RPDM_Chapter11

May 2002

Road Planning and Design Manual Chapter 11: Horizontal Alignment

11Chapter 11

HorizontalAlignment

May 2002

Table of Contents11.1 General 11-1

11.2 Movement on a Circular Path 11-1

11.3 Maximum Side Friction (fmax) 11-211.3.1 General 11-2

11.3.2 Pavement Surfacing 11-3

11.4 Curve Superelevation 11-311.4.1 General 11-3

11.4.2 Maximum Values of Superelevation and Increases inSide Friction 11-4

11.4.3 Minimum Superelevation and Adverse Superelevation 11-5

11.4.4 Superelevation on Horizontal Curves with Radius > Rmin 11-6

11.4.5 Minimum Length of Superelevation on Horizontal Curves 11-8

11.4.6 Superelevation Development Length 11-8

11.4.7 Positioning of the Axis of Rotation 11-13

11.4.8 Positioning of Superelevation Development 11-13

11.5 Lengths of Horizontal Curves 11-14

11.6 Reverse Curves 11-16

11.7 Broken Back Curves 11-22

11.8 Compound Curves 11-23

11.9 Transition Curves 11-2311.9.1 General 11-23

11.9.2 Use of Transitions 11-24

11.9.3 Types of Transitions 11-25

11.10 Curve Widening 11-2611.10.1 General 11-26

11.10.2 Application of Curve Widening 11-29

11.11 Radius to Meet Sight Distance Requirements 11-30

11.12 Curve Perception Issues 11-30

11.13 Curves on Steep Down Grades 11-32

References 11-33

Relationship to Other Chapters 11-33

Appendix 11A: Clothoid Spiral Transition Curves 11-35

Chapter 11: Horizontal Alignment Road Planning and Design Manual

11

Chapter 11 Amendments - May 2002

Revision Register

Issue/ Reference Description of Revision Authorised DateRev No. Section by

1 First Issue. Steering OctCommittee 2000

2 11.1 New paragraph re coordination of horizontal and verticalalignment.

11.2 Last paragraph changed to draw attention to limitationsof Equation 11.1. Definition of superelevation added.

11.3 New paragraph and Table 11.1B giving maximum side friction factors for trucks.New paragraph and Table 11.1C giving maximum side friction factors for unsealed roads.

11.4 Addition of explanation of the purpose of superelevation and effect of adverse superelevation.

11.4.1 Retitled and inclusion of guidelines on acceptableincreases in side friction. Note re superelevation onbridges added to Table 11.2.

11.4.2 Retitled and additional material on adverse superelevation.11.4.3 Additional explanation at start.11.4.5 Revised definition of superelevation development length. Steering May

Added definitions of superelevation runoff length and Committee 2002tangent runout length. Cross reference to Tables 11.5 and 11.6 added to Fig. 11.1.

11.4.7 New paragraph covering when superelevation can bedeveloped entirely on the tangent.

11.5 New paragraph on desirable minimum curve length.11.6 Equation for reverse circular curve spacing corrected

(Case 4A). Additional cases of reverse curves.New paragraph on design speed for superelevationdevelopment when curve speeds vary.

Fig. 11.2A & Fig. 11.2 divided into Fig. 11.2A with 6 cases of reverseFig. 11.2B curves and Fig. 11.2B for similar (broken back) curves.Figs. 11.4, New Figures 11.4 and 11.5. Previous Figure 11.4 now 11.6.11.5 & 11.6 Figure 11.6 - case for trucks on right hand curves added.11.8 Additional constraints on the use of compound curves.11.9.2 Additional explanation of the basis of transition

curve length.11.10 Table 11.8 - renumbered. Additional note re maintenance

of clearances on the straight. Two paragraphs added reapplication of curve widening on reverse curves.

11.12 New section - Curve Perception Issues.11.13 New section - Curves on Steep Downgrades.References Additional references (McLean, Tziotis and Gunitallake;

Rahmann; Kanellaidis).New Section Relationship to other chapters.

May 2002 iii


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iv May 2002


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11.1 General

The horizontal alignment of a road is usually aseries of straights (tangents) and circular curveswhich may or may not be connected by transitioncurves. A suitable horizontal alignment is bestchosen by the procedure described in Chapter 6(Speed Parameters). The following sectionsoutline various design criteria which also need tobe considered when developing a horizontalalignment.

In addition, the horizontal alignment must bedesigned in conjunction with the verticalalignment and both properly coordinated inaccordance with the principles stated inChapter 10.

Standard curve notation that is used in variousfigures in this Chapter is described in Appendix A,Table 11A.1.

11.2 Movement on aCircular Path

As a vehicle traverses a circular curve, it is subjectto forces associated with the circular path.According to the principle of inertia, in theabsence of forces, a moving body will travel in astraight line. A force must be applied to changedirection. For a circular change of direction, theforce is called centripetal force and, in roaddesign, this is provided by side friction developedbetween the tyres and the pavement, and bysuperelevation.

