLASSIFICATION OF THIS PAGE Unclassified 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassified 20. LIMITATION OF ABSTRACT NSN 7540-01-280-5500Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. 239-18 298-102 iii PREFACE InsupportoftheFederalRailroadAdministrations(FRA)TrackSafetyResearch Program,theJohnA.VolpeNationalTransportationSystemsCenter(VolpeCenter) has been conducting and managing research to develop technical information that can beusedtosupportrationalcriteriaforthepreservationofsafeoperationsonrailroad tracks. TheFederalRailroadAdministration(FRA)convenesbiannualmeetingsofthe TechnicalResolutionCommittee(TRC)toaddressissuesinvolvingtheapplicationof FRAsafetyregulations.ThemissionoftheTRCistoachieveconsensusonthe interpretationofexistingregulations.TheTRCcharterallowsforsomediscussionon whethersafetystandardsshouldberevised,butsuchtopicsmustbereferredtothe Rail Safety Advisory Committee (RSAC) for action.The RSAC was formed to facilitate negotiated rule-making for the purpose of revising the FRA safety standards. OneareaofinteresttotheRSACistheeffectofrail-headwearonrailstrengthand structuralintegrity.Thepurposeofthisreportistoprovidetechnicalinformation regarding rail-wear limits developed on the basis of engineering analyses. Theanalysesdescribedinthisreportconsiderrailstrengthaseitherresistanceto permanentplasticbendingorresistancetofracture.Theresultsfromtheseanalyses show that estimates for rail-wear limits based on the fracture mechanics approach are morerestrictive(i.e., moreconservative)thanthosebasedonpermanentplastic bending.Therefore,forsafeoperationsonrailroadtracks,allowablerail-wearlimits should be estimated on the basis of fracture strength. iv METRIC/ENGLISH CONVERSION FACTORSENGLISH TO METRICMETRIC 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)=0.09square 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 CONVERSION1 0 2 3 4 5InchesCentimeters0 1 3 4 5 2 6 11 10 9 8 7 13 12 QUICK FAHRENHEIT - CELSIUS TEMPERATURE CONVERSION -40 -22 -4 14 32 50 68 86 104 122 140 158 176 194 212

FC -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 10286Updated 6/17/98 v TABLE OF CONTENTS SectionPage 1.INTRODUCTION.................................................................................................. 1 2.STRENGTH ANALYSIS FOR WORN RAIL ......................................................... 3 2.1Vertical Bending Component ..................................................................... 4 2.2Lateral Bending Component ...................................................................... 5 2.3Strength-Based Wear-Limit Estimation...................................................... 5 3.FRACTURE MECHANICS ANALYSIS FOR WORN RAIL................................... 9 3.1Stress Intensity Factor for Detail Fractures................................................ 9 3.2Stress Analysis for Rails with Detail Fractures......................................... 11 3.3Fracture-Mechanics-Based Wear-Limit Estimation ................................. 13 4.WORST-CASE LOAD ESTIMATION................................................................. 15 5.RESULTS........................................................................................................... 23 6.DISCUSSION AND CONCLUSIONS ................................................................. 27 REFERENCES.............................................................................................................. 29 Appendix A. Section Properties for New or Unworn Rail............................................. 31 Appendix B. Approximation of Worn Rail Section Properties...................................... 33 Appendix C. Stress-Gradient Magnification Factor for Fracture Analysis ................... 37 vi LIST OF FIGURES FigurePage 1.Actual and Idealized Rail Cross-Sections............................................................. 3 2.Idealized Rail-Head Wear Patterns ...................................................................... 3 3.Stress-Strain Curve for Elastic-Perfectly-Plastic Material Behavior...................... 6 4.Elastic and Fully-Plastic Stress Distributions in an Idealized Rail Section............ 7 5.Schematic for Estimation of Head-Height Wear Limit .......................................... 8 6.Schematic of a Detail Fracture in the Rail Head................................................... 9 7.Bending Moment Distribution for a Single Wheel Load...................................... 12 8.Schematic for Estimating Rail Wear as a Function of Critical DF Size............... 13 9.Schematic for Estimating Rail Wear as a Function of DF Size after 20 MGT from Initial Size of 5% HA..................................................................... 14 10.Estimate of Rail-Wear Limit Based on Fracture Strength................................... 14 11. Schematic of a Two-Sided Gaussian Distribution Function................................ 15 12. Speed Dependence of Coefficient of Variation and Dynamic Load Increment... 17 13.Histogram of Extreme Wheel Loads Measured on the Northeast Corridor ........ 18 14.Regression for Occurrence Rate versus Peak Load .......................................... 19 15.Wear-Limit Estimates for Head-Height Loss Based on Rail Strength ................ 24 16.Wear-Limit Estimates for Side Wear Based on Rail Strength ............................ 24 17.Wear-Limit Estimates for Head-Height Loss Based on Fracture Strength ......... 25 18.Wear-Limit Estimates for Side Wear Based on Fracture Strength..................... 25 A-1.Dimensions for a Generic Rail Section............................................................... 31 B-1.Location of Centroids in Rail with Gage-Face Side Wear .................................. 35 vii LIST OF TABLES TablePage 1.Mo/Me Ratios for Vertical Head-Height Loss ......................................................... 7 2.Mo/Me Ratios for Gage-Face Side Wear ............................................................... 7 3.Coefficients of Variation for Dynamic Vertical Load ........................................... 16 4.Summary of Freight-Car Wheel-Load Histogram............................................... 18 5.Representative Worst-Case Load Estimates with Corresponding Foundation Stiffness and Rail Sections.............................................................. 21 A-1.Section Properties for New or Unworn Rail ........................................................ 32 B-1.Equivalent Rail-Head Height and Width for Various Rail Sections..................... 33 C-1.Stress-Gradient Magnification Factors for 132 RE Rail Section and Vertical Head-Height Loss.................................................................................. 39 C-2.Stress-Gradient Magnification Factors for 132 RE Rail Section and Gage-Face Side Wear........................................................................................ 40 viiiEXECUTIVE SUMMARY InitsApril1996meeting,theFRATechnicalResolutionCommittee(TRC)discussed thequestionofhowtrackinspectorsshoulddealwithheavilywornrail.Althoughthe TrackSafetyStandardsdonotaddressrailwear,theTRCrequestedthattechnical informationbedevelopedonquantitativeestimationofwearlimitsbasedonrail strength.The Volpe Center was asked to perform this task, and to do so has used an approximate method that was previously developed in support of the FRA Track Safety ResearchProgramtodeterminerailsectionpropertiesneededforrailstressanalysis.Thecorrespondingwearlimitsarebasedonstrengthcriteriathatconsiderfailureas eitherpermanentplasticbendingorrailfracture.Rail-wearlimitsbasedonfracture strengthassumethepresenceofaninternaltransversedefectknownasthedetail fracture.Thestudyresultsshowthatthe defect-based limits are more restrictive than those based on plastic bending.In other words, the fracture mechanics approach gives moreconservativewear-limitestimatesthantheanalysisbasedonpermanentplastic bending.Therefore,forsafeoperationsonrailroadtracks,allowablerail-wearlimits shouldbeestimatedonthebasisoffracturestrength.Forallbutthelightestrail sectionsconsidered,thelimitsforallowablewearwereestimatedas0.5inchhead-heightlossor0.6inchgage-faceloss,undertheassumptionthattherailisinspected for internal defects every 20 million gross tons (MGT). 11.INTRODUCTION Rail-wear limits have traditionally been based on strength to ensure that the rail can adequately support revenue service traffic without failure.In this report, wear limits areestimatedfromstrengthanalysesthatconsidertwofailuremodes: (1) permanent plastic bending, and (2) rail fracture from the presence of an internal transversedefectknownasthedetailfracture.1Themechanicalpropertiesofrail steel required in these analyses are yield strength and fracture toughness. Intheanalysesdescribedinthisreport,therailisassumedtobehaveasa beam supportedbyalinearelasticfoundation [1].Thelongitudinalrail-bendingstressis calculatedfromtheapplicationofasingleverticalwheelload.Inthestrength analysis,thecritical-loadestimateisbasedonaplastic-collapsemodelinwhich attainmentofyieldingthroughthefulldepthoftherailisassumed.Inthefracture strengthanalysis,thestressintensityfactorfora detailfractureisequatedtothe nominal value of fracture toughness for rail steel.Worst-case vertical wheel loads are assumed in the determination of wear limits based on plastic-bending strength. Here,theworst-caseloadisdefinedasthewheelloadexpectedto occur no more thanonceper105to106wheelpassages.Thiscriticalorworst-caseloadis estimated from a correlation between actual freight-traffic load data and the AREA-recommendedformulafordynamicloadmagnificationfactor.Lowerloads correspondingroughlytothemaximumexpectedoncepertrainpassage(oronce per400wheelpassages),areusedtoestimatetheeffectsofrailwearoncritical detail fracture (DF) size for rail failure by fracture.Rail-section properties needed in thebeam-theorystressanalysisarecalculatedusingamethodologythatassumes weartooccurfromlossofmaterial.Twowearpatternsareconsidered:uniform loss of material from either the top of rail (referred to as vertical head-height loss) or the gage side of the rail (called gage-face side wear). Vertical head-height loss occurs in virtually all track, and is the predominant mode of wear in tangent and shallow curves.Wear on the gage face of the rail is caused by contactloadsappliedtothesideoftherailheadbywheelflanges,andismost pronounced on curves greater than 3 to 4 degrees. Inthisreport,rail-wearlimitsareestimatedforFRAtrackclasses 2through 5.Seven particular rail sections are considered:70 ASCE, 85 ASCE, 100 RE, 115 RE, 132 RE,136 RE,and140RE.Sections2and3describetheyieldstrengthand fracture strength analyses, respectively, for worn rail.The estimation of worst-case wheelloadsisdescribedinSection 4.Theresultsfromtheseanalysesare presentedinSection 5.A discussionoftheresultsandconclusionsfromtherail strength investigations are given in Section 6.

