USArmyCorpsofEngineers

AfghanistanEngineerDistrict

AEDDesignRequirements:

VerticalCurveDesign

VariousLocations,Afghanistan

MARCH2009

TABLEOFCONTENTS

AEDDESIGNREQUIREMENTSFOR

VERTICALCURVEDESIGNVARIOUSLOCATIONS,AFGHANISTAN

SectionPage

1. General1

2. VerticalCurves1

3. CrestVerticalCurveStoppingSightDistance1

4. CrestVerticalCurvePassingSightDistance3

5. SagVerticalCurves5

6. DesignConsiderations7

7. As-Builts7

Exhibits

Exhibit1. TypesofVerticalCurves1

Tables

Table1. StoppingSightDistance1

Table2. DesignControlforStoppingSightDistanceandforCrestVerticalCurves 2

Table3. PassingSightDistanceforDesignofTwo-LaneHighways3

Table4. DesignControlsforCrestVerticalCurvesBasedonPassingSight4

Distance

Table5. DesignControlforSagVerticalCurves6

1.General

ThepurposeofthisdocumentistoproviderequirementstoContractorsforanyprojectrequiringthedesignandconstructionofverticalcurveroaddesign.

2.VerticalCurves

Verticalcurvesareparaboliccurvesusedtoachieveagradualchangebetweentangentgrades(G1and

G2)andmaybeeitheracrestcurveorasagcurveasshowninExhibit1.

Exhibit1.TypesofVerticalCurves

3.CrestVerticalCurvesStoppingSightDistance

Themajorcontrolforsafeoperationonacrestverticalcurveisthesightdistancerequired.Attheminimum,thestoppingsightdistancefortheroaddesignspeedprovidedinTable1shouldbeprovidedforallcrestverticalcurves.Whereverpractical,largerstoppingsightdistancesshouldbeused.

Table1.StoppingSightDistance

Equations3-1and3-2providethegeneralequationsforcalculatingtheminimumlengthofcrestverticalcurvesbasedontherequiredsightdistanceandthealgebraicdifferenceingrade.Equation3-1istobeusediftherequiredsightdistanceislessthanthelengthoftheverticalcurveandEquation3-2istobeusediftherequiredsightdistanceisgreaterthanthelengthoftheverticalcurve.

Equation1L=(AS)2/100((2h1)1/2+(2h2)1/2)2(S<L)

1/2

1/22

Equation2L=2S-(200(h1

+h2

))/A)(S>L)

Where:

L=lengthofverticalcurve(m)S=sightdistance(m)

A=algebraicdifferenceingrades(%)h1=heightofeyeaboveroadwaysurface(m)h2=heightofobjectaboveroadwaysurface(m)

Whentheheightoftheeyeandtheheightoftheobjectare1.08metersand0.6metersrespectively,asusedforstoppingsightdistance,generalequation3-1and3-2becometherequirescrestcurvelengthforstoppingsightasshowninequations3-3and3-4respectively.

Equation3L=AS2/658(S<L)Equation4L=2S-(658/A)(S>L)

Where:

L=lengthofverticalcurve(m)S=sightdistance(m)

A=algebraicdifferenceingrades(%)

Therateofverticalcurvature(K)isequaltothelengthoftheverticalcurve(L)dividedbythealgebraicdifferenceinthetangentgrades(A)inpercent(K=L/A).ForagivendesignspeedtheminimumlengthofthecrestverticalcurveforstoppingsightdistancecanbeverifiedbydeterminingtherateofverticalcurvatureandcheckingthisvalueagainsttherateofverticalcurvatureprovidedinTable2forthedesignspeedoftheroad.Analternativemethodtodeterminingtheminimumlengthofacrestverticalcurve(L)forstoppingsightdistanceistomultiplytherateofverticalcurvature(K)forthedesignspeedoftheroadwaybythealgebraicdifferenceinthetangentgrades(A)inpercent(L=K*A).