Superelevation is the crossfall that is provided onthe pavement on a horizontal curve in order toassist a vehicle to maintain a circular path.

For normal values of superelevation, side frictionand radius, the following formula is accepted:

(11.1)

where

e = pavement superelevation (m/m or tangent ofangle). This is taken as positive if thepavement falls towards the centre of thecurve.

f = coefficient of side friction force developedbetween the vehicle tyres and the roadpavement. This is taken as positive if thefrictional force on the vehicle acts towardsthe centre of the curve.

g = acceleration due to gravity = 9.8 m/s²

v = speed of vehicle (m/s)

V = speed of vehicle (km/h)

R = curve radius (m)

Where f equals zero in the formula, all of thecentripetal force is provided by the superelevation.This condition can occur on large radius curveswith positive superelevation or for slow movingvehicles on curves of any radius. At low speeds, fcan be negative, and the curve is then over-superelevated for that speed. Curves are generallydesigned, however, so that a positive f is requiredfor the range of vehicle speeds likely to occur.

On short length horizontal curves, the radius ofthe vehicle path can be considerably larger thanthe centreline or edgeline radius. In these cases,the curve radius ‘R’ in Equation 11.1 should bemade equal to the vehicle path radius in lieu of thehorizontal curve radius.

Equation 11.1 can be rearranged as follows:

R127V

gRvfe

22==+

May 2002 11-1


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Chapter 11

Horizontal Alignment

11-2 May 2002

(11.2)

It then follows that:

(11.3)

where

Rmin = minimum horizontal curve radius (m)

emax = maximum superelevation for theprevailing conditions (see Table 11.2)

fmax = maximum coefficient of side frictionalforce developed between the vehicle tyresand the road pavement. However, thedesirable maximum coefficient of sidefriction (or even a value close to it) shouldbe used in preference to the absolutemaximum value whenever possible (seeTables 11.1).

Equation 11.1 is commonly called the ‘pointmass’ formula. This is because the vehicle isconsidered as a single point (at its centre ofgravity) which means all points of the vehicledescribe the same radius. This also means that theapplication of the equation is not relevant forexample, for large articulated vehicles turning atintersections.

11.3 Maximum SideFriction (fmax)

11.3.1 General

The maximum side friction values to be adoptedfor use in Equations 11.2 and 11.3 are given inTables 11.1.

The values of fmax in Table 11.1A are forpassenger cars on sealed pavements. Research hasshown that articulated vehicles may roll over atvalues of side friction in the range 0.2 (even lessfor some vehicles carrying livestock) to 0.35. Theabsolute maximum for trucks turning at low speedis usually taken as 0.25 (see McLean, Tziotis and

Gunitallake, 2001). Table 11.1B sets out themaximum side friction factors to be used fortrucks in situations that require specific checkingfor truck operation (see Chapter 6, Section 6.3.3).

Table 11.1A Maximum Design Values of SideFriction Demand for Cars on Sealed Pavements

Design Absolute Maximum Desirable MaximumSpeed Coefficient of Coefficient ofkm/h Side Friction Side Friction

40 0.35 0.3050 0.35 0.3060 0.33 0.2470 0.31 0.1975 0.29 0.1880 0.26 0.1685 0.22 0.1590 0.20 0.1395 0.18 0.13

100 0.16 0.12105 0.12 0.12110 0.12 0.12120 0.11 0.11130 0.11 0.11

Table 11.1B Maximum Design Values of SideFriction Demand for Trucks on Sealed Pavements

Design Coefficient of Side Friction for TrucksSpeed Absolute Desirable(km/h) Maximum Maximum

50 0.25 0.2160 0.24 0.1770 0.23 0.1480 0.20 0.1390 0.15 0.11

100 0.12 0.12110 0.12 0.12

120 0.11 0.11

Source: McLean, Tziotis and Gunitallake (2001).

Table 11.1C sets out the maximum side frictionfactors to be used on unsealed road surfaces.

)fe(127VR

maxmax

2

min +=

)fe(127VR

2

+=


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Table 11.1C Maximum Side Friction Factors onUnsealed Road Surfaces

Speed 50 60 70 80 90 100 110

Max.coeff. 0.12 0.11 0.10 0.10 0.09 0.09 0.08of sidefriction

Source: Unsealed Roads Manual - ARRB 1993.