1Detail fractures are the most common transverse rail defect found in continuous welded rail (CWR). 32.STRENGTH ANALYSIS FOR WORN RAIL Rail-wearlimitscanbeestimatedonthebasisofyieldorultimatestrengthwhen longitudinalrailbendingstressesarecalculated.Inthecaseofwearintermsof head-heightloss,onlytheverticalcomponentofthelongitudinalbendingstressis considered.Since gage-face wear occurs from lateral wheel-rail contact loads, the lateral load component of the longitudinal bending stress is considered in this wear case in addition to the vertical component. AmethodologywasdevelopedinReference[2]toestimatesectionpropertiesfor wornrail.Inthismethodology,theactualrailcross-sectionisapproximatedbyan idealizedsectionconsistingofthreerectangularareas representing the head, web, andbaseoftherail(Figure 1).Rail-headwearisassumedtooccurbya uniform lossofmaterialfromeitherthetopofthehead(referredtoasverticalhead-height loss) or the gage-side face (referred to as gage-face side wear).These two different wear patterns are illustrated in Figure 2 for an idealized rail section.Therefore, the physicallossofmaterialfromweartranslatestoageometricchangeintherail cross-section in the engineering analyses.In other words, rail strength is influenced bywearthroughchangesinrailsectionpropertiesduetowear.Varioussection propertiesforneworunwornrailarelistedinAppendix A.Thespecificequations used to determine section properties for worn rail are given in Appendix B. Figure 1.Actual and Idealized Rail Cross-Sections (a) Vertical head-height loss(b) Gage-face side wear Figure 2.Idealized Rail-Head Wear Patterns 42.1Vertical Bending Component For the present, it is convenient to assume that the entire rail cross-section remains elastic.Underthisassumption,thelongitudinalbendingstressduetoverticalrail bending is calculated from VVyyxM x cI( )( )(1) whereMV(x)istheverticalbendingmomentasafunctionoflongitudinalposition alongtherail,cisthedistancefromthecentroidoftheentirerailtothepointof interest,andIyy is the second area moment of inertia with respect to the horizontal axis through the rail centroid. Inthepresentstudy,bendingstressesarecalculatedatthetopoftherail; specifically, the upper gage corner of the rail head.Thus, c h hN (2) where h is the total rail height, and hN is the distance from the bottom of the rail to the rail centroid.In addition, wear in terms of head-height loss is assumed to affect the rail strength through the geometric parameters in equation (1), namely, c and Iyy. Mathematically,theserail-sectionparametersaretreatedasfunctionsofhead-height loss or gage-face side wear. Thebendingmomentinequation (1)iscalculatedfrombeam-on-elastic-foundation theory [1], or M xVe x xVVxV V( ) (cos sin ) 4 (3) where V is the vertical wheel load.It is evident from this equation that the maximum bending moment occurs directly beneath the wheel.Also, in this equation, VvyykEI44 (4) where E is the modulus of elasticity for rail steel (a value of 3 107 psi is assumed), andkvistheverticalfoundationstiffness(whichisassumedtovarydependingon FRA track classification). 52.2Lateral Bending Component Similar equations to those given for vertical bending can be written to determine the lateralcomponentofthelongitudinalrail-bendingstress.Themagnitudeofthe lateral bending component at the upper gage corner of the rail is given by LL HzzxM x wI( )( )2(5) whereIzzisthesecondareamomentofinertiawithrespecttotheverticalaxis through the rail centroid, and wH is the width of the rail head.In addition, the lateral bending moment as a function of longitudinal position along the rail is M xLe x xLLxL LL( ) (cos sin ) 4 (6) where L is lateral wheel load and LvzzkEI08544..(7) Inthisequation,thelateralfoundationstiffnessisassumedtobe0.85timesthe vertical foundation stiffness [4]. 2.3Strength-Based Wear-Limit Estimation TherailbendingequationspresentedinSections 2.1and 2.2canbecombinedto deriveexpressionsformaximumelasticload,Ve,underwhichthegreateststress magnitudereachesthematerialyieldstrength,YLD.Forverticalbendingonly (Section 2.1), combining equations (1) to (3) gives ( )Vh h IeYLDN V yy / 4(8) Iflateralbending(Section 2.2)isincluded,acombined-stressexpressionfrom equations (1) to (3), (5), and (6) gives Vh hIwILVeYLDNV yyHL zz+