Table2.DesignControlsforStoppingSightDistanceandforCrestVerticalCurves

Example1:Withatwo-lanecrestverticalcurvewithenteringandexitingtangentgradesof

+2.00%and-3.75%respectivelyandadesignspeedof100km/h,calculatetheminimumverticalcurvelengthforstoppingsightdistance.

FromTable2witha100km/hdesignspeed,therequiredstoppingsightdistanceis185metersandtherateofverticalcurvatureis52.UsingEquation3-3thelengthoftheverticalcurvecanbedetermined.

L=AS2/658=[(2.00+3.75)*1852]/658=299.08meters.

Sincethesightdistance(185meters)islessthatthelengthoftheverticalcurve(299.08meters)wecanverifythattherateofverticalcurvaturemeetsthedesignrequirements.

K=L/A=299.08/(2.00+3.75)=52.0152

Therateofverticalcurvatureforthe299.08meterlongverticalcurvemeetsorexceedstherequiredrateofverticalcurvaturefromTable2thecurvelengthissatisfactory.

Example2:Withatwo-lanecrestverticalcurvewithenteringandexitingtangentgradesof

+8.00%and+4.15%respectivelyandadesignspeedof80km/h,calculatetheminimumverticalcurvelength.

FromTable2withan80km/hdesignspeed,therequiredstoppingsightdistanceis130metersandtherateofverticalcurvatureis26.UsingEquation3thelengthoftheverticalcurvecanbedetermined.

L=AS2/658=[(8.00-4.15)*1302]/658=98.88meters.

Sincethesightdistance(130meters)islargerthatthelengthoftheverticalcurve(98.88meters)

calculatedwithEquation3therequiredlengthoftheverticalcurveiscalculatedwithEquation4.L=2S-(658/A)=2*130-(658/(8.00-4.15)=89.09meters.

Withthecalculatedsightdistanceknown,wecanverifythattherateofverticalcurvaturemeetsthedesignrequirements.

K=L/A=89.09/(8.00-4.15)=23.1426

Sincetherateofverticalcurvatureforthe89.09meterlongverticalcurvedoesnotmeetstherequiredrateofverticalcurvaturefromTable2theverticalcurvelengthisdeterminedbytherateofverticalcurvature.

L=KA=26*(8.00-4.15)=100.10meters.

Theminimumverticalcurvelengthshouldbe100.10meters.

4.CrestVerticalCurvePassingSightDistance

Designvaluesofcrestverticalcurvesforpassingsightdistancedifferfromthoseforcreststoppingsightdistancebecauseofthedifferentsightdistanceandobjectheightcriteria.TherequiredpassingsightdistanceforvariousdesignspeedscanbeobtainedfromTable3shownbelow.

Table3.PassingSightDistanceforDesignofTwo-LandHighways

Theheightoftheobjectforpassingsightdistanceincreasesto1.08metersfrom0.60metersforstoppingsightdistance.Thesetwofactorsresultincrestpassingsightdistancesthataresubstantiallylongerthanthecreststoppingsightdistance. Whentheheightoftheeyeandtheheightoftheobjectareboth1.08meters,asusedforpaddingsightdistance,Equation1andEquation2becometherequirescrestcurvelengthforpassingsightasshowninEquation5andEquation6respectively.

Equation5L=AS2/864(S<L)Equation6L=2S-(864/A)(S>L)

Where:

L=lengthofverticalcurve(m)S=sightdistance(m)

A=algebraicdifferenceingrades(%)

Again,therateofverticalcurvature(K)isequaltothelengthoftheverticalcurve(L)dividedbythealgebraicdifferenceinthetangentgrades(A)inpercent(K=L/A).ForagivendesignspeedtheminimumlengthofthecrestverticalcurveforpassingsightdistancecanbeverifiedbydeterminingtherateofverticalcurvatureandcheckingthisvalueagainsttherateofverticalcurvatureprovidedinTable4forthedesignspeedoftheroad.Analternativemethodtodeterminingtheminimumlengthofacrestverticalcurve(L)forpassingsightdistanceistomultiplytherateofverticalcurvature(K)forthedesignspeedoftheroadwaybythealgebraicdifferenceinthetangentgrades(A)inpercent(L=K*A).