11.3.2 Pavement Surfacing

Horizontal curves on which high values of sidefriction are likely to be demanded should beprovided with a pavement surfacing capable ofproviding good skid resistance. This isparticularly important on horizontal curves inconstrained areas such as mountainous terrain.The following treatments will help decrease thelikelihood of skidding:

• Surfacing with materials using aggregates witha high Polished Aggregate Friction Value andCrushing Value;

• Applying larger aggregates with sprayed seals(provides good drainage routes if appliedproperly (i.e. texture depths up to 2.5mm));

• Using polymer modified surfacings (helpsstone retention in asphalt and sprayed seals);

• Applying specially designed surfacings (e.g.open graded asphalts, tar based binders, epoxyresins, slurry treatments);

• Reducing drainage paths by designingappropriate crossfalls and providing drainageinlets/structures; and

• Adopting pavement designs that reduce ruttingduring the life of the pavement.

11.4 Curve Superelevation

11.4.1 General

It is normal practice for horizontal curves to besuperelevated. This allows a component of thevehicle weight to provide some of the centripetal

force that is needed for the vehicle to move in acircular path.

If a curve is not superelevated, the curve is said tohave adverse or negative superelevation.Therefore, ‘e’ is then expressed as a negative valuein equation 11.1. With adverse superelevation,there is a component of the vehicle weight that actsopposite to the centripetal force that is needed forthe vehicle to move in a circular path. This in turnrequires greater side friction than for a curve ofgiven radius with positive superelevation if thevehicle is to take the curve at the same speed.

The amount of superelevation is chosen primarilyon the basis of safety, but other factors arecomfort and appearance. The superelevation thatis applied to a horizontal curve should take intoaccount the following:

• tendency to increase the tracking of the rearwheels of slow moving vehicles towards thecentre;

• stability of high laden commercial vehicles;

• stability of vehicle loads;

• difference between inner and outer formationlevel, especially in flat country;

• length available to introduce the necessarysuperelevation;

• the need to avoid major changes in side frictiondemand between successive horizontal curves;

• the amount of centripetal force provided bysuperelevation versus that provided by sidefriction (see 11.4.4); and

• the need to keep superelevation below about5% on a floodway that is located on ahorizontal curve.

The amount of superelevation required for a givenradius, speed and coefficient of friction can becalculated by rearranging Equation 11.1 asfollows:

(11.4)fR127

Ve2

−=

May 2002 11-3


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11-4 May 2002

The absolute maximum and desirable maximumvalues of the coefficient of side friction ‘f’ can beobtained from Tables 11.1.

Values of superelevation obtained from Equation11.4 are subject to the maximum and minimumvalues given in Sections 11.4.2 and 11.4.3respectively.

11.4.2 Maximum Values ofSuperelevation andIncreases in Side Friction

Maximum values of superelevation are shown inTable 11.2. These values also take into account therelevant factors that are listed in Section 11.4.1.The maximum values are consistent with practicethroughout Australia. The values for rural roadsare suitable for road trains carrying livestock. Thevalues for urban roads reflect property boundarycontrols and traffic operation in congestedconditions.

As equation 11.3 shows, the minimum radiushorizontal curve (Rmin) that is suitable for a givenspeed requires the use of maximum allowablesuperelevation and maximum allowablecoefficient of side friction. With horizontal curvesthat are larger than Rmin, it is normal practice toprovide the same proportions of superelevationand side friction that apply to Rmin (see Section11.4.4). This practice helps (partly but notcompletely) ensure that any increase in sidefriction demand between successive horizontalcurves is kept within acceptable limits.

Large increases in side friction demand over whatdrivers have become accustomed to, can lead to achange in vehicle response that is not anticipatedby some drivers. Furthermore, these situationsmay create hazards for motorcyclists because theyhave difficulty in maintaining control when thereis a sudden change in side friction demand.Therefore, the need to limit changes in sidefriction demand becomes a desirable secondarycontrol for ensuring geometric consistency (seeSection 6.6). However, the limits on decrease inspeed between horizontal alignment elements thatare set out in Section 6.6 are the primaryrequirement for geometric consistency. The

desirable maximum changes in speed will oftenensure that changes in side friction demand areacceptable.

Increases in side friction are considered to be asecondary control because drivers select theirspeed through their perception of the horizontalcurvature. In turn, the selected speed reflectsdriver experience with a number of factorsrelating to vehicle control, including side frictiondemand.

It is not possible to prescribe simple limits on theincrease in side friction demand from onehorizontal curve to another. However, thefollowing guidelines are provided:

• The desirable maximum increase in sidefriction demand from one horizontal curve tothe next is about 25%. This is based on anAustroads (1989) criterion for compoundcurves and is therefore not a precise value.However, this condition only needs to applywhen the side friction factor on the latter curveis greater than about 0.12. This is because 0.12is a comfortable limit for drivers, even at highoperating speeds (see Table 11.1).

• If a curve has a ‘generous’ radius for its designspeed (this can still easily occur in a restrictedspeed environment due to preceding curvature),then a larger increase than the desirable 25% islikely to occur with the next curve. Even whenthe latter curve has a side friction demandcomfortably below the maximum for its designspeed, the increase may be greater than 25%.Therefore, an increase of more than 25% isacceptable when a curve follows a ‘generous’curve. However, it will be necessary to checkon any increase in side friction between thelatter curve and any curves prior to the'generous' radius curve. The purpose is to checkthe side friction demand against the level thatdrivers have become aware of.