_,

1]11 4 8(9) whereL/Visaspecifiedratiooflateral-to-verticalwheelloadratio.Comparisonof equations (8)and (9)showsthatVeforthecombined-loadingcaseisalways somewhatlessthanVe forpurelyverticalbending.Inthefollowingdevelopment, 6purely vertical bending will be assumed for the analysis of vertical head-height loss, but combined bending will be assumed for the analysis of gage-face side wear. Maximum elastic load is a convenient quantity to calculate from beam theory, but is too low to be a realistic measure of rail strength.A better quantity for this purpose is theplastic-collapseload,Vo,underwhicharailwouldbeexpectedtoforma large permanentkink.PlasticitytheorycanbeappliedtomakeestimatesofVounder certain simplifying assumptions. Forexample,wellestablishedresultspresentedbyHodge [3]givesimpleformulas forthemaximumelasticandplasticcollapseloads;intermsofverticalbending moments, Me and Mo; for beams of rectangular cross-section, and thin, wide-flanged I-sectionsassumedtoconsistofelastic-perfectly-plasticmaterial(Figure 3).The ratioofMo/Meisfoundtobe1.05to1.1forI-sectionsofheightandwidth comparabletorailsections.Withsucharatioavailable,inprinciple,onemay calculateVeandestimateVo=(Mo/Me)Vebecausetheproportionalitybetween applied load and applied bending moment remains linear even as the beam yields. YLDStrainStressE1 Figure 3.Stress-Strain Curve for Elastic-Perfectly-Plastic Material Behavior TheequivalentsectionmodelsshowninFigure 2,andsummarizedinAppendix B, havebeenusedtofollowasimilarprocedureinthepresentcase.Figure 4is a schematic of the stress distributions for the maximum elastic moment, Me, and the plastic collapse moment, Mo.2Strictly speaking, the estimates indicated in Figure 4 arevalidonlyforsymmetricalwear(head-heightloss)andpurelyverticalbending.However, the ratio of Mo/Me based on vertical bending has also been applied to the side-wear case, neglecting the small effects of lateral bending and the cross-product terms associated with asymmetrical side wear.

2InFigure4,thelocationoftheneutralaxisfortheelasticstressdistributioncoincideswiththe location of the rail centroid.For the fully-plastic case, however, the neutral axis shifts to a location that divides the rail cross-sectional area into two equal halves. 7YLDYLDYLD Figure 4.Elastic and Fully-Plastic Stress Distributions in an Idealized Rail Section Ratios of Mo/Me were found to range roughly from 1.4 to 1.6 for the case of vertical head-heightloss,andfrom1.4to2.0forthecaseofgage-sidewear.Tables 1 and 2 summarize these estimates for the two different wear patterns and the various rail sections considered in this report. Table 1. Mo/Me Ratios for Vertical Head-Height Loss Wear (%HA) 70 ASCE85 ASCE100 RE115 RE132 RE136 RE140 RE 01.371.361.381.401.391.391.38 201.401.401.411.431.431.421.41 401.421.441.441.481.471.471.45 601.431.471.481.541.521.521.51 801.461.541.591.651.611.611.60 Table 2. Mo/Me Ratios for Gage-Face Side Wear Wear (%HA) 70 ASCE85 ASCE100 RE115 RE132 RE136 RE140 RE 01.371.361.381.401.391.391.38 201.461.461.461.491.481.481.47 401.541.561.561.601.581.591.58 601.631.681.671.741.721.731.72 801.791.881.911.991.941.951.97 Strength-based limits on rail wear are estimated as follows.Consider first the case of head-height loss, h, for which purely vertical bending is assumed in the analysis.WiththerailsectionpropertiesmodifiedasdescribedinAppendix B,calculations aremadeforthecorrespondingmaximumelasticloadVeasafunctionofhfrom 8equation (8), and the moment ratio Mo/Me as outlined above.The rail collapse load is then estimated as VMMVooee (10) AsindicatedinFigure 5,Vodecreasesasthehead-heightloss,h,increases,and thewearlimitisdeterminedbytheintersectionoftheVoversushcurvewiththe worst-case load line, Vmax (See Section 4 for Vmax). Vertical load, VHead-height loss, hhlimitWorst-case load, VmaxPlastic collapse load, Vo Figure 5. Schematic for Estimation of Head-Height Wear Limit A similar procedure is followed to estimate the limit for gage-face side wear.In this case,equation (9)isusedtocalculateVewithalateral-to-verticalloadratio L/V = 0.2,assumedtorepresentaveragetrackcurvaturesoflessthan3 degrees.Theside-wearlimitisthendeterminedfromaplotsimilartotheschematicin Figure 5. 93.FRACTURE MECHANICS ANALYSIS FOR WORN RAIL Itisalsoworthwhiletoestimatehowrailwear affects critical size for rail failure, on the basis of fracture strength.In this case, the rail is assumed to contain an internal transversedefect.Linearelasticfracturemechanicsprinciplesareappliedto determine the amount of wear that would result in rail fracture when a rail containing a detail fracture (DF) of a given (assumed) size experiences a once-per-train wheel load.Wearisassumedtoapproachalimitwhendetailfracturescangrowfrom a barely detectable size to a critical size in less than one-half an inspection period.Thegrowthbehaviorofdetailfracturesinrailshasbeenstudiedinprevious research [4]. 3.1Stress Intensity Factor for Detail Fractures Forsizessmallerthan50%oftherail-headarea(%HA),thedetailfractureis assumed to be an embedded elliptical flaw located in the vicinity of the upper gage corneroftherailhead(Figure 6).Thedimensionsofthedefectarecharacterized bythesemi-majorandsemi-minoraxesoftheellipse,aandb,respectively.Examinations of rails containing detail fractures have revealed that the aspect ratio, b/a, is typically equal to 0.7.The center of the detail fracture is characterized by its location relative to the unworn running surface, z*, and the vertical mid-plane, y. 2a2bz*y Figure 6. Schematic of a Detail Fracture in the Rail Head ArelationshipbetweenthelocationoftheDFcenterandDFsizein136RErail sections was derived empirically in the previous study [4] yAAAAH H

_, +

_, 11874 2 9523 343062. . . (11) 10zAAAAH H*. . . +

_,

_, 0 6213 17580 179332(12) where A is the size of the detail fracture in terms of area (A = 0.7a2), and AH is the cross-sectionalareaofanunwornornewrailhead.Theparabolictrend characterizedbytheseequationsfor136RErailwasassumedtobeconstantfor other rail sizes.Also, the depth of the DF center below the rail running surface and inward from the gage face was assumed to be independent of rail size; i.e., varying only with the defect area ratio, A/AH. ThestressintensityfactororKformulafortheembeddedellipticalflawshown schematically in Figure 6 has the following mathematical form K MbaM a aI s

_,

21 ( ) (13) whereisthelongitudinaltensilestressatthedefectcenterduetorailbending and other effects, a is the semi-major axis of the elliptical flaw, Ms is a magnification factortoaccountfortheellipticalflawshape,andM1isa magnificationfactorto accountforfiniteboundaries.Themagnificationfactorsfordetailfractureswere derived in Reference [4].For instance, for an elliptical flaw aspect ratio of b/a = 0.7, Ms = 0.984.Thefinite-sectionmagnificationfactorwasmodifiedfromtheoriginal formulation to account for loss of rail-head area because of wear, and is given by M aXaAXaAXaA XaAXHHH H1 222 23211000 72 11000 70 632 0211000 70 37 12 11000 72 11000( ).tan... .. sin.cos.

