UseofSideFrictioninHorizontalCurveDesign:AMarginofSafety1 

Assessment2 

3 

4 EricDonnell,Ph.D.,P.E.5 AssociateProfessor6 DepartmentofCivilandEnvironmentalEngineering7 ThePennsylvaniaStateUniversity8 231SackettBuilding9 UniversityPark,PA1680210 Phone:(814)863‐705311 E‐mail:edonnell@engr.psu.edu12 13 JonathanWood14 Ph.D.Candidate15 DepartmentofCivilandEnvironmentalEngineering16 ThePennsylvaniaStateUniversity17 212SackettBuilding18 UniversityPark,PA1680219 Phone:(435)760‐278120 E‐mail:jsw277@psu.edu21 22 ScottHimes,Ph.D.23 TransportationAnalyst24 VanasseHangenBrustlin,Inc.(VHB)25 4000WestChaseBoulevard26 Suite53027 Raleigh,NC2760728 Phone:(919)334‐560829 E‐mail:shimes@vhb.com30 31 DarrenTorbic,Ph.D.32 PrincipalTrafficEngineer33 MRIGlobal34 2332RavenHollowRoad35 StateCollege,PA1680136 Phone:(814)237‐883137 E‐mail:dtorbic@mriglobal.org38 

39 

Numberofwordsinabstract:24740 

Numberofwordsinnarrativeofpaper:4,70341 

Numberoffiguresandtables:942 

Numberofreferences:2143 

44 

Submittedforconsiderationto5thInternationalSymposiumonHighwayGeometricDesign45 

Vancouver,BritishColumbia,Canada46 

Re‐submittedonMarch20,201547 

48 

Abstract1 

2 

Currentengineeringpracticeusesapoint‐massmodeltodesignhorizontalcurveson3 

highwaysandstreets.Inthismodel,amaximumsidefrictionfactorisused,incombination4 

withtheselecteddesignspeedandmaximumrateofsuperelevation,todeterminethe5 

minimumradiusofcurveforanalignment.Thelimitingvalueforsidefrictionusedin6 

designwasestablishedinthe1940’sandisbasedondrivercomfortthresholds.Thelateral7 

frictionavailableatthetire‐roadwayinterfaceisameasureoffrictionsupplyandis8 

dependentonthepavementsurfacetypeandcondition,vehicleoperatingspeedand9 

decelerationcharacteristics,vehiclelaneposition,andtiretype.Driversselectindividual10 

operatingspeedsonaroadway,whichresultsinasidefrictiondemandwhentraversinga11 

horizontalcurve.Thepurposeofthispaperisthreefold.First,keysidefrictionconceptsin12 

horizontalcurvedesignaredescribed.Thisincludesdefinitionsof,andthefundamental13 

principlesassociatedwiththeapplicationofsidefrictionfactorsinhorizontalcurvedesign14 

policy.Thesecondpartofthepaperprovidesananalysisofthemarginofsafetythatexists15 

inhorizontalcurvedesignpolicy.Thisanalysisconsidersvariousvehicletypes,pavement16 

surfacetypes,andoperatingspeeddistributions.Comparisonsbetweenfrictionsupply,17 

demand,anddesignsidefrictionfactorsaremadeinthissectionofthepaper.Thefinal18 

objectiveofthepaperistodescribeaframeworktomoreeffectivelyconsiderthecurrent19 

vehiclefleet,rangeofpavementconditions,andvehiclespeeddistributioninhorizontal20 

curvedesignpolicy.21 

1 

INTRODUCTION1 

2 

GeometricdesignpolicyintheUnitedStates(U.S.)issetbytheAmericanAssociationof3 

StateHighwayandTransportationOfficials’(AASHTO)APolicyonGeometricDesignof4 

HighwaysandStreets,hereinreferredtoastheGreenBook.Thephysicsofuniformcircular5 

motionwereusedtodevelophorizontalcurvedesign,whereavehicleisassumedto6 

operateasapointmass.Asavehicletravelsahorizontalcurve,itundergoescentripetal7 

accelerationequaltothesquareofthevehiclespeeddividedbytheradiusofthecurved8 

path.Thisaccelerationisbalancedbyacombinationofsuperelevationandthefriction9 

betweentheroadsurfaceandtiresofthevehicle.10 

11 

HorizontalcurvesdesignedinaccordancewithGreenBookcriteriahavebeengenerally12 

showntoprovidesubstantialmarginsofsafetywithrespecttovehicleskiddingand13 

rollover,forbothpassengercarsandtrucks(Harwoodetal.,1989;HarwoodandMason,14 

1994;Harwoodetal.,2003).Previousresearch,however,hasconsideredfrictiondata15 

measuredinthe1930’sand1940’s,whichwereusedtodeveloplimitingvaluesoffriction16 

usedinhorizontalcurvedesignpolicy.Sincethen,thevehiclefleethaschanged17 

considerably,ashastiredesign,pavementdesign,andfrictionmeasurementmethods.18 

19 

Tworecentresearchstudieswerecompletedthataffordedtheopportunitytocollect20 

asphaltpavementsurfacefrictiondataontwo‐laneandmulti‐laneruralhighwaysinthe21 

easternU.S.Thesestudiesalsoincludedprovisionstocollectpassengercarandtruck22 

operatingspeeddataatthelocationswherefrictionmeasurementswererecorded.This23 

paperpresentstire‐pavementfrictionplotsandcomparesthemtofrictiondemandderived24 

fromtheoperatingspeeddataandpoint‐massmodel.Themarginsofsafetyagainst25 

skiddingandrolloverforpassengercarsandtrucksarecomputedatvariousspeedswithin26 

theoperatingspeeddistribution.27 

28 

Theremainderofthispaperisorganizedintofivesections.Thefirstprovidesbackground29 

informationonthepoint‐massmodelanddefinesseveralkeytermsusedthroughoutthe30 

manuscript.Thenextsectionprovidesbackgroundliteraturerelatedtofrictionconcepts,31 

whichwasusedtoinformthedatacollectionmethodology.Then,thedatacollectioneffort32 

andsitecharacteristicsaredescribedinthethirdsection.Thefourthsectiondescribesthe33 