• Sections of existing road or sections of road thatinvolve a reduction in desired speed (see 6.1.3),may involve larger increases in side friction.With the latter, at least, drivers will be alerted tothe need for increases in side friction. In thesecases, the increase will be acceptable when the


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side friction demand does not exceed thedesirable maximum for the curve design speed.

• Curves with a side friction demand greater thanthe desirable maximum for the curve designspeed should be preceded by a curve that alertsdrivers to the required side friction.

• A higher value of superelevation than thatshown for a particular radius/speedcombination in Tables 11.5 and 11.6, may beused to limit an increase in side friction.

Table 11.2 Maximum Values of Superelevation

Urban Rural RoadsRoads

Desirable Number of Design Speed Heavy VehiclesRange km/h ≥≥20 veh/day <20 veh/day

5% >70 6% 6%

5% ≤70 7% 10%*

* Use with caution.Notes:1. Higher values (up to 10%) may be used in specialcases such as rural roads in mountainous terrain or thereuse of existing pavement or kerb lines.2. On bridges, the maximum superelevation should be

5% on urban roads and 6% on rural roads.

11.4.3 Minimum Superelevationand AdverseSuperelevation

It is normal practice to superelevate all curves toa value that is at least equal to the normal crossfallon straights. This is normally 3% in order toensure adequate surface drainage. However, thereare situations where the application ofsuperelevation can cause pavement drainageproblems. In some situations when the grade isnearly flat, water will not run off the road properlyat places where the crossfall is also nearly flat.These areas are also likely to experiencemaintenance problems. In other situations,drainage paths for the pavement surface will beunacceptably long because the grade will be steepenough to have the water flow being mainly downthe grade and switching from one side of the road

to the other due to the change in the direction ofthe crossfall (see also 12.2.4). These situationsshould always be checked with the aid ofpavement surface contours. These contours can bereadily generated by road design modellingsoftware.

Problems with drainage of the pavement surfacemay be overcome by modifying the combinationof the grading, the superelevation and theapplication of the superelevation. If the horizontalcurve radius is large enough, there may be scopeto leave the curve unsuperelevated (that is, haveadverse superelevation). There are also othersituations as discussed later that may warrant theuse of adverse superelevation on large radiuscurves. However, the use of adversesuperelevation must be considered to beexceptional practice.

In addition to the problems with adversesuperelevation that are given in section 11.4.1,further problems with using adversesuperelevation are:

• Greater tendency of vehicles to move towardsthe outside (in terms of radius) of the trafficlane on multilane roads.

• Greater instability of vehicle loads. This is alsoexacerbated by greater suspension movementdue to the weight component acting in theoutwards direction.

• On two-lane, two-way roads, the normalcrowned pavement structure is carried throughthe curve. Vehicles on the outer lane will haveadverse superelevation and vehicles on theinner lane will have positive superelevation.Hence, a vehicle that crosses from the innerlane to the outer lane will experience a rapidincrease in side friction demand with increasedprobability of the driver losing control.

These problems with adverse superelevation limitits use on rural roads and motorways to:

• Large radius curves that can be regarded asstraights and where pavement drainage can beimproved. See Table 11.3A.

• Temporary roadways. See Table 11.3B.

• Side tracks. See Table 11.3B.

May 2002 11-5


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11-6 May 2002

• Temporary connections. See Table 11.3B.

On urban roads adverse superelevation can betolerated more than on rural roads. This is mainlydue to lower vehicle speeds and driver familiaritywith the road. Common urban situations that mayrequire the use of adverse superelevation are:

• Property access controls

• Channel drainage controls

• Grading restrictions

• The need to maintain visibility of the roadsurface - especially near intersections

• Turns at at-grade intersections

• Roundabouts.

For any road, adverse superelevation should notexceed 3%. In line with common practicethroughout the world, the maximum side frictionvalues that can be used in conjunction withadverse superelevation are half the absolutemaximum values that are used for curves withpositive superelevation. However, this limit onside friction values does not apply to turns at at-grade intersections.

Minimum curve radii for the use of adversesuperelevation on urban roads are given in Table11.3B. This table may also be used for temporaryroads, and side tracks and temporary connectionson rural roads or motorways.

Table 11.3A Rural Roads - Minimum HorizontalCurve Radii that can have Adverse Crossfall Equalto the Normal Crossfall on Straights

Speed MinimumEnvironment Radius

(km/h) (m)70 90080 125090 1500

100 2000110 3000120 4000130 5000

Note: For normal rural roads. Does not apply tointersections where higher f demand may beconsidered (Austroads 1988 - Intersections at Grade -Part 5).