_,

_,

_,

1]1111+

_,

_, +

_,

_,

1]1111

1]1111

_,

72aAH

_,

1]1111

1]11111111111(14) where X is a measure of wear in terms of percent rail-head area (%HA).The aspect ratio of 0.7 for typical detail fractures has been included in equation (14). TheKformulagiveninequation(13)appliestoremoteuniformtensiononly.Therefore,astress-gradientmagnificationfactormustalsobeappliedtotheK 11formula if the rail head is subjected to a non-uniform stress field, which is the case whentherailissubjectedtocombinedbendingfrombothverticalandlateral loading.The stress-gradient or non-uniform stress magnification factor depends on the ratio of lateral to vertical wheel load as well as defect size relative to the unworn rail head area.The formalism to calculate the stress-gradient or non-uniform stress magnificationfactorisgiveninAppendix C.Appendix Calsoincludestabulated valuesofthestress-gradientmagnificationfactorsfora132RErailsectionwith various levels and types of wear, lateral-to-vertical load ratios, and DF sizes. 3.2Stress Analysis for Rails with Detail Fractures The magnitude of the stresses that drive the growth of detail fractures is assumed to bea linearsuperpositionofresidual,thermal,andbendingstresses.Thus,the stress intensity factor formula for a detail fracture is given by [ ]K MbaM a M a aI s R T G B

_,

+ + 21 ( ) ( ) (15) where R is the residual stress, T is the thermal stress, MG is the stress gradient or non-uniform stress magnification factor, and B is the bending stress. A relationship between the magnitude of tensile residual stress in the rail head and theDFsizehasbeendevelopedonthebasisofexperimentaldataobtainedfrom two separate tests [4], [5] conducted at the Facility for Accelerated Service Testing (FAST) RH HH HAAHAAAHAAAAAHA

_,

_,

_,

_,

>'30 2125 100 0 100 1010 0125 100 100 10%. % %.forfor(16) where (A/AH 100) represents the DF size in percentage of rail-head area (%HA). Thermal stresses for fully restrained CWR in tangent track can be calculated from TE T (17) where is the linear coefficient of thermal expansion, E is the modulus of elasticity, andTisthetemperaturedifferencebetweenthein-servicetemperatureandthe neutralorstress-freetemperature.Inthepresentcalculations,Tisanassumed variable. 12Rail-bendingstressescompriseboththeverticalandlateralbendingcomponents, as described previously in Section 2 for the rail strength analysis.For a rail modeled as a beam on elastic foundation, the maximum tensile stress in the rail head occurs atsomedistanceawayfromthepointofloadapplication.Thisphenomenonis referred to as reverse bending.In terms of absolute value, the maximum bending moment occurs directly beneath the wheel load.Referring to Figure 7, the location of the maximum reverse moment for vertical bending relative to the wheel position is defined by xV02(18) where V is calculated from equation (4). x0V Figure 7. Bending Moment Distribution for a Single Wheel Load Therefore,thebendingstressusedinequation (15)forthefractureanalysisis calculated from ( )zzH LyyN VBIw x MIh h x M2) ( ) (0 0+ (19) wheretheverticalandlateralbendingmomentsaredefinedbyequations (3) and (6), respectively.3

3The small effect of the cross-product terms associated with asymmetrical bending has been neglected in equation (19). 133.3 Fracture-Mechanics-Based Wear-Limit Estimation Rail-wearlimitsbasedonthefracturemechanicsapproachareestimatedby combiningresultsfromtwoseparatesetsofcalculations.Inthefirstsetof calculations, rail wear is determined as a function of critical DF size.In other words, equation (15)issetequaltothefracturetoughnessofrailsteelfordifferentcritical DF sizes and different levels of rail wear.Results from this numerical procedure are shown schematically in Figure 8.In the present analysis, the fracture toughness for railsteel,KIC,isassumedtobe35ksi-in1/2.Also,anextremetemperature differencefromtheneutralorstress-freetemperatureof50Fisassumed.In addition,a dynamicmagnificationisappliedtomagnify the magnitude of a static wheelloadduringcalculationofthestressintensityfactor.Detailsofthedynamic wheelloadcalculationforthispurpose(referredtoasthe0.8-loadlevel)are described in Section 4. Head-height loss, hStress Intensity Factor, KIVarying aCRKICHead-height loss, hCritical DF size, aCR Figure 8. Schematic for Estimating Rail Wear as a Function of Critical DF Size When we consider only the results from this first set of calculations, the question is: whatcritical-sizedefectshouldbeconsideredinestimatingtherail-wearlimit?To answerthisquestion,weassumethatrailwearbecomescritical(i.e., reaches a limit) when detail fractures can grow from a barely detectable size (assumed to be 5%HA)toacriticalsizeinlessthanonetypicalinspectioninterval.Forthis purpose, we assumed 20 MGT.Thus, a second set of calculations is performed to determinetheDFsizethatwillbereachedafter20-MGTtrafficaccumulationfor variouslevelsofrailwear,assuminganinitialDFsizeof5%HA.Inthese DF-growthcalculations,a moderatetemperaturedifferentialof15Fisassumed.Figure 9showsa schematicoftheresultsfromthissecondsetofcalculations.Reference [4] describes propagation analyses of detail fractures, which are used in the present calculations. 14DF Size, aHead-height loss, hDF size at 20 MGT, afMGT20 MGT 5%HAVarying h Figure 9.Schematic for Estimating Rail Wear as a Function of DF Size after 20 MGT from Initial Size of 5% HA The results from the two sets of calculations described above can be combined, as shownintheschematicdiagraminFigure 10.Theintersectionofthetwocurves; onebasedonfracturetoughnessandtheotherbasedonDF-growthrate,defines thelimitforrailwear.AlthoughtheschematicdiagramsshowninFigures 8 through 10considerrailwearashead-heightloss,thenumericalprocedures described in this section are also applicable for the case of gage-side wear. Head-height loss, hDF size, ahlimitCurve based onDF-growth rateCurve based onfracture toughness Figure 10. Estimate of Rail-Wear Limit Based on Fracture Strength 154.WORST-CASE LOAD ESTIMATION Car body and truck dynamic motions (pitch, bounce, and rocking) cause wheel loads tovaryatfrequenciesupto10 Hz.Foragivenstaticwheelload,V,thisdynamic loadVtVDcanbemodeledasaGaussianrandomprocesswithprobabilitydensity functions: p V Vc VVc VDvDv( ) exp( )( )t t

1]112 222(20) wherethecoefficientofvariation,cv,scalestheroot-mean-squaredynamic incrementintermsofthestaticload.Inmeasurementsofactualdata,however, differenceshavebeenobservedbetweendynamicloadincrementsand decrements.The statistical variation of dynamic loads should then be modeled with twoone-sidedGaussiandistributionstoaccountfortheseobserveddifferences.4