resultsofthefrictionsupplyanddemandcomparisons.Finally,thepaperconcludeswitha34 

frameworktoreconsiderthehorizontalcurvedesignpolicyusedintheGreenBook.35 

36 

KEYTERMSANDHORIZONTALCURVEDESIGNCONCEPTS37 

38 

Thefollowingarealistofkeytermsanddefinitionsusedthroughoutthispaper:39 

40 

CentripetalAcceleration(ar):anobjectthatmovesinacircularpath(i.e.,horizontal41 

curve)withaconstantspeedfollowsapathtangenttothecurve.Becausethevelocity42 

vectorundergoesachangeindirection,theobject(i.e.,vehicle)undergoesanacceleration43 

perpendiculartothepathandtowardthecenterofthehorizontalcurve.Thecentripetal44 

2 

accelerationisequaltothesquareofthevehiclespeeddividedbytheradiusofthecircular1 

path.2 

3 

LateralAcceleration:atermusedbyhighwayengineersthatisequivalenttocentripetal4 

accelerationforthepurposesofhorizontalcurvedesign.5 

6 

SideFrictionSupply(ftire‐pavement):frictionavailablebetweenthepavementsurfaceand7 

vehicletirestopreventskiddingonahorizontalcurve,alsoreferredtoasthecoefficientof8 

friction.Themaximumsidefrictionsupplyisutilizedwhenavehicleisatthepointof9 

impendingskid.10 

11 

SideFrictionFactor(f):theunbalancedportionoflateralaccelerationortheportionof12 

lateralaccelerationthatisnotbalancedbysuperelevation.Thesidefrictionfactor13 

representsdemandsidefriction,andisalsoreferredtoasnetlateralaccelerationinthe14 

point‐massmodel.15 

16 

RolloverThreshold(frollover):maximumlateralaccelerationthatavehiclecanexperience17 

withoutoverturning.18 

19 

MaximumSideFriction(fmax):themaximumsidefrictiondemandsetforthinthe20 

AASHTOGreenBookforuseinhorizontalcurvedesign.Themaximumsidefrictionisbased21 

ondrivercomfortlevels(i.e.,toleranceforlateralacceleration),andisalsoreferredtoas22 

thelimitingsidefrictionfactor.23 

24 

LateralFrictionMargin:thedifferencebetweentheavailabletire‐pavementfrictionand25 

thefrictiondemandofthevehicleasittracksthecurve[i.e.,sidefrictionsupply(ftire‐26 

pavement)–sidefrictionfactor(f)].Thisfrictionmarginrepresentstheadditionallateral27 

accelerationthatavehiclecouldundergowithoutskidding.Apositivemarginindicatesa28 

vehiclecanundergoadditionallateralaccelerationwithoutskidding,whileanegative29 

marginindicatesthevehicletireswillskidgiventheleveloffrictionsuppliedbetweenthe30 

tireandpavementfortheconditioninquestion.31 32 

FromthebasiclawsofNewtonianphysics,considerapoint‐masstravelingalongacurved33 

roadwaywithaconstantradius(R)andaconstantvelocity(V),asshowninFigure1.The34 

pointmassundergoesacentripetalacceleration,whichactstowardsthecenterof35 

curvature.Thecentripetalaccelerationis:36 

37 

(1)38 

2

raV

R

3 

1 

Figure1.Point‐MassModel:VehicleTravelingOnaHorizontalCurve2 

3 

Theacceleration,assumingthatthepoint‐massisavehicle,isbalancedbythesidefriction4 

developedbetweenthepavementsurfaceandtires,thecomponentofvehicle'sweight5 

actingparalleltotheroadduetosuperelevation,oracombinationofboth,asshownin6 

Figure2.Letthebankingangleofroadwaybeα(radians.).Thesuperelevation(e)is7 

typicallydefinedbytherise(changeinelevation)infeetper100feetacrosstheroad(i.e.,8 

inthetransversedirection).Hence,e/100=tanα.Therearethreeforcesactingonthepoint9 

massasshowninFigure2:10 

11 

1. Normalreactionfromtheroad(N),12 

2. Thetire‐pavementfrictioncorneringforceactingattheroadtowardsthecenterof13 

therotation(Fc),and14 

3. Vehicleweight(W=mg;wheremisthemassandgisthegravitationalacceleration).15 

16 

Figure2.LateralForcesActingonPointMassduringCornering17 

18 

4 

Performingaforcebalanceinthey‐axisdirection(referringtoaxissystemshownin1 

Figure2),thefollowingresults:2 

(2)3 

Andinthez‐axisdirection:4 

(3)5 

Theforcecomponentsareasfollows:6 

7 

(4)8 

Expressionsforthefrictionfactorandsuperelevationare:9 

10 

(5)11 

Equation3canbesolvedformassbysubstitutingvaluesfromEquation4toobtain12 1 ∙ .SubstitutingthisintoEquation2,andthensimplifying13 

theresultbysubstitutingexpressionsfromEquation5,thefollowingresults:14 

15 

(6)16 

Rearrangingterms,thebasichorizontalcurveformularesults:17 

18 

(7)19 

Theproduct inthedenominatorisusuallysmallandisgenerallyignored.The20 

simplifiedformulacanbeusedtosolveforthecurveradiusasafunctionofthefriction21 

factor,thedesignspeed,andthesuperelevation.22 

23 

(8)24 

cyyy

yr

FWNR

Vm

Fam

2

czzzz FWNF0

,0,

cos,sin

sin,cos

yz

ccyccz

yz

WmgW

FFFF

NNNN

NFf c tan100

e

2 +tan 100=

1 . tan 1 . 100

efV fegR f f

2 +0.01=

1 0.01

V f e

gR fe

/100f e

2

R= ( +0.01 )

V

g f e

5 

Thelimitingfactorforroaddesignisthesidefrictionfactorf.Also,thesuperelevationrate1 

foracurvewillnotexceedamaximumvalueselectedbythedesigner,whichisoftenbased2 

onregionalclimate(i.e.,locationsexperiencingregularsnoworicyconditionsinthewinter3 

monthswilloftenhavelowermaximumratesofsuperelevationthanlocationsthathavea4 

warmerclimate).Hence,foragivendesignspeedofaroadway,practicallowerlimitson5 

theradiusofcurvature,Rmin,aregivenby:6 

7 

(9)8 

Here,fmaxisthemaximumdemandsidefrictionfactorusedinhorizontalcurvedesign,and9 

emaxisthemaximumsuperelevationrateforagivendesignspeed,VDS.AASHTOuses10 

Equation9fordeterminingtheminimumradiusofcurvature.Thisusageisgenerally11 

justifiedsinceitprovidesamoreconservativedesignthanEquation8.12 

13 

ThebasicsidefrictionformulacanbeobtainedbyrearrangingtermsinEquation8as14 

follows:15 

16 

(10)17 

InAASHTOpolicy,fiscalledthe“sidefrictionfactor”whichrepresentstheportionof18 

lateralaccelerationthatisnotbalancedbysuperelevation.Thetermfrepresentsafriction19 