Table 11.3B Minimum Horizontal Curve Radiiwith Adverse Superelevation for Urban Roads,temporary roads, and side tracks and temporaryconnections on rural roads or motorways

Curve Minimum AbsoluteDesign Radius MaximumSpeed (m) Coefficient of(km/h) Side Friction

40 85 0.1850 130 0.1860 200 0.1770 295 0.1680 505 0.1390 910 0.10

100 1575 0.08120 3780 0.06130 4435 0.06

Based on a maximum adverse superelevation of 3%.

11.4.4 Superelevation onHorizontal Curves withRadius > Rmin

It is normal practice to have superelevationprovide a significant amount of the centripetalforce that is needed for a vehicle to move in acircular path. This is because superelevation is apositive and permanent feature, whereas sidefriction is subject to roadway surface and vehiclevariations (Rahmann, 1981). Even so, it iscommon to have two to three times as muchcentripetal force provided by side friction as thatprovided by superelevation.

There are a number of methods to determine thesuperelevation (and hence resultant side friction)for curves with a radius larger than the minimumradius for a given design speed. Main Roads hasused at least two different methods in the past. Itmust also be reiterated that the length of suchcurves should be checked to ensure that the lengthdoes not cause the operating speed to increasebeyond the curve design speed when the designspeed is less than 110 km/h.

The method that is currently favoured byAustroads and which will normally be used fornew works, is for the superelevation to be variedlinearly from 0 for R = infinity to emax for Rmin.This then means that all curves that are designed


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for a given speed will have the same proportionsof superelevation and side friction demand, notwithstanding the following practicalconsiderations:

• For construction expediency, superelevationvalues are normally rounded (upwards) to amultiple of 1% so that there is a correspondingadjustment of side friction.

• The perceived benefits of uniformity willusually only be possible on high speed ruralroads (i.e. where the design or operating speedexceeds 100 km/h) and urban roads withenforced speed limits because curve operatingspeeds vary on intermediate and low speedrural roads.

• Other methods have been used in the past sothat there are likely to be many cases where thereuse of existing pavement will dictate adifferent superelevation. This is acceptable ifthe resultant side friction is suitable for thecurve design speed and consistent with that forany adjacent curves (see 11.4.2).

With the “linear distribution method”, thesuperelevation ‘e’ for a curve of radius R that isgreater than Rmin is given by equation 11.5.

(11.5)

Note that fmax may be either the absolutemaximum value or the desirable maximum valuefor the design speed V.

The value of ‘e’ is usually rounded upwards (e.g.4.0% stays 4.0% but 4.1% becomes 5%) and thecorresponding coefficient of side friction iscalculated from

(11.6)

With different possibilities for emax (see Table11.2) and fmax (absolute maximum vs. desirablemaximum) different values of superelevation maybe attributed to a given combination of radius anddesign speed. However, the subjective basis of the

“linear distribution method” (and indeed mostother methods) and the practice of rounding thesuperelevation value, allows a practicalrationalization to be made. Hence Tables 11.5 and11.6 cover the majority of rural and urban casesrespectively. Tables 11.5 and 11.6 also showcorresponding superelevation runoff requirementsand transition lengths.

For rural areas, rationalization has been achievedby distributing from the combination of 6%superelevation and the desirable maximum valueof coefficient of side friction. This practice issupported by the following points:

• For speeds in the range of 60 to 90 km/h, it isfortuitous that it is possible to distribute 'up' tothe combination of 10% superelevation andabsolute maximum coefficient of side friction.

• For speeds greater than 100 km/h, the desirablemaximum and absolute maximum coefficientsof side friction are the same. Hence, curveswith superelevation between 6% and 10% usethe same coefficient of side friction. This isconsistent with long standing practice for thesedesign speeds.

• The maximum superelevation for design speedsof 120 km/h and 130 km/h is 6%.

• Except for large radius curves that are notsuperelevated (see Section 11.4.3), curves aresuperelevated at least to a value equal to thenormal crossfall on straights. This further limitsthe range of superelevation that needs to bedistributed.

• In areas where 7% maximum superelevation isused, Table 11.5 promotes the use of acoefficient of side friction that is sufficientlyclose to the desirable maximum in conjunctionwith 7% superelevation. Given the subjectivenature of the desirable maximum coefficientsof side friction, this is acceptable.

For urban areas, rationalization has been achievedby distributing from the combination of 5%maximum superelevation and the desirablemaximum value of coefficient of side friction. Fordesign speeds less than or equal to 100 km/h,curves with a radius less than that corresponding

roundede

RV

f −=127

2

)fe(R127eV

emaxmax

max2

+=

May 2002 11-7


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to 5% superelevation and the desirable maximumcoefficient of side friction, maintain the 5%superelevation with increasing side friction untilfmax. This practice helps ensure that the relativelylow maximum superelevation is applied beforevehicles have to make use of increasing sidefriction.