A schematic of these distribution functions is shown in Figure 11.The coefficient of variationcorrespondingtothedynamicloadincrement,cv+,isapplicablein determining the worst-case load. p(VtVD)cv-cv+VVtVD Figure 11. Schematic of a Two-Sided Gaussian Distribution Function Physically, the worst-case load may be caused by a combination of several factors thatinclude:(1) trackirregularitiesandirregulartrackstiffnessduetovariable characteristics and settlement of the ballast; (2) discontinuities at welds, joints, and switches;(3) irregularrail-runningsurface(e.g., corrugatedrail);(4) defectsin vehiclessuchaswheelflatsandwheeleccentricity;and (5) vehicle dynamics such asnaturalvibrationsandhunting.Mathematically,theworst-caseloadisthe averagestaticwheelloadmultipliedbyanextremedynamicloadmagnification

4In addition, separate distributions are usually required to match the dynamic behavior of lightly and heavily loaded cars.To simplify the present analysis, however, fully loaded cars are considered. 16factor.Thedynamiceffectisassumedtoincreasewithtrainspeed,andits mathematical relationship to the coefficient of variation is discussed in the following text. ProceduresaredescribedinOrringer,etal.,[4]todeterminethecoefficientsof variation from load data given in the form of either exceedance curves or cumulative probability curves. Results from applying these procedures to freight-traffic load data from various sources are listed in Table 3.Table 3 is an abstract, from Orringer, et al.,[4]ofthoseresultsforwhichcv+valuescanberelatedtoloadedfreighttrain speeds. Table 3. Coefficients of Variation for Dynamic Vertical Load Environment Description [Ref.]Average Wheel Load (kips) Speed (mph) cv for +VD 1FAST, concrete tie, 1977 [7].30.540 to 450.26 2Loadedcoalhoppercarover1,900 miles on six midwestern and eastern railroads [8]. 30.015 to 30 30 to 45 45 to 60 0.20 0.17 0.22 3DOT-112A,33,000-gal tank car over 114milesofamidwesternrailroad mainline [8]. 30.015 to 30 30 to 45 45 to 60 0.17 0.25 0.60 4Hopper car loaded with crushed rock over 182 miles on a western railroad mainline [8]. 30.015 to 30 30 to 45 45 to 60 > 60 0.08 0.10 0.11 0.12 5FAST, concrete tie, circa 1977 [9].27.740 to 450.30 6NortheastCorridor,freight,concrete tie, Edgewood, MD, 1984 [10]. 24.17 24.32 45 to 700.31 0.33 In the present analysis, the worst-case load can be defined in terms of the number ofstandarddeviationsabovethemeanvalue.Thecoefficientofvariation,cv,is related to the standard deviation by the static wheel load, V, as given in the following expression cVv (21) As indicated in Table 3, the coefficient of variation appears to vary with speed.This isconsistentwiththeAmericanRailroadEngineeringAssociation(AREA)formula for dynamic load factor [6] 17DLFvD + 133100(22) wherevisthetrainspeed(inmilesperhour)andDisthewheeldiameter(in inches). For the purpose of load estimation, cv can be considered as comparable to the term 33v/100D in the AREA formula.Figure 12 compares this term with the cv data from Table 3.With only one exception, the formula term bounds the data in Table 3, and is therefore a reasonable representation of the service environment. 0 10 20 30 40 50 60 7000.10.20.30.40.50.6Train Speed, v (mph)Coefficient of Variation, cv444433322266 51Dynamic load incrementfrom AREA-recommendedformula for D = 36 inches Figure 12.Speed Dependence on Coefficient of Variation and Dynamic Load Increment Figure13,reproducedfromOrringer,etal.,[4]showsthenumbersofextreme-valued peak dynamic wheel loads measured in a field test on the Northeast Corridor in the early 1980s [10].Table 4 summarizes the results for freight trains, which were typically operated at 60 mph through the instrumented site.Taking v = 60 mph and assuming that the extreme loads came from loaded 100-ton cars (corresponding to a static load, V, equal to 33 kips, and a wheel diameter, D, equal to 36 inches), we may apply the previous analysis to estimate the standard deviation as = cvV = 18.2 kips.Thus,therangeof55to80kipsforextremeloadslistedinTable 4canbe interpreted as roughly 1.2- to 2.6- events. 18 55 60 65 70 75 80012Wheel Loads in 5-Kip BinsOccurrences per 1000 axlesFreightPassenger Figure 13.Histogram of Extreme Wheel Loads Measured on the Northeast Corridor Table 4. Summary of Freight-Car Wheel-Load Histogram Peak LoadVpeak (kips) Occurrences per1000 Axles, n 551.50 600.60 660.50 700.25 750.25 800.10 Although the peak values in Table 4 occur infrequently, they cannot be assumed to boundtheworstcasebecausetheloadsweremeasuredonlyfora limitedtimeat a singlesite.Therefore,itisnecessarytoextrapolatefromtheavailabledatato estimateworst-caseloadsforvariousoperatingspeeds.Forthispurpose,the regression formula Vnpeak 2 40 04210. log.(23) 19hasbeenderivedfromaleast-square-erroranalysisofthedatalistedinTable 4.Figure 14 compares equation (23) with the data points. 50 60 70 800.010.1110Peak wheel load, Vpeak (kips)Occurrences per 1000 axles, n Figure 14. Regression for Occurrence Rate versus Peak Load For randomly occurring events such as peak loads, the generally accepted definition ofa worst-caseeventforthepurposeofriskanalysisisoneexpectedtobe exceedednomoreoftenthanoncein105to106times.5Forexample,whenthe Gaussian distribution is used to model a random process, the 5- level is sometimes usedtodefinetheworstcase.The5-levelofaGaussianprocesshasan exceedance rate of 1.6 10-6, which lies within the range mentioned above.There is no justification for applying the Gaussian model to extreme wheel loads, but it is reasonabletoadoptthe1.6 10-6exceedancerateperaxlepassageasa worst-casecriterion.Also,sincethereisnosignificantnumericaldifferencebetween exceedanceandoccurrenceratesattheseextremes,wemaytakethe correspondingoccurrencerateasn = 1.6 10-3perthousandaxlesandapply equation (23) to estimate the worst-case load as Vmax. log ( . ). 2 4 16 1030 04212410kips (24)