“demand”whichmustberesistedbytheavailable“supply”offrictiongeneratedatthetire‐20 

pavementinterface.Inaddition,theunbalancedlateralaccelerationcreatesanoverturning21 

momentonthevehiclethatmustberesistedbythevehicle’srollstability,whichdepends22 

onvehicledesign,loading,andsuspensioncharacteristics.Theterm“sidefrictionfactor,”as23 

usedintheGreenBook,representsfrictiondemand,notfrictionsupply.24 

25 

AASHTOdesignpolicyforhorizontalcurvesisbasedontheassumptionthatthevalueoff26 

canbedeterminedasafunctionofvehiclespeed,curveradius,andsuperelevation.An27 

inherentassumptionisthatvehiclesfollowthecurvedpathexactly,yetthereisresearchto28 

suggestthatdriverswillelongateahorizontalcurvewhileothersmaydriveasmaller29 

radiusthantheactualcurveradius(WongandNicholson1992).30 

31 

Thetire‐pavementinterfacecansupplyfriction(ftire‐pavement)toresistthetendencyofthe32 

vehicletoskidduetolateralaccelerationasthevehicletraversesacurvedpath.The33 

pavementfrictiongeneratedatthetire‐pavementinterfaceisproportionaltothenormal34 

loadtransmittedtothetirethroughthevehiclesuspensionwhichdependsontireand35 

pavementproperties.Fromtheviewpointofapoint‐massmodel,thevehiclewillskidiff>36 

ftire‐pavement,whereftire‐pavementrepresentsthemaximumamountoffrictionthatcanbe37 

generatedatthetire‐pavementinterfacetocounteractlateralaccelerationandprevent38 

skidding.39 

40 

maxmax

2

min 01.0 efg

VR DS

eRg

Vf DS

01.0

2

6 

Similarly,fromtheviewpointofapoint‐massmodel,thevehiclewilloverturniff>frollover,1 

wherefrolloverrepresentsthemaximumlateralaccelerationthatavehiclecanexperience2 

withoutoverturning.frolloverisreferredtoasthe“rolloverthreshold”ofthevehicle.Rollover3 

thresholdsareacharacteristicofvehicledesignandloadingthatcanbeestimatedfrom4 

statictests,butarebestdeterminedfromdynamictests.5 

6 

TheGreenBookdesigncriteriaforhorizontalcurvesarenotbasedonanyformal7 

assumptionsaboutthemagnitudesofftire‐pavementandfrollover.Rather,horizontalcurve8 

designisbasedonlimitingthevalueofftobelessthanorequaltoaspecifiedvalue,fmax,9 

whichhasbeenselectedbasedondrivercomfortlevels(i.e.,drivertoleranceforlateral10 

acceleration).Afurtherassumption,statedbutnotexplicitlydemonstratedinAASHTO11 

policy,isthatthevaluesoffmaxusedindesignhavebeenselectedsuchthatfmax<ftire‐pavement12 

andfmax<frollover.Thefirstcriterion,fmax<ftire‐pavement,isaddressedinGreenBookFigures3‐13 

4and3‐5,whichshowsthatthevaluesoffmaxusedindesignarelessthanthevaluesofftire‐14 

pavement.Thesecondcriterion,fmax<frollover,isassertedbutnotdemonstratedintheGreen15 

Book.Researchbyothers,includingHarwoodetal.(1989)andHarwoodetal.(2003)have16 

shownthattheassumptionsoffmax<ftire‐pavementandfmax<frolloverdoappeartobegenerally17 

applicabletobothpassengervehiclesandtrucksforhorizontalcurvesdesignedin18 

accordancewithAASHTOpolicy.19 

20 

LITERATUREREVIEW21 

22 

Theliteraturereviewisorganizedintotwosections–thefirstdescribestire‐pavement23 

frictionstudiesandthesecondsectiondescribesdrivercomfortthresholdstudies.24 

25 

FrictionStudies26 

27 

Thebasicsidefrictionformula(Equation10)providesanestimateofthesidefrictionfora28 

vehiclemaneuveronahorizontalcurve.Oneoftheearlieststudiesonmeasuringthe29 

coefficientoffriction(ftire‐pavement)atthepointofimpendingskidonaroadwaywasdoneby30 

Moyer(1934).Theauthorfoundthat,ondryconcretesurfaces,theskiddingside(lateral)31 

frictionlimitsrangedfrom1.01at5mphto0.89at30mphondryconcretepavements.On32 

wetconcretesurfaces,thelimitsofsideskiddingfrictionrangedfrom0.78at5mphto0.6433 

at30mph.34 

35 

Fancheretal.(1986)developedvaluesforthecoefficientofroadadhesion(ftire‐pavement)for36 

differenttrucktiresondryandwetconcretepavementsat40mphspeeds.Theaverage37 

skiddingfrictionwas0.54ondrysurfacesand0.47onwetsurfaces.Theaveragepeak38 

frictiononadryconcretepavementwas0.76,whiletheaveragepeakfrictiononawet39 

concretepavementwas0.64.40 

41 

Ithasalsobeenobservedthatfordrypavementsthereisnosignificantchangeinthetire‐42 

pavementfrictionasspeedsincrease.Butthereisanoticeablechangeinfrictiononwet43 

7 

surfaces.Onwetpavements,themaximumandslidingfrictionhasbeenshown(Wong,1 

2008)todecreaseby30percentormorebetween20and60mphforworntires.2 

3 

Finally,Lammetal.(1999)reportedthatlateralfrictionatthetire‐pavementinterfaceis4 

about7.5percentsmallerthanthelongitudinalfriction.Thisinterrelationshipbetween5 

lateralandlongitudinalforcesisreferredtoasthefrictionellipse,whichisdescribedin6 

detailbyGillespie(1992).7 

8 

ComfortThresholds9 

10 

AkeyconsiderationinAASHTO'spolicyinselectingmaximumsidefrictionfactors(fmax)for11 

useindesignistheleveloflateralaccelerationsufficienttocausedriverstoexperiencea12 

feelingofdiscomfortandtoreactinstinctivelytoavoidhigherspeeds.Itisassumedthatat13 