Table4.DesignControlsforCrestVerticalCurvesBasedonPassingSightDistance

Example3:Withatwo-lanecrestverticalcurvewithenteringandexitingtangentgradesof

+2.00%and-3.75%respectivelyandadesignspeedof100km/h,calculatetheminimumverticalcurvelengthforstoppingsightdistance.

FromTable4witha100km/hdesignspeed,therequiredpassingsightdistanceis670metersandtherateofverticalcurvatureis520.UsingEquation5thelengthoftheverticalcurvecanbedetermined.

L=AS2/864=[(2.00+3.75)*6702]/864=2987.47meters.

Sincethesightdistance(670meters)islessthatthelengthoftheverticalcurve(2987.47meters)wecanverifythattherateofverticalcurvaturemeetsthedesignrequirements.

K=L/A=2987.47/(2.00+3.75)=519.56520

Sincetherateofverticalcurvatureforthe2987.47meterlongverticalcurvedoesnotmeetstherequiredrateofverticalcurvaturefromTable4theverticalcurvelengthisdeterminedbytherateofverticalcurvature.

L=KA=520*(2.00+3.75)=2990.00meters.

Theminimumverticalcurvelengthshouldbe2990.00meters.

5.SagVerticalCurves

Atleastfourdifferentcriteriaforestablishingthelengthofsagverticalcurvesarerecognizedtosomeextent.Theseareheadlightsightdistance,passengercomfort,drainagecontrol,andgeneralappearance.Ofthesefourcriteria,theheadlightsightdistanceisthebasisfordeterminingthelengthofsagverticalcurves.Equation7andEquation8showthegeneralequationsforsagverticalcurvestoppingsightdistancebasedonaneyeandobjectheightsof1.08metersand0.6metersrespectively.

Equation7L=AS2/[200(h1+S(tanz))](S<L)Equation8L=2S-[(200(h1+S(tanz)))/A](SL)

Where:

L=lengthofverticalcurve(m)S=sightdistance(m)

A=algebraicdifferenceingrades(%)

h1=heightofheadlight(m)

z=upwarddivergenceofheadlightbeam(o)

Aheadlightheightof0.60metersanda1-degreeupwarddivergenceofthelightbeamfromthelongitudinalaxisofthevehiclearecommonlyassumed.Equations7and8becomeEquations9and10respectively,withtheknownrelationshipbetweenthelengthofthesagverticalcurve(L)inmeters,thealgebraicdifferenceingrades(A)inpercentandthedistancebetweenthevehicleandpointwherethe1-degreeupwardangleofthelightbeamintersectsthesurfaceoftheroadway(s)inmeters.

Equation9L=AS2/(120+3.5S)(S<L)Equation10L=2S-[(120+3.5S)/A](S>L)

Where:

L=lengthofverticalcurve(m)S=sightdistance(m)

A=algebraicdifferenceingrades(%)

Foroverallsafety,asagverticalcurveshouldbelongenoughthatthelightbeamdistanceinnearlythesameasthestoppingsightdistance.Accordingly,itisappropriatetousethestoppingsightdistancesfordifferentdesignspeedsasthevalueofSintheaboveequations.Asinthecaseofcrestverticalcurves,

itisconvenienttoexpressthedesigncontrolintermsoftherateofverticalcurvature(K).Againtherateofverticalcurvatureisequaltothelengthoftheverticalcurve(L)dividedbythealgebraicdifferenceinthetangentgrades(A)inpercent(K=L/A).ForagivendesignspeedtheminimumlengthofthesagverticalcurvecanbeverifiedbydeterminingtherateofverticalcurvatureandcheckingthisvalueagainsttherateofverticalcurvatureprovidedinTable5forthedesignspeedoftheroad.Analternativemethodtodeterminingtheminimumlengthofasagverticalcurve(L)istomultiplytherateofverticalcurvature(K)forthedesignspeedoftheroadwaybythealgebraicdifferenceinthetangentgrades(A)inpercent(L=K*A).