5 Forexample,thecurrentU.S.Armyfatigue-lifespecificationfornewrotorcraftisbasedonthe so-called six-nines reliability, or probability of failure of one in a million [11]. 20 The above worst-case load estimate applies to the Northeast Corridor field-test site, forwhich = 18.2 kipswasderivedearlierasthestandarddeviationoftheentire dynamic load range.Thus, the 5- dynamic load increment above the static wheel loadappearstobeareasonablecriterionforestimatingworst-casewheelloadsat operating speeds and/or wheel diameters other than those associated with the test site.Table 5summarizestheworst-caseloadsestimatedforthemaximum operatingspeedsallowedontrackclasses 2through5,assuminga 33-kipstatic loadanda 36-inchwheeldiameter.Theright-handcolumnlistssuggestionsfor typicalandminimumrailsectionsrepresentativeoftrackconstructionforthe correspondingclass.Inthewear-limitanalyses,verticalfoundationstiffnessis assumed to vary with different track classifications.The assumed values for vertical foundation stiffness for each track class are also listed in Table 5. The load level selected as a basis for wear-limit estimation depends on the mode of failureassumed.Forastrength-basedlimit,thewearprocessreducesplastic bendingstrengthmoreorlessuniformlyalongmanyraillengths.Therefore,one must expect the reduced strength at a point where the worst possible combination of trackirregularityandvehicledynamicsmayoccur,anditfollowsthatthe5-load levelisanappropriatebasis.Forafracture-strength-basedlimit,however,the assumedmodeoffailureispropagationofafatiguecracktofracture.To realistically treat this case requires one to recognize that sparsely distributed cracks (typically,0.25to2pertrackmile)arenotnecessarilyfoundwheretheworst dynamicloadsoccur.A realisticextremeloadbasisforthiscaseisthelevel expectedoncepertrainpassage;i.e.,aboutonceper400axles,assuming a 100-car freight train.The corresponding value of n in equation (23) would be 2.5 per1000axles,whichgivesavalueofVmaxequalto47.7 kips,correspondingtoa 0.8-loadlevel(assuminga 33-kipstaticwheelload).Table5alsolistsvaluesof worst-case loads based upon the 0.8- load level, which are subsequently applied in the fracture mechanics calculations for estimating rail-wear limits. 21Table 5.Representative Worst-Case Load Estimates with Corresponding Foundation Stiffness and Rail Sections FRA Track Class Max. Train Speed v (mph) (a) Coefficient ofVariation cv (b) Standard Deviation (kips) (c) Worst-Case Load Vmax (kips) (d) Vertical Foundation Stiffness kv (psi) Rail Section 2250.237.639/711,00070 ASCE 85 ASCE 100 RE 115 RE 3400.3712.143/942,00085 ASCE 100 RE 115 RE 132 RE 4600.5518.248/1242,000115 RE 132 RE 136 RE 140 RE 5800.7324.152/1545,000115 RE 132 RE 136 RE 140 RE NOTES: (a) Maximum train speeds are based on FRA Track Safety Standard 213.9 for freight traffic. (b) Coefficient of variation calculated from: cv = 33v/100D, where D = 36 inches. (c) Standard deviation calculated from: = cvV, where V = 33 kips. (d) Worst-case loads are calculated from: Vmax = V + 0.8 and Vmax = V + 5.0. The loads corresponding to 0.8- above the static value were applied to the fracture mechanics analysis, and those for 5- were used in the estimates based on permanent bending strength. 235.RESULTS Figure15showsrail-wearestimatesofverticalhead-heightlossbasedonrail strength.Four different rail sizes are considered for each track class.As the track classincreasesfromClass 2toClass 4(implyingincreasedmaximumtrainspeed, higherworst-caseloads,andimprovedfoundationstiffness),thewearlimitfor a givenrailsectiondecreases.TheestimatedwearlimitsforClass 4and 5are practically equal.In these results, the yield strength for rail steel was assumed to be 90 ksi. Rail-wearlimitsforsidewearestimatedonthebasisofrailstrengthareshownin Figure 16.As in the case for vertical head-height loss, limits on side wear for both Class 4 and Class 5 track are effectively the same. Rail-wear limits for head-height loss based on the fracture mechanics analyses are shown in Figure 17.In these analyses, the fracture toughness of rail steel has been assumed to be 35 ksi-in1/2.The wear-limit estimates based on the fracture-strength approachareroughly30 to40%lessthanthosebasedontheelastic-plastic bendinganalysis.Similarly,resultsfortheside-wearlimitareshowninFigure 18.Again, the wear-limit estimates based on the fracture analysis are about one-third of those based on permanent rail bending. 24 Class 2 Class 3 Class 4 Class 50.00.51.01.52.0FRA Track ClassificationHead-height loss, h (inches)70 ASCE 85 ASCE 100 RE 115 RE 132 RE 136 RE 140 RE Figure 15. Wear-Limit Estimates for Head-Height Loss Based on Rail Strength Class 2 Class 3 Class 4 Class 50.00.51.01.52.02.53.03.54.0FRA Track ClassificationGage-side wear, w (inches)70 ASCE 85 ASCE 100 RE 115 RE 132 RE 136 RE 140 RE Figure 16. Wear-Limit Estimates for Side Wear Based on Rail Strength 25Class 2 Class 3 Class 4 Class 50.00.20.40.60.81.0FRA Track ClassificationHead-height loss, h (inches)70 ASCE 85 ASCE 100 RE 115 RE 132 RE 136 RE 140 RE Figure 17.Wear-Limit Estimates for Head-Height Loss Based on Fracture Strength Class 2 Class 3 Class 4 Class 50.00.20.40.60.81.01.21.41.6FRA Track ClassificationGage-side wear, w (inches)70 ASCE 85 ASCE 100 RE 115 RE 132 RE 136 RE 140 RE Figure 18. Wear-Limit Estimates for Side Wear Based on Fracture Strength 276.DISCUSSION AND CONCLUSIONS Wearlimitswereestimatedonthebasisoftwostrengthcriteria:oneconsiders failureaspermanentplasticyielding,andtheotherconsidersfailurewhentherail fractures.Theanalysesthatconsiderrailfractureassumedthepresenceofan internal defect (known as the detail fracture). Worst-casewheel loads were estimated using actual freight car load data from the NortheastCorridorthatwerecorrelatedwiththeAREAformulafordynamicload magnification.Regressionanalyseswereconductedtoextrapolateworstcase dynamicwheelloadsforoperatingconditionsotherthanthoseontheNortheast Corridor.Theworstcasewheelloadsforpermanentrailbendingareexpectedto occuronceinevery105to106wheelpassages.Ifa Gaussiandistributionis assumed for the dynamic load increment, the worst case load corresponds to a 5- event.Worstcasewheelloadsforthefracturemechanicsanalysisassumed a maximumwheelloadtooccuronceineverytrainpassage(orroughlyonceper 400 wheel passages). Ofthetwocases,thefracturemechanicsapproachgivesthemorerestrictive(i.e., conservative)limits.Therefore,forsafeoperationsonrailroadtracks,allowable