lowspeeds,driversaremoretoleranttodiscomfortandhencehighervaluesofsidefriction14 

aresought.ThisassumptionisbasedonresearchconductedbyBarnett(1936),who15 

definedthesafespeedas“...theminimumspeed,atwhichthecentrifugalforce,createdby16 

themovementofthevehiclearoundthecurve,causesthedriverorpassengertofeelaside17 

pitchoutward.”Inthiswork,900roadtestreportswerestudiedandthesidefriction18 

factorwasobservedtolieinthe0.10‐0.20range.Barnettassumedatrendoftheside19 

frictionfactorof0.16forspeedsof60mphandless,anda0.01decreaseforeach5mph20 

furtherincreaseinthespeed.21 

22 

Theballbankindicatoristypicallyusedtomeasurelateralaccelerationstosetthedesign23 

speedsonthecurvesthatwillavoiddiscomfort.Oneoftheearliestball‐bankindicator24 

studieswasdonebyMoyerandBerry(1940).Theyrecommendedmaximumdesignside25 

frictionfactorsof0.21forspeedsof20mphorlower,0.18forspeedsof25and30mph,26 

and0.15forspeedsequaltoorgreaterthan35mph.27 

28 

29 

Meyer(1949)recommendedthatagreatermarginofsafetyshouldbeusedathigher30 

speedsthansuggestedbycomfortconsiderationsalone,recommendinganexponential31 

curvetypevariationforsidefrictionvs.speeddata.StonexandNoble(1940)performed32 

highspeedtestsonthePennsylvaniaTurnpike,recruitingskilleddriverstodetermineside33 

frictionvaluesforuseindesign.Theauthors’recommendedamaximumsidefriction34 

factorof0.10forhorizontaldesign.35 

36 

ItisunclearwhichofthestudieswereusedtodevelopfmaxvaluesintheGreenBook.The37 

trendofsidefrictionfactor(f)vs.designspeed(V)showninFigure3‐6ofthemostcurrent38 

policy(AASHTO,2011)indicatesthat,forspeedsabove45mph,themaximumdesignside39 

frictionfactorisdecreasedataconstantrate,byavalueof0.01foreachadditional5mph40 

asrecommendedbystudiesfromthe1940’s.Also,thecurvepassesthroughavalueof0.1041 

at70mphasrecommendedbyStonexandNoble(1940).Forlow‐speeddesign(45mphor42 

less),theAASHTOGreenBookmaximumsidefrictionfactors(fmax)usedindesignare43 

basedonintersectioncurvefrictionconceptsdevelopedinthe1940’sand1950’s44 

(Cysewski,1949;George,1952),whichindicatethatdrivercomfortthresholdsare45 

8 

significantlyhigheratlowspeedswhencomparedtohightravelspeeds.Wheneachof1 

thesestudiesisconsideredinaggregate,itislikelythattheyhavecollectivelyshapedGreen2 

Bookmaximumsidefrictionfactorsusedinhorizontalcurvedesign.3 

4 

DATACOLLECTIONANDANALYSISMETHODOLOGY5 

6 

Thissectionofthepaperdescribesthedatacollectionandanalysismethodsusedto7 

developcomparisonsbetweenfrictionsupply(ftire‐pavement)andfrictiondemandorthe8 

lateralfrictionfactor(f),whichwasderivedfromvehicleoperatingspeeddata.9 

10 

DataCollection11 

12 

FrictionMeasurements13 

14 

Frictionsupplydata(ftire‐pavement)werecollectedontheapproachtangentandwithinthe15 

limitsofahorizontalcurvealong7two‐laneruralhighwaysinWestVirginia(referto16 

Boodlaletal.,2014forcompletestudy)andalong8multi‐lanerural,dividedhighwaysin17 

MarylandandWestVirginia(seeTorbicetal.,2014forcompletestudy).Allpavement18 

surfaceswereasphaltcementconcrete,andweresubjectivelyratedasbeinginfairto19 

excellentcondition.Thefrictiondatawerecollectedusingacombinationofaportable20 

dynamicfrictiontester(DFTester)andcirculartexturemeter(CTmeter).Alltestingusing21 

theDFTesterwasdoneinaccordancewithAmericanSocietyforTestingandMaterials22 

(ASTM)standardE1911‐09a(ASTM,2009).Figure3isaphotographoftheDTTester.The23 

testmethodproducespavementsurfacefrictionpropertiesasafunctionofspeed.Testing24 

involvesspinningadisk,fittedwiththreespring‐loadedrubbersliders,ataninitialspeed25 

of55mph.Thediskrotationalspeeddecreasesasaresultoffrictionbetweenthesliders26 

andpavedsurface.Waterisdeliveredtotheunitsuchthata0.04‐inchwetfilmcoatsthe27 

pavedsurfacewhenthemeasurementsareinitiated.TheoutputfromthetestisaDF28 

Testernumberat20km/h(12mph)andawet‐pavementspeedconstant(Sp),whichcan29 

beusedtocomputetheInternationalFrictionIndex(IFI).ThedataoutputfromtheDF30 

Testerwasreportedinmetricunits,sothesedatawereconvertedtoU.S.unitstobe31 

consistentwiththeoperatingspeeddata.32 

33 

Togeneratefrictionvs.speedplots,thepavementmacrotexturewasdeterminedusingaCT34 

meter.Thisdeviceusesalasersensortodevelopprofilesofthepavementsurfaceonthe35 

samecircumferenceastheDFTester(Saitoetal.,2001).Theoutputfromthisdeviceisthe36 

meanprofiledepth(MPD)ofthemeasuredsurface.37 

38 

9 

 1 

Figure3.DynamicFrictionTester2 

3 

Collectively,theDFTesterandCTMeterdatawereusedtodeterminethewetpavement4 

frictionforall15datacollectionlocations.ASTME1960‐07(ASTM2011)recommendsthe5 

followingstandardpracticetocalculateIFIofapavementsurface:6 

7 

MPDS p 7.892.14 (11)8 

9 

)/40exp(732.0081.060 20 pSDFTF (12)10 

11 

]/)60exp[(60 pSSFFS (13)12 

13 

where: Sp=speedconstantofwetpavementfriction14 

MPD=meanprofiledepth(mm)15 

F60=calibratedwetfrictionat60km/h16 

DFT20=DFTnumberat20km/hperASTME191117 

FS=frictionatanotherslipspeedS18 

19 

Thefrictionformulainequations(11)through(13)applytospeedsuptoapproximately20 

50mph,thusvaluescomputedabovethisareextrapolationsabovethetypicalbounds.All21 

frictionmeasurementswerecollectedatalocationthatcorrespondedtotheoutsidetire22 

pathonhorizontalcurves.Thislocationisgenerallyconsideredtobewherethepavement23 

ismostpolishedand,therefore,willsupplylessfrictionthantheinsidewheelpath.A24 

typicalfrictionmeasurementprotocolisshowninFigure4.Onmulti‐lane,divided25 

highways,theapproachtangenttothecurveincludedthreemeasurementlocations,while26 

thecurveitselfincludedfourmeasurementlocations(PC,one‐thirdcurvelength,two‐27 

thirdscurvelength,andPT).Threemeasurementswererecordedateachlocation,soa28 

totalof12frictionmeasurementswererecordedwithinthecurvelimitsonmulti‐lanerural29 

highways.Nofrictiondatacollectedontheapproachtangentwereusedinthisstudy.30 

31 

Ontwo‐laneruralhighways,thesamefrictionmeasurementprotocolwasused,onlytwo32 

measurementswererecordedateachlocation.Assuch,therewereatotalof8friction33 

10 

measurementsrecordedwithinthelimitsofasinglehorizontalcurve.OncetheDFTester1 

andCTMeterdatawererecorded,equations(11)through(13)wereusedtocomputethe2 

frictionsupplyatalltwo‐laneruralandmulti‐lane,dividedruralhighways.Equation(13)3 

wasusedtocomputethefrictionsupplyformostoftheGreenBookdesignspeed4 

continuum(25to80mph).Lowerspeedswerenotconsideredastheoperatingspeeddata5 

collectedinthefield(seebelow)weregenerallynotlowerthan25mph.6 

7 

8 Figure4.FrictionMeasurementLocations.9 

10 

OperatingSpeedMeasurements11 

12 

Onthemulti‐lanedividedhighways,operatingspeeddataforfree‐flowpassengercarsand13 

truckswererecordedontheapproachtangentandthroughthebeginningofthehorizontal14 

curveusinglaserguns.Free‐flowvehiclesweredefinedasthosehavingtimeheadwaysof15 

atleast5seconds.Allspeeddatacollectedusinglasergunswerepost‐processedsothat16 

onlyspeedsmeasuredwithinthelimitsofthehorizontalcurvewereusedtocompute17 

frictiondemand(f)data.Atthetwo‐laneruralhighwaylocations,datawerecollected18 

usingNu‐metricsHi‐staron‐pavementsensors.Thesensorswerepositionedatthemid‐19 

pointofeachhorizontalcurve.Passengercarandtruckspeedswereidentifiedusing20 

informationaboutthewheelbaseofthevehicles,whilethetimestampdatawereusedto21 

identifyfree‐flowvehicles(headwaysofatleast5seconds).22 

23 

11 

SiteCharacteristicData1 

2 

Inadditiontothefrictionandspeeddata,sitecharacteristicdatawerecollectedusing3 

recorddrawingsandfieldmeasurements.Thesitecharacteristicdataforallstudysitesare4 

showninTable1.Mostsiteswereondowngradeswithpostedspeedlimitsof55mphor5 

higher.Severalcurveshadcurveadvisoryspeedwarningsignsandtheradiiranged6 

considerablyamongthesites.Allofthehorizontalcurvesweresuperelevatedatleast4.57 

percent,whilemanyhadsuperelevationratesofatleast7percent.8 

9 

Table1.SiteCharacteristicData.10 

11 

SiteRoute(direction)

CountyGrade(%)

CurveRadius(ft)

PostedSpeedLimit(mph)

AdvisorySpeed(mph)

Max.Super(%)

CurveDirection

MD1* I‐68(WB) Garrett ‐4.1 1,909 65 None 6 LeftMD2* I‐68(WB) Washington +6.0 1,909 65 None 5.5 RightMD3* I‐68(WB) Washington ‐5.7 1,900 65 None 4.5 RightWV1* I‐77(SB) Mercer ‐4.9 1,206 70 50 8 LeftWV2* I‐68(WB) Monongalia ‐5.7 1,909 70 50 7.8 LeftWV3* I‐79(SB) Kanawha ‐3.7 1,146 70 50 8 LeftWV4* I‐77(NB) Kanawha ‐5.2 1,041 60 50 8 RightWV5* I‐64(EB) Kanawha ‐5.0 1,637 60 None 7.2 LeftWV6 WV32 Randolph ‐1.7 1073 55 50 7 RightWV7 WV32 Randolph ‐0.2 680 55 40 8 RightWV8 US33 Pendleton ‐8.0 192 55 25 8 LeftWV9 US33 Pendleton ‐7.5 274 55 25 8.5 LeftWV10 US219 Randolph ‐2.0 605 55 30 11 RightWV11 US219 Randolph ‐3.0 273 55 25 12 RightWV12 US219 Randolph ‐6.5 545 55 30 12 Right

*Multi‐lane,DividedHighway

12 

13 

AnalysisMethodology14 

15 

Frictionsupplyvalueswerecomputedusingequations(11)through(13)fordesignspeeds16 

rangingfrom25to80mph.Separatefrictionsupplycurvesforruraltwo‐laneandmulti‐17 

lanehighwaysweregenerated,aswereseparatecurvesforpassengercarsandtrucks.It18 

shouldbenotedthattheDFTester(ASTM,2009)iscapableofprovidingfriction19 

measurementatspeedsupto55mph,sofriction‐speedcomputationsabovethislevelwere20 

extrapolatedbeyondthefield‐measuredvalues.21 

22 

Thepoint‐massmodel(seeequation[10])wasusedtodevelopfrictiondemandcurvesfor23 

thetwo‐laneandmulti‐lanehighwaysites.Insteadofusingthedesignspeed(VDS)to24 

computethesidefrictiondemand(f),eachindividualvehicleoperatingspeedwasused.25 

TheradiusofcurveandsuperelevationdatainTable1werealsousedtocomputetheside26 

frictiondemand.Asummaryofthepassengercarandtruckoperatingspeeddataforeach27 

siteisshowninTable2.28 

29 

30 

12 

Table2.PassengerCarandTruckOperatingSpeedData.1 

2 

SitePassengerCarObservations

TruckObservations

PassengerCarCurveMeanSpeed

(mph)

PassengerCarCurveSpeedDeviation(mph)

TruckCurveMeanSpeed(mph)

TruckCurveSpeed

Deviation(mph)