Table5.DesignControlforSagVerticalCurves

Example4:Withatwo-lanesagverticalcurvewithenteringandexitingtangentgradesof-2.50%and+4.00%respectivelyandadesignspeedof100km/h,calculatetheminimumverticalcurvelength.

FromTable5witha100km/hdesignspeed,therequiredstoppingsightdistanceis185metersandtherateofverticalcurvatureis45.UsingEquation9thelengthoftheverticalcurvecanbedetermined.

L=AS2/(120+3.5S)=[(2.50+4.00)*1852]/(120+(3.5*185)=289.85meters.

Sincethesightdistance(185meters)islessthatthelengthoftheverticalcurve(289.85meters)wecanverifythattherateofverticalcurvaturemeetsthedesignrequirements.

K=L/A=289.85/(2.50+4.00)=44.5952

Sincetherateofverticalcurvatureforthe289.85meterlongverticalcurvedoesnotmeetstherequiredrateofverticalcurvaturefromTable5theverticalcurvelengthisdeterminedbytherateofverticalcurvature.

L=KA=45*(2.50+4.00)=292.50meters.

Theminimumverticalcurvelengthshouldbe292.50meters.

Example5:Withatwo-lanesagverticalcurvewithenteringandexitingtangentgradesof-8.00%and-5.30%respectivelyandadesignspeedof80km/h,calculatetheminimumverticalcurvelength.

FromTable5withan80km/hdesignspeed,therequiredstoppingsightdistanceis130metersandtherateofverticalcurvatureis30.UsingEquation9thelengthoftheverticalcurvecanbedetermined.

L=AS2/(120+(3.5S)=[(8.00-5.30))*1302]/(120+(3.5*130)=79.36.

Sincethesightdistance(130meters)islargerthatthelengthoftheverticalcurve(79.36meters)

calculatedwithEquation9therequiredlengthoftheverticalcurveiscalculatedwithEquation3-10.L=2S-[(120+(3.5S))/A]=2*130-[(120+(3.5*130))/(8.00-5.30)]=47.03meters.

Withthecalculatedsightdistanceknown,wecanverifythattherateofverticalcurvaturemeetsthedesignrequirements.

K=L/A=47.03/(8.00-5.30))=17.4230

Sincetherateofverticalcurvatureforthe47.03meterlongverticalcurvedoesnotmeetstherequiredrateofverticalcurvaturefromTable5theverticalcurvelengthisdeterminedbytherateofverticalcurvature.

L=KA=30*(8.00-5.30)=81.00meters.

Theminimumverticalcurvelengthshouldbe81.00meters.

6.DesignConsiderations

Thefollowingdesignconsiderations,inadditiontothecriterialistedabove,shouldbereviewedforallhorizontalcurvestoendureasafedesign.

The“roller-coaster”typeofprofileshouldbeavoided.Suchprofilesgenerallyoccuronrelativelystraighthorizontalalignmentswheretheroadwayprofilecloselyfollowsarollingnaturalgroundline.Thistypeofprofileisavoidedbytheuseofhorizontalcurvesorbymoregradualgrades.

A“broken-back”gradeline(twoverticalcurvesinthesamedirectionseparatedbyashorttangentsection)shouldbeavoided,particularlyinsags.“Broken-back”gradelinescanbeavoidedbychangingthegradelinesorthelengthsoftheverticalcurves.

Sagverticalcurvesshouldbeavoidedincutsectionsunlessadequatedrainagecanbeprovided.

7.As-Builts

Uponcompletionofconstructionoftheroadway,TheContractorshallsubmiteditableCADformatAs-Builtdrawings.Thedrawingshallshowthefinalproductasitwasinstalledinthefield,withtheexactdimensions,locations,materialsusedandanyotherchangesmadetotheoriginaldrawings.RefertoContractSections01335and01780Aofthespecificprojectforadditionaldetails.