rail-wearlimitsshouldbeestablishedonthebasisoffracturestrength.Withthe exceptionofthelightestrailsectionsconsidered(70 ASCEonClass 2trackand 85 ASCEonClasses 2and 3track),thefracture-strength-basedlimitscanbe summarizedas:(1) maximumallowablehead-heightlossof0.5 inch;or (2) maximum allowable gage-face wear of 0.6 inch.The basis for estimating these limitsalsoincludedanassumptionofa20-MGTinspectioninterval,which representstypicalindustrypractice.Thesamegeneralbasiswouldleadtolarger amounts of allowable wear if more frequent inspection were assumed. It should be noted that 70 ASCE and 85 ASCE rail sections are typically found more often in bolted joint rail (BJR) track rather than in CWR track.The most common rail defects encountered in BJR track are cracks emanating from bolt holes rather than detail fractures.In principle, it would appear that more reasonable rail wear limits for theselighterrailsizesmaybeestimatedbymodifyingthefracturemechanics analysistoconsiderboltholecracksratherthandetailfractures.Buta potential problem in such analysis is that, to date, a validated model to determine the growth rate of bolt hole cracks has not been developed. 29REFERENCES [1]M.Hetenyi,BeamsonElasticFoundation,UniversityofMichiganPress,Ann Arbor, MI, 1983. [2]D.Y. Jeong, et al., Propagation Analysis of Transverse Defects Originating at the Lower Gage Corner of Rail, Final Report, DOT/FRA/ORD-98/06, 1996. [3]P.G.Hodge,Jr.,PlasticAnalysisofStructures,McGraw-Hill,NewYork,NY, 1959. [4]O.Orringer,etal.,CrackPropagationLifeofDetailFracturesinRails,Final Report, DOT/FRA/ORD-88/13, 1988. [5]P.ClaytonandY.H.Tang,DetailFractureGrowthinCurvedTrackatthe FacilityforAcceleratedServiceTesting,ResidualStressesinRails,Vol.1, Kluwer Academic Publishers, The Netherlands, pp. 37-56 (1992). [6]G.M. McGee, Calculations of Rail Bending Stresses for 125 Ton Tank Cars, AAR Research Center Report No. 19506, Chicago, IL, April 1965. [7]D.R.Ahlbeck,M.R.Johnson,H.D.Harrison,andJ.M.Tuten,Measurements ofWheel/RailLoadsonClass5Track,FinalReport,DOT/FRA/ORD-80/19, 1980. [8]M.R.Johnson,SummarizationandComparisonofFreightCarTruckLoad Data,ASMETransactions,JournalofEngineeringforIndustry100,60-66 (1978). [9]A.M.Zarembski,EffectofRailSectionandTrafficonRailFatigueLife, American Railway Engineering Association Bulletin 673, 514-527 (1979). [10]J.M.TutenandH.D.Harrison,Design,Validation,andApplicationofa MonitoringDeviceforMeasuringDynamicWheel/RailLoads,Proceedingsof the ASME Winter Annual Meeting, Paper No. 84-WA/RT-10 (1984). [11]R.A.Everett,Jr.,F.D.Bartlett,Jr.,andW.Elber,ProbabilisticFatigue MethodologyforSafeRetirementLives,JournaloftheAmericanHelicopter Society, 41-53 (1992). 31APPENDIX A.SECTION PROPERTIES FOR NEW OR UNWORN RAIL ZZY YhBhNhHh Figure A-1. Dimensions for a Generic Rail Section NOMENCLATURE Rail Dimensions (in) hTotal rail height hHDistance from the bottom of the rail to the centroid of the rail head hNDistance from the bottom of the rail to the centroid of the entire rail hBDistance from the bottom of the rail to the centroid of the rail base wHRail head width Cross-Sectional Areas (in2) ARCross-sectional area of the entire rail AHCross-sectional area for the rail head only AWCross-sectional area for the rail web only ABCross-sectional area for the rail base only Second Area Moments of Inertia(in4) IyyVertical bending inertia for the entire rail IzzLateral bending inertia for the entire rail IyyHVertical bending inertia for the rail head only IzzHLateral bending inertia for the rail head only 32 Table A-1. Section Properties for New or Unworn Rail 70 ASCE 85 ASCE100 RE115 RE132 RE136 RE140 RE h4.6255.1886.006.6257.1257.3137.313 hH4.014.465.235.806.36.396.35 hN2.222.472.752.983.23.353.37 hB0.2990.3210.3940.4110.4360.4350.436 wH2.43752.56252.68752.71883.002.93753.00 AR6.818.339.9511.2612.9513.3513.8 AH2.813.493.803.914.424.865.00 AW1.412.252.233.163.673.623.94 AB2.592.593.924.194.864.874.86 Iyy19.730.0749.065.688.194.996.8 Izz4.866.959.3510.414.214.514.7 IyyH0.3290.5580.7140.7290.8371.171.38 IzzH1.241.752.122.132.843.033.14 33APPENDIX B.APPROXIMATION OF WORN RAIL SECTION PROPERTIES Inthisreport,wearisassumedtooccurfromuniformlossofmaterialfromtherail head.In this appendix, the amount of wear is quantified by a percentage of the rail- headarea,whichcanberelatedtohead-heightlossorgage-facesidewear.Moreover, the equations to determine the section properties that are needed in the stressanalysisofwornrailsarelistedinthisappendix.Theseequationswere derivedfromidealizingtheactualrailcross-sectionasthreerectangularsections representing the head, web, and base of the rail.The specific details in deriving the equationsforverticalhead-heightlossandgage-facesidewearcanbefoundin Jeong, et al., [2]. Equivalence between the actual and idealized rails is achieved by matching section properties for both cross-sections.In particular, the second area moments of inertia for the rail-head about the vertical and horizontal axes through the centroids for the actual and idealized rail-heads are related by I h wyyH eq eq 1123I h wzzH eq eq 1123(B-1) whereheqandweqaretheequivalentrail-headheightandwidth.Aftersome algebraicmanipulations,expressionsfortheequivalentrail-headheightandwidth can be found hIIeqyyHzzH 14438wIIeqzzHyyH 14438 .(B-2) Numerical values of the equivalent rail-head heights and widths for the rail sections examined in this report, as determined from equations (B-2), are listed in Table B-1.Thetablealsoshowsthemagnitudeoftherail-headcross-sectionalarea approximatedbytheproductoftheseequivalentdimensions,andthepercentage difference between the approximate area and the actual rail-head area. Table B-1. Equivalent Rail-Head Height and Width for Various Rail Sections 70 ASCE 85 ASCE100 RE115 RE132 RE136 RE140 RE heq1.191.391.491.501.531.721.82 weq2.322.472.572.572.822.772.74 heqweq2.773.443.843.874.304.754.99 AH2.813.493.803.914.424.865.00 % diff. in AH -1.5%-1.4%+1.1%-1.1%-2.6%-2.2%-0.2% 34Thesectionpropertiesforwornrailarecalculatedwiththeaidoftheseequivalent dimensions and by assuming a wear pattern; i.e., head-height loss or gage-face side wear. Head-Height Loss The following equations are applied to determine section properties for rail affected bywearintermsofhead-heightloss.Thelossofrail-headheightandthe equivalent rail-head height are related by hXheq