MD1* 66 47 64.9 4.8 60.9 3.7MD2* 34 45 60.3 8.7 37.4 11.9MD3* 66 30 67.7 5.4 63.8 5.5WV1* 65 45 66.6 5.6 61.5 5.3WV2* 66 30 67.3 5.6 54.2 6.3WV3* 97 50 66.7 5.3 63.0 4.9WV4* 72 50 61.9 5.3 56.7 3.8WV5* 24 38 66.3 4.5 64.2 3.5WV6 176 10 52.7 7.3 49.0 3.8WV7 234 7 47.9 5.5 34.5 4.7WV8 221 21 32.1 4.4 26.8 9.0WV9 197 22 36.1 6.6 34.5 6.1WV10 122 11 36.4 7.4 34.5 4.7WV11 112 14 34.5 6.4 32.4 3.6WV12 68 8 41.4 4.6 39.4 4.0

*Multi‐laneHighway

3 

Thedifferencebetweenthesidefrictiondemand(f),computedusingtheoperatingspeed4 

datainthepoint‐massmodel,andthesidefrictionsupply(ftire‐pavement)fromtheDFTester5 

andCTMeter,wascomputedforeachvehicletypeandroadwaytype.Thelateralmargins6 

werethenassessedfromthesecomparisons.7 

8 

RESULTS9 

10 

Thefrictiondatausedinthisanalysiswerecollectedintwoseparateprojects(Boodlalet11 

al.,2014;Torbicetal.,2014).Atotalof56frictionmeasurementsfortwo‐laneroadsand12 

96frictionmeasurementsformulti‐laneroadswereusedtocomputethemeanand13 

standarddeviationofthefrictionsupply(ftire‐pavement)withinthelimitsofahorizontal14 

curve.AsummaryoftheSp,MPD,DFT20,andavailablelateralfrictionvalues(see15 

equations(11)through(13))forvariousspeedsareprovidedinTable3.Thelateral16 

frictionwasestimatedas0.925timesthelongitudinalfrictionperLammetal.(1999).17 

18 

Sidefrictiondemand(f)wascomputedforpassengercars(includingvansandpick‐up19 

trucks)andlargetrucks(i.e.,tractorsemi‐trailers)onbothtwo‐laneandmulti‐lane20 

highways.SeveralfrictioncomparisonsareshownforpassengercarsinFigure5,andfor21 

largetrucksinFigure6.Ineachfigure,thefollowingfrictionvaluesareshown:22 

23 

Themeanand10th‐percentilewet‐pavementfrictionsupplyfromtheDFTesterand24 

CTMetermeasurements.Separateplotsfortwo‐laneandmulti‐lanehighwaysare25 

shownasthepavementsurfacequalityvariedbetweenthesesites.26 

TheAASHTOGreenBook(2011)maximumsidefrictionfactorsfordesign.27 

Thelargetruckrolloverthreshold(frollover)inFigure6.28 

13 

Pointestimatesofthemeanand90th‐percentilefrictiondemandsbasedonobserved1 

operatingspeeds.2 

Twofrictiondemandcurves,whicharebasedonthedesignspeedplus5mph,at3 

minimumradiiforsuperelevationratesof2%and8%.Thesecurvesarethebest4 

approximationofalinetofittheobservedfrictiondemandpoints.5 

6 

Table3.FrictionData.7 

8 

VariableTwo‐Lane Multi‐Lane

Mean St.Dev. Mean St.Dev.Sp 86.319 24.674 88.273 30.143MPD 0.804 0.275 0.826 0.336DFT20 0.443 0.059 0.514 0.072

Speed(mph)LateralFriction(Two‐Lane) LateralFriction(Multi‐Lane)Mean St.Dev. Mean St.Dev.

25 0.439 0.385 0.495 0.40030 0.396 0.348 0.448 0.37035 0.358 0.313 0.405 0.34140 0.324 0.279 0.366 0.31145 0.293 0.249 0.332 0.28250 0.266 0.221 0.300 0.25455 0.242 0.197 0.274 0.22860 0.219 0.173 0.247 0.20265 0.199 0.153 0.224 0.17870 0.181 0.136 0.204 0.15775 0.165 0.120 0.185 0.13780 0.15 0.106 0.168 0.120

9 

FromFigures5,themeanfrictionsupplyonmulti‐lanehighwaysrangesfrom10 

approximately0.5at25mphtoapproximately0.17at80mph.The10th‐percentilefriction11 

supplycurveformulti‐lanehighwaysrangesfrom0.40at25mphtoabout0.12at80mph.12 

Ontwo‐laneruralhighways,themeanfrictionsupplyrangesfrom0.44at25mphtoabout13 

0.15at80mph,whilethe10th‐percentilefrictionsupplyontwo‐laneruralhighwaysranges14 

from0.37at25mphto0.09at80mph.Thefrictionsupplyvaluesarebasedonthefriction15 

measurementprotocoldescribedabove,whichisrepresentativeofwetpavement16 

conditions.Assuch,thefrictionsupplymeasuresondrypavementsurfaceswillexceedthe17 

curvesshowninFigure5.18 

19 

Thefrictiondemandatthedesignspeedplus5mph,withasuperelevationof2percent,20 

rangesfrom0.40atadesignspeedof25mphto0.10atadesignspeedof80mph,whichis21 

closetothe10th‐percentilefrictionsupplyontwo‐laneruralhighwaysinthepresentstudy.22 

Atthesamespeed(designspeedplus5mph),with8percentsuperelevation,thefriction23 

demandcurverangesfrom0.43atadesignspeedof25mphto0.11atadesignspeedof8024 

mph,whichissimilartothe10th‐percentilemulti‐lanehighwayfrictionsupplyatlow25 

speeds(designspeedslessthan35mph)andathighspeeds(designspeedsgreaterthan6526 

mph).Thefrictiondemandatthedesignspeedplus5mphwith8percentsuperelevation27 

isclosetothe10th‐percentiletwo‐laneruralhighwayfrictionsupplycurvebetweendesign28 

