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.