_,

100(B-3) whereXisthepercentageofwornrail-headarea.Thesecondareamomentof inertia for vertical bending for a worn rail is calculated from [ ][ ]I X I A h h X h wh w h h h Xyy Y Y R N N eqeq N( ) ( ) ( )( ) .' ' +

1]1011212232 !(B-4) Inthisnotation,IYYistheverticalbendinginertiaoftheneworunwornrail,hN(0) refers to the distance from the bottom of the rail to the centroid of the new or unworn rail, and hN(X) is the distance from the bottom of the rail to the centroid of the worn rail, which is calculated from h XA h A hXA h XA AXANR N H H H HB W H( )( ) ( ) ( ) +

_,

1]1+ +

_,

1]10 0 11001100.(B-5) whereARisthecross-sectionalareaoftheunwornrail.Thedistancefromthe bottomoftherailtothecentroidfortherailheadonlyina wornrailiscalculated from h X h hH( ) 12 .(B-6) where h is the total height of the rail. 35Gage-Face Side Wear The following equations are used to calculate section properties for rail affected by gage-facesidewear.Theamountofgage-facesidewearisrelatedtothe equivalent head width by wXweq

_,

100 .(B-7) The second area moments of inertia for vertical and lateral bending of a worn rail, in this case, are calculated, respectively, from [ ][ ]I X I A h h X h wh w h h h Xyy Y Y R N N eqeq eq N( ) ( ) ( )( )' ' +

1]1011212232"(B-8) [ ] ( )I X I A y Xhwh w w w y Xzz Z Z R Neqeq eq N( ) ( )( )' ' +

1]12 321212 "(B-9) Aconsequenceofgage-facesidewearisthattherailcross-sectionisasymmetric withrespecttothemid-planecenterlineoftherail.Further,thecoordinatesofthe centroid for the entire rail and for the rail head only are not located on the mid-plane centerline, as they are for a new or unworn rail (Figure B-1). zHzNyNyH Figure B-1. Location of Centroids in Rail with Gage-Face Side Wear 36 The location of the centroid for the entire worn rail is defined by y XXA y XA AXANH HB W H( )( )

_,

1]1+ +

_,

1]111001100(B-10) h XA h A hXA AXANR N H HB W H( )( ) ( )

_,

+ +

_,

1]10 01001100.(B-11) The vertical location for the centroid of the worn rail head only is assumed to remain unchanged from that of the new or unworn rail head.The horizontal location of the centroid for the worn rail head only is calculated by y X wH( ) 12 (B-12) 37APPENDIX C.STRESS-GRADIENT MAGNIFICATION FACTORFOR FRACTURE ANALYSIS Thehandbookformulasforthestressintensityfactorassociatedwithanelliptical flawofaspectratiob/ainacombinedbendingfieldwithflawcenterlocation(y,z)6 can be expressed as K a MI( ) ( ) 2(C-1) where is the far field stress, and M() is a function of angle around the crack.This function for an elliptical flaw in the rail head is given in Hodge, J., [3] as MMyzIILVMyzIILVvVLyyzzLVLyyzz( )( ) ( )+ + 1(C-2) where1/Vand1/Larethecharacteristicverticalandlateralbendingwavelengths fortherailonagivenfoundation,whicharedefinedinequations(4)and(7), respectively.Also,Iyy andIzzarethesecondareamomentsoftherailsectionfor vertical and lateral bending, L/V is the ratio of lateral to vertical load, and Mb z EE EvII II( )( / ) ( )sin( ) ( ) ( ) ( ) ++ + 11 122 2(C-3) and Ma y EE ELII II( )( / ) ( ) cos( ) ( ) ( ) ( ) + 11 2 122 2(C-4) where 2=1-(b/a)2.In equations (C-3) and (C-4),EIand EIIare the complete elliptic integrals of the first and second kind, respectively, which are defined by E dI( ) sin/ 12 202EdII( )sin/ 12 202(C-5)

6 Inthisformulation,thecoordinates(y,z)refertothelocationofthestresspointrelativetothe centroidal axes of the rail section. 38The dependence of the stress intensity factor on position along the crack front can be eliminated by calculating a simple average value.This concept was generalized inthecrack-growthanalysesofdetailfractures [3]tocalculatethePthroot-mean valuewherePistheexponentinthecrack-growthrateequation.Thestress-gradient or non-uniform stress magnification factor is defined mathematically as MM ddGPP

1]11111( ) 02021(C-6) where +bb a sin ( / ) cos2 2(C-7) is the radius to the perimeter of the ellipse.Equation (C-6) includes the special case of P = 1, which is the calculation for a simple average.In the crack-growth analyses presentedinOrringer,etal.,[4]andinthepresentanalyses,Pisassumedtobe equal to 4. The influence of wear on the stress-gradient magnification factor is affected throughequation (C-2).Particularly,thesecondareamomentsofinertiaandthe characteristicwavelengthsdependonthelevelofwear.Therefore,thestress gradientmagnificationfactordependsnotonlyontheleveloramountofwearbut also on the type of wear (i.e., vertical head-height loss or gage-side wear). Table C-1 summarizes the stress-gradient magnification factors for 132 RE rail with variouslevelsofverticalhead-heightloss.Themagnificationfactoralsovaries dependingonthelateral-to-verticalloadratioandsizesofthedetailfracture.Similarly,Table C-2liststhestress-gradientmagnificationfactorsfor132 RErail with gage-face side wear. 39 Table C-1.Stress-Gradient Magnification Factors for 132 RE Rail Section and Vertical Head-Height Loss (a) DF Size =10% HA Worn AreaLateral-to-Vertical Load Ratio, L/V (%HA)0.050.100.200.300.400.50 01.0011.0011.0041.0071.0111.014 201.0011.0011.0041.0061.0101.013 401.0011.0011.0031.0061.0081.011 601.0011.0011.0021.0051.0071.009 801.0011.0011.0021.0031.0051.007 (b) DF Size =15% HA Worn AreaLateral-to-Vertical Load Ratio, L/V (%HA)0.050.100.200.300.400.50 01.0011.0021.0071.0121.0181.023 201.0011.0021.0061.0111.0161.021 401.0011.0021.0051.0091.0141.018 601.0011.0021.0041.0071.0111.015 801.0011.0011.0031.0051.0081.011 (c) DF Size =20% HA Worn AreaLateral-to-Vertical Load Ratio, L/V (%HA)0.050.100.200.300.400.50 01.0011.0031.0091.0171.0251.034 201.0011.0031.0081.0151.0231.031 401.0011.0031.0071.0131.0201.027 601.0011.0021.0061.0101.0161.022 801.0011.0021.0041.0081.0121.016 40 Table C-2.Stress-Gradient Magnification Factors for 132 RE Rail Section and Gage-Face Side Wear (a) DF Size =10% HA Worn AreaLateral-to-Vertical Load Ratio, L/V (%HA)0.050.100.200.300.400.50 01.0011.0011.0041.0071.0111.014 201.0011.0011.0041.0081.0131.019 401.0011.0011.0041.0081.0151.023 601.0001.0011.0031.0071.0141.024 801.0001.0011.0021.0051.0101.018 (b) DF Size =15% HA Worn AreaLateral-to-Vertical Load Ratio, L/V (%HA)0.050.100.200.300.400.50 01.0011.0021.0071.0121.0181.023 201.0011.0021.0071.0141.0221.032 401.0011.0021.0061.0141.0251.039 601.0011.0011.0051.0121.0241.041 801.0001.0011.0031.0081.0161.031 (c) DF Size =20% HA Worn AreaLateral-to-Vertical Load Ratio, L/V (%HA)0.050.100.200.300.400.50 01.0011.0031.0091.0171.0251.034 201.0011.0031.0101.0201.0321.047 401.0011.0031.0091.0201.0371.059 601.0011.0021.0071.0171.0351.062 801.0011.0011.0041.0111.0241.047