speedsof35and65mph.TheAASHTOmaximumsidefrictionfactorsareatleast0.0529 

belowthedemandfrictionvaluesatdesignspeedsupto40mph,butthemarginsbetween30 

14 

theAASHTOmaximumsidefrictionanddemandfrictioninthepresentstudyarenear1 

equalatthehighestspeedsshowninFigure5.Thissuggeststhatpassengercardrivers2 

travelnearthecomfortthresholdsusedintheAASHTOGreenBookhorizontalcurvedesign3 

policyathigherdesignspeeds,butexceedcomfortthresholdsatlowerspeeds.4 

5 

ThetruckfrictionsupplyanddemandcurvesshowninFigure6aresimilartothosefor6 

passengercarsinFigure5.Fordesignspeedslessthan45mph,therolloverthresholdfor7 

largetrucksislowerthanthemeanfrictionsupplycurves,butexceedsthe10th‐percentile8 

frictionsupplycurvesontwo‐laneandmulti‐lanehighways.Thefrictiondemandatthe9 

designspeedplus5mph,witheightpercentsuperelevation,exceedsthelargetruck10 

rolloverthresholdatdesignspeedslessthan30mph.Thefrictiondemandatthedesign11 

speedplus5mph,with2percentsuperelevation,isnearlyequaltothelargetruckrollover12 

thresholdatdesignspeedslessthan30mph.TheAASHTOmaximumsidefrictionfactors13 

fordesignareslightlylowerthanthedemandfrictionvaluesathighspeeds,butare14 

significantlylowerthanthedemandfrictionvaluesatlowerspeeds.15 

16 

Themeanfrictionsupplyonmulti‐lanehighwaysshowninFigure6exceedsthetruck17 

rolloveratdesignspeedsequaltoorlowerthan35mph.Ontwo‐laneruralhighways,the18 

meanfrictionsupplyexceedsthetruckrolloverthresholdatdesignspeedsof30mphor19 

lower.The10th‐percentilefrictionsupplyontwo‐andmulti‐lanehighwaysisbelowthe20 

truckrolloverthresholdatalldesignspeedsshowninFigure6.Thelateralmargins21 

againstrolloverforlargetrucksrangefromapproximately0.10at35mphto22 

approximately0.25at65mph,whencomparingthefrictiondemand(f)curvestothe23 

rolloverthreshold.24 

25 

26 

15 

1 Figure5.FrictionSupplyandDemandforPassengerCars.2 

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

25 35 45 55 65 75

Side Friction

Design Speed

Passenger Cars

Observed Mean Demand

Observed 90th Percentile Demand

Two‐Lane Friction Supply (Mean)

Two‐Lane Friction Supply (10th Percentile)

Multi‐Lane Friction Supply (Mean)

Multi‐Lane Friction Supply (10th Percentile)

AASHTO Maximum Friction

Demand @ Design Speed + 5 mph with 2% Superelevation

Demand @ Design Speed + 5 mph with 8% Superelevation

16 

1 Figure6.FrictionSupplyandDemandforLargeTrucks.2 

3 

4 

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

25 35 45 55 65 75

Side Friction

Design Speed

Large Trucks

Observed Mean Demand

Observed 90th Percentile Demand

Two‐Lane Friction Supply (Mean)

Two‐Lane Friction Supply (10th Percentile)

Multi‐Lane Friction Supply (Mean)

Multi‐Lane Friction Supply (10th Percentile)

Large Truck Rollover Threshold

AASHTO Maximum Friction

Demand @ Design Speed + 5 mph with 2% Superelevation

Demand @ Design Speed + 5 mph with 8% Superelevation

17 

CONCLUSIONSANDRECOMMENDATIONSFORFUTUREWORK1 

2 

Conclusions3 

4 

Thispaperdescribesvariousfrictionconceptsusedinhorizontalcurvedesign,and5 

comparesthelateralfrictionmarginsforvariousvehicletypesandoperatingspeedsto6 

frictionsupplycurvesdevelopedbasedonfieldmeasurementsontwo‐laneandmulti‐lane7 

ruralhighways.Itwasfoundthatdriverstravelatspeedsthatnearlyapproximate8 

AASHTOmaximumsidefrictionfactorsonruralhighwayswithhighdesignspeeds(i.e.,9 

greaterthan45mph).Atlowerdesignspeeds,however,theobservedfrictiondemandof10 

driversinthepresentstudyoftenexceededtheAASHTOmaximumsidefrictionfactors11 

usedinhorizontalcurvedesign.Itwasfoundthattheobservedfrictiondemand,which12 

werebasedonspeedscollectedondrypavementconditions,wasgenerallyatleast0.0513 

belowmeanfrictionsupplycurves(basedonwetpavementconditions)forpassengercars14 

ontwo‐laneandmulti‐lanehighwaysincludedinthepresentstudy,exceptatlowdesign15 

speeds,whentheobserved90th‐percentiledemandfrictionvaluesinsomecaseswere16 

coincidentwithmeanfrictionsupplyvalues.Observeddemandfrictionvalueswerealmost17 

alwaysatleast0.05belowmeanfrictionsupplycurvesatalldesignspeedsforlargetrucks18 

ontwo‐andmulti‐lanehighwaysinthepresentstudy.Atlowdesignspeeds(35mphor19 

less),itappearsthatlargetrucksaremorelikelytorolloverthanskidwhenfriction20 

demandexceedsfrictionsupply.21 

22 

Thefrictionsupplylevelsmeasuredinthefieldindicatethatfrictionvariesbetweenrural23 

two‐laneandmulti‐lanehighways.Itisassumedthatthisvariabilityistheresultof24 

pavementsurfacedifferences.Passengercarsaremorelikelytoskidbeforerollingoveron25 

horizontalcurves,whilelargetrucksaremorelikelytorolloveronhorizontalcurvebefore26 

skiddingonlow‐speedroads.27 

28 

RecommendationsforFutureWork29 

30 

Thereareseverallimitationsofthepoint‐massmodelinhorizontalcurvedesign.These31 

include:32 

33 

themodeldoesnotaccountfordifferencesinvehicledynamicsbetweenpassenger34 

vehiclesandtrucks,andthemodelignorestireforcedifferencesbetweenthe35 

front/rearorleft/righttiresofavehicle(i.e.,theforcesactingonalltiresare36 

assumedtobethesame).37 

themodelignoresthecombinedcharacteristicsofthehighwayalignmentsuchthat38 

thehorizontalalignmentisdesignedinisolationwithoutaccountingforthe39 

overlappingverticalalignment.40 

thepoint‐massmodelassumesthatvehiclestraversecurvesfollowingapathof41 

constantradiusequaltotheradiusofthecurve;however,itisunlikelythatvehicles42 

willsteerthecurveinaconstantradius.43 

thepoint‐massmodelassumesvehiclestraversethecurveataconstantspeedand44 

doesnotconsiderspeedchanges.45 

18 

1 

Futureresearchshouldconsiderhowthelateralmarginsofsafetyagainstskiddingand2 

rolloverchangewhenconsideringmodelsthatconsiderthelimitationsofthepoint‐mass3 

model.Examplesofalternativehorizontalcurvedesignmodelsincludethemodifiedpoint‐4 

mass,whichconsidersverticalgrades;bicyclemodel(treatseachaxleofavehicle5 

separately);or,amulti‐bodymodel(treatseachtireseparately).Futureresearchisalso6 

recommendedtodeveloppavement‐frictionperformancecurves.Amethodologyto7 

comparefrictiondemandtotheavailablesupplyisanalternativemethodtoassessthe8 

lateralmarginsagainstskiddingandrollover.Finally,thelateralmarginsforpassenger9 

carsandlargetrucksaresmallatlowdesignspeeds,thusfutureworkisneededto10 

determineifrevisedgeometricdesigncriteriaforhorizontalcurvesareneededtolimitthe11 

probabilityofskiddingorrolloveronsmallradiicurves.Thefrictiondemandobservedat12 

highdesignspeedsapproximatelymatchedAASHTOmaximumsidefrictionfactors.Thus,13 

thecomfortthresholdsusedtoestablishhigh‐speedhorizontalcurvedesigncriteriaappear14 

tocloselyapproximatedriverspeedchoice.Futurefrictionassessmentsshouldseekto15 

considerthedifferencesbetweenpassengercarandtrucktires,aswellaswetversusdry16 

pavementdemandandsupplyconditions. 17 

19 

REFERENCES1 

2 

AmericanAssociationofStateHighwayandTransportationOfficials.APolicyonGeometric3 

DesignofHighwaysandStreets,Washington,D.C.,2011.4 

5 

American Society for Testing andMaterials. StandardTestMethod forMeasuringPaved6 

SurfaceFrictionalPropertiesUsingtheDynamicFrictionTester.ASTMInternationalE1911‐7 

09a,WestConshohocken,PA,2009.8 

9 

AmericanSocietyforTestingandMaterials.StandardPracticeforCalculatingInternational10 

FrictionIndexofaPavementSurface. ASTMInternationalE1960‐07,WestConshohocken,11 

PA,2011.12 

13 

Barnett, J. SafeSideFrictionFactorsandSuperelevationDesign.HighwayResearchBoard14 

16,Washington,D.C.,1936.15 

16 

Boodlal,L.,E.T.Donnell,R.J.Porter,D.Garimella,T.Le,K.Croshaw,S.Himes,P.Kulis,17 

andJ.Wood.FactorsAffectingSpeedandSafetyonRuralandSuburbanRoads.Report18 

No. KLS‐14‐001, Federal Highway Administration, McLean, VA, July 2014, 359 pp.19 

(reportinpress)20 

21 

Cysewski,G.R. Urban IntersectionalRightTurningMovements. TrafficEngineering,Vol.22 

20,No.1,October1949,pp.22‐37.23 

24 

Fancher, P.S., R.D. Ervin, C.B.Winkler, and T.D. Gillespie. AFactBook of theMechanical25 

PropertiesoftheComponentsforSingle‐UnitandArticulatedHeavyTrucks.ReportNo.DOT26 

HS 807 125, National Highway Traffic Safety Administration, U.S. Department of27 

Transportation,Washington,D.C.,1986.28 

29 

Gillespie,T.FundamentalsofVehicleDynamics. SAE InternationalPress,Warrendale,PA,30 

1992.31 

32 

George,L.E. CharacteristicsofLeft‐turningPassengerVehicles. HighwayResearchBoard33 

31,Washington,D.C.,1952,pp.374‐385.34 

35 

Harwood,D.W.andJ.M.Mason.HorizontalCurveDesignforPassengerCarsandTrucks.36 

TransportationResearchRecord 1445, Transportation Research Board,Washington, D.C.,37 

1994.38 

39 

Harwood, D. W., J. M. Mason, W. D. Glauz, B. T. Kulakowski, and K.Fitzpatrick, Truck40 

Characteristics for Use inHighway Design and Operation, Vol.I. Research Report. FHWA41 

ReportFHWA‐RD‐89‐226,Washington,D.C.,1989.42 

43 

20 

Harwood,D.W.,D.J.Torbic,K.R.Richard,W.D.Glauz,andL.Elefteriadou,NCHRPReport1 

505:ReviewofTruckCharacteristicsasFactorsinRoadwayDesign.TransportationResearch2 

Board,Washington,D.C.,2003.3 

4 

Lamm,R.,B.Psarianos,andT.Mailaender.HighwayDesignandTrafficSafetyEngineering5 

Handbook.McGrawHillHandbooks,NewYork,NY,1999.6 

7 

Meyer,C.F.RouteSurveying,FirstEdition.InternationalTextbookCo.,Scranton,PA,1949.8 

9 

Moyer, R.A. Skidding Characteristics of Automobile Tires on Roadway Surfaces and Their10 

Relation toHighway Safety. Bulletin No. 120, Ames, Iowa, Iowa Engineering Experiment11 

Station,1934.12 

13 

Moyer,R.A.andD.S.Berry.MarkingHighwayCurveswithSafeSpeedIndicators.Highway14 

ResearchBoard20,Washington,D.C.,1940.15 

16 

Saito,K.,A.Tamai,S.Kameyama,andS.Nisiyama.DevelopmentofTestersforMeasuring17 

SkidResistanceandTextureofPavedSurfaces,andtheirApplicationforDeterminationof18 

theInternationalFrictionIndex(IFI).JournaloftheEasternAsiaSocietyforTransportation19 

Studies,Vol.4,No.1,October2001,pp.397‐411.20 

21 

Stonex,K.A.andC.A.Noble.CurveDesignandTestsonthePennsylvaniaTurnpike.Highway22 

ResearchBoard20,HighwayResearchBoard,Washington,D.C.,1940.23 

24 

Torbic,D.J.,M.K.O’Laughlin,D.W.Harwood,K.M.Bauer,C.M.Bokenkroger,L.M.25 

Lucas,J.R.Ronchetto,S.N.Brennan,E.T.Donnell,A.Brown,andT.Varunjikar.26 

NCHRPReport774:SuperelevationCriteriaforSharpHorizontalCurvesonSteep27 

Grades.NationalCooperativeHighwayResearchProgram,Washington,DC,2014.28 

29 

Wong,J.Y.,TheoryofGroundVehicles,JohnWileyandSons,Inc.2008.30 

31 

Wong,Y.andA.Nicholson.DriverBehavioratHorizontalCurves:RiskCompensationand32 

theMarginofSafety.AccidentAnalysisandPrevention,Vol.24,No.4,1992,pp.425‐436.33