11.4.5 Minimum Length ofSuperelevation onHorizontal Curves

In constrained situations such as mountainousterrain or urban roads, curves should be fullysuperelevated even if only instantaneously. But itis desirable that there be at least 30m of fullysuperelevated curve.

11.4.6 SuperelevationDevelopment Length

Superelevation is developed by rotating theroadway cross-section about some axis; mostcommonly the horizontal control line.

Superelevation development length is definedas the length required to rotate the pavement fromthe point of normal crossfall on the approachtangent (straight) to the point where the fullsuperelevation for the curve is attained. In turn,this superelevation development length has twocomponents:

• Superelevation runoff length. This is thelength from the point where the pavement hasbeen rotated to zero crossfall to the point wherethe full curve superelevation has been attained.

• Tangent runout. This is the residual lengthfrom the point of normal crossfall to the pointof zero crossfall. This component lies on theapproach tangent.

There are two criteria that are used to determinethe length of superelevation development:

1. Maximum rate of rotation of crossfall.

2. Relative grade between the edge of thecarriageway and the control line.

The superelevation development length to beadopted will generally be the longer valuecalculated for the two criteria above,notwithstanding normal minimum lengths ofsuperelevation runoff. The maximum rate ofrotation criterion is a mandatory standard thatmust be adopted as a minimum. The relative gradecriterion is for appearance purposes and should beobtained at all locations unless economic or safetyconsiderations dictate otherwise e.g. betweenreverse curves on steep grades.

Except for constrained situations such asmountainous areas or urban roads, it is normalpractice to recognize minimum lengths forsuperelevation runoff for both transitioned anduntransitioned curves. This is due to the fact thatmost vehicles describe some transition path whenentering or leaving a curve with the minimumlength of the transition path being about 30 to 50 m.

This in turn relates to the practices of:

• Basing transition curve lengths onrecommended superelevation runoff lengths(see Tables 11.5 and 11.6; see also 11.9.2 forfurther explanation) and hence, matching thesuperelevation runoff with the transition (see11.4.8 Positioning of SuperelevationDevelopment); and

• Not providing transitions when there issufficient room within a traffic lane for vehiclesto make their own transition path (see 11.9.2Use of Transitions).

Therefore, the normal minimum lengths forsuperelevation runoff for a curve are:

• 40 m for transitioned curves, which ties to a 40m minimum transition length.

• 50 m for an untransitioned curve when thecurve design speed is greater than 80 km/h,with this length normally being equidistantabout the curve tangent point.

• 30 m for an untransitioned curve when thecurve design speed is less than or equal to 80km/h, with this length normally beingequidistant about the curve tangent point.

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Maximum Rate of Rotation ofCrossfall

The following maximum rates of rotation areapplicable:

• 0.025 radians per second for most road types.In particular, roads with vehicles that carrylivestock.

• 0.035 radians per second in low speed areas(≤70km/h). Typically this includes roads insteep terrain and geometry consisting of reversecurves with little or no straight between thecurves.

• 0.04 radians per second in low speed areas(<60km/h) with extremely constrictedhorizontal alignment i.e. sections of roads insteep terrain and geometry consisting of smallradius reverse curves with little or no straightbetween the curves.

The minimum superelevation development lengthbased on the above criteria is given by Equation11.7.

(11.7)

where

Le = superelevation development length (m)

V = design speed (km/h)

e1, e2 = crossfall or superelevation at ends ofdevelopment length (m/m); +ve whensloping upwards from the control line;–ve when sloping downwards from thecontrol line

r = rate of rotation of the road crossfall(radians/second)

Note that for the range of crossfalls involved,m/m ≈ radians.

Relative Grade

The relative grade is the percentage differencebetween the grade of the edges of the carriagewayand the grade of the axis of rotation. Thisdifference should be kept below the values shown

in Table 11.4 to achieve a reasonably smoothappearance.

Table 11.4 Maximum Relative Grades BetweenEdge of Carriageway and Control Line inSuperelevation Development for AppearanceCriterion

Design Relative Grade - Per centSpeed One Two Morekm/h Lane1 Lanes2 than two

Lanes3

40 or under 0.9 1.3 1.760 0.6 1.0 1.380 0.5 0.8 1.0

100 0.4 0.7 0.9120 or over 0.4 0.6 0.8

1 Applies to normal two-lane roadway with control lineon centreline.

2. Applies to: two-lane roadway with control along oneedge; four-lane roadway with control on centreline;two-lane roadway with climbing lane and control oncentreline of basic two lanes.

3. Applies to multilane roadway with more than twolanes between the control and the edge of runninglanes.

4. Relative grade values may be interpolated fordesign speeds not listed in the table.

The minimum superelevation development lengthbased on the criteria in Table 11.4 is given byEquation 11.8.

(11.8)

whereLe = superelevation development length (m)W = maximum width from axis of rotation to

edge of running lane (m)e1, e2 = crossfall or superelevation at ends of

development length (m/m); +ve whensloping upwards from the control line;–ve when sloping downwards from thecontrol line

Gr = allowable relative grade from Table 11.4(%).

GreeW100L 12

e−=

reeV278.0L 12

e−=

May 2002 11-9


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11-10 May 2002


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Table 11.5 Horizontal Curve Design Parameters for Rural Roads

Curve Radius1,2,3 Super- Friction Plan Super. DesirableDesign m elevation1,9 coefficient9 Transition Criteria min. curveSpeed % Type & Length1,5 Satisfied6,7 length8

km/h m m44 10 0.35 S60 R,g1,g2 14045 9 0.35 S60 R,g1,g2,g3 14047 7 0.35 S40 R,g1,g2 100

50 55 6 0.30 S40 R,g1,g2,g3 8065 5 0.25 U30 R,g1,g2 8082 4 0.20 U30 R,g1,g2,g3 80

109 3 0.15 U30 R,g1,g2,g3 8066 10 0.33 S60 R,g1 14071 8 0.32 S60 R,g1,g2 14081 7 0.28 S60 R,g1,g2,g3 140

60 95 6 0.24 S40 R,g1,g2 100113 5 0.20 S40 R,g1,g2,g3 100142 4 0.16 U30 R,g1,g2,g3 100189 3 0.12 U30 R,g1,g2,g3 10094 10 0.31 S80 R,g1,g2 180

103 9 0.29 S80 R,g1,g2 180116 8 0.25 S60 R,g1,g2 140132 7 0.22 S60 R,g1,g2 140

70 154 6 0.19 S40 R,g1 140185 5 0.16 S40 R,g1,g2 140222 5 0.13 U40 R,g1,g2,g3 140232 4 0.13 U40 R,g1,g2,g3 140309 3 0.13 U30 R,g1,g2,g3 140140 10 0.26 S100 R,g1,g2 240153 9 0.24 S80 R,g1,g2 200172 8 0.21 S80 R,g1,g2 200196 7 0.19 S80 R,g1,g2,g3 200

80 204 7 0.18 S60 R,g1,g2 180230 6 0.16 S60 R,g1,g2 180275 5 0.13 S60 R,g1,g2,g3 180300 5 0.12 U50 R,g1,g2,g3 180344 4 0.11 U50 R,g1,g2,g3 180440 4 0.07 U50 R,g1,g2,g3 180441 4 0.07 U50 R,g1,g2,g3 180458 3 0.08 U30 R,g1,g2,g3 180213 10 0.20 S100 R,g1,g2224 9 0.20 S100 R,g1,g2,g3252 8 0.17 S80 R,g1,g2288 7 0.15 S80 R,g1,g2,g3 230

90 336 6 0.13 S60 R,g1,g2400 5 0.11 U50 R,g1,g2440 5 0.09 U50 R,g1,g2441 5 0.09 U50 R,g1,g2500 4 0.09 U50 R,g1,g2,g3671 3 0.07 U50 R,g1,g2,g3

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Table 11.5 Horizontal Curve Design Parameters for Rural Roads (contd.)


km/h m m303 10 0.16 S120 R,g1,g2,g3315 9 0.16 S100 R,g1,g2328 8 0.16 S100 R,g1,g2,g3375 7 0.14 S80 R,g1,g2,g3

100 437 6 0.12 S80 R,g1,g2,g3 280441 6 0.12 S80 R,g1,g2,g3525 5 0.10 S60 R,g1,g2,g3650 4 0.08 U50 R,g1,g2,g3875 3 0.06 U50 R,g1,g2,g3433 10 0.12 S120 R,g1,g2,g3454 9 0.12 S120 R,g1,g2,g3476 8 0.12 S100 R,g1,g2,g3502 7 0.12 S100 R,g1,g2,g3

110 529 6 0.12 S80 R,g1,g2,g3 340635 5 0.10 U60 R,g1,g2,g3794 4 0.08 U60 R,g1,g2,g3

1059 3 0.06 U60 R,g1,g2,g3667 6 0.11 S80 R,g1,g2,g3800 5 0.09 S80 R,g1,g2,g3

120 875 5 0.08 S80 R,g1,g2,g3 4001000 4 0.07 U60 R,g1,g2,g31334 3 0.06 U60 R,g1,g2,g3785 6 0.11 S100 R,g1,g2,g3939 5 0.09 S80 R,g1,g2,g3

130 1174 4 0.07 U60 R,g1,g2,g3 4701566 3 0.06 U60 R,g1,g2,g3

Notes:1. The circular curve radii listed in the table correspond to changepoints in superelevation and/or transition length. In

practice, rounded values of curve radius will usually be used and for any radius falling between two radii listed fora given design speed, the superelevation and transition values for the smaller radius will apply.

2. The minimum radius listed for a given design speed is the absolute minimum radius possible, being that whichcorresponds to the maximum possible superelevation and maximum coefficient of side friction.

3. The radius corresponding to 6% superelevation is the desirable minimum for most design situations, being that whichcorresponds to a desirable maximum superelevation of 6% and the desirable maximum coefficient of side friction.

4. The shaded rows in the table represent cases that should be avoided where possible.5. Plan Transition Type: S denotes a Spiral transition is required for the application of superelevation; U denotes that the

curve is untransitioned with the superelevation runoff being applied over the given length in accordance with Figure 11.1.6. Transition Criteria Satisfied: R denotes rate of rotation criterion; g1 denotes relative grade for 1 lane; g2 denotes

relative grade for 2 lanes; g3 denotes relative grade for 3 or more lanes. See Table 11.4. For the 50, 60 and70km/h design speeds, the g1 criterion yields longer transition lengths than the rotation criterion (0.035 rad/s) sothat the rounded transition lengths end up being sufficient for a rotation of 0.025 rad/s. 0.025 rad/s is used for allcurve design speeds > 70 km/h.

7. If a required transition criterion is not satisfied, use a plan transition length 20m longer than shown in the table.8. Desirable minimum curve length is for appearance. It includes the length of any spiral transitions.9. Other combinations of superelevation and coefficient of side friction are possible for a given radius and

curve design speed. Other engineering requirements are drainage of the pavement surface, constraint onchange in superelevation between adjoining curves, temporary connections etc. However, the combinationshould be supported by some engineering requirement. In particular, the need to reuse existing pavementmay mandate a different superelevation with the coefficient of side friction being checked against desirableand/or absolute maximum limits and for consistency with any adjacent horizontal curves.

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Table 11.6 Horizontal Curve Design Parameters for Urban Roads


km/h m m32 5 0.35 U25 R,g1,g2 40

40 36 5 0.30 U20 R,g1 4045 4 0.24 U20 R,g1,g2 4060 3 0.18 U15 R,g1,g2 4049 5 0.35 U25 R,g1 5056 5 0.30 U30 R,g1,g2 50

50 56 5 0.30 U25 R,g1 5070 4 0.24 U25 R,g1,g2 7094 3 0.18 U20 R,g1,g2,g3 7075 5 0.33 S40 R,g1,g2,g3 100

60 98 5 0.24 S40 R,g1,g2,g3 100122 4 0.19 U30 R,g1,g2,g3 100163 3 0.14 U20 R,g1,g2 100107 5 0.31 S40 R,g1,g2 140

70 161 5 0.19 S40 R,g1,g2 140200 4 0.15 U30 R,g1,g2 140268 3 0.11 U30 R,g1,g2,g3 140163 5 0.26 S60 R,g1,g2,g3 180240 5 0.16 S60 R,g1,g2,g3 180

80 300 4 0.13 U40 R,g1,g2,g3 180400 3 0.10 U30 R,g1,g2,g3 180440 3 0.08 U30 R,g1,g2,g3 180441 3 0.08 U30 R,g1,g2,g3 180255 5 0.18 S60 R,g1,g2,g3 230354 5 0.13 U50 R,g1,g2 230

90 440 5 0.10 U50 R,g1,g2,g3 230443 4 0.10 U50 R,g1,g2,g3 230600 3 0.08 U50 R,g1,g2,g3 230375 5 0.16 S60 R,g1,g2,g3 280

100 463 5 0.12 S60 R,g1,g2,g3 280579 4 0.10 U50 R,g1,g2,g3 280772 3 0.07 U50 R,g1,g2,g3 280560 5 0.12 S80 R,g1,g2,g3 340

110 700 4 0.10 U60 R,g1,g2,g3 340934 3 0.07 U60 R,g1,g2,g3 340709 5 0.11 S80 R,g1,g2,g3 400

120 886 4 0.09 U60 R,g1,g2,g3 4001200 3 0.07 U60 R,g1,g2,g3 400

Notes:1. The circular curve radii listed in the table correspond to changepoints in superelevation and/or transition length.

In practice, rounded values of curve radius will usually be used and for any radius falling between two radii listedfor a given design speed, the superelevation and transition values for the smaller radius will apply.

2. The minimum radius listed for a given design speed is the absolute minimum radius possible, being that whichcorresponds to the maximum possible superelevation and maximum coefficient of side friction.

11.4.7 Positioning of the Axisof Rotation

The adopted position of the axis of rotation, i.e.the point about which the crossfall is rotated todevelop superelevation, depends on:

• the type of road facility

• total road cross section adopted

• terrain

• the location of the road.

On a two-lane, two-way road, the superelevationis normally developed by rotating each half of thecross section (including shoulders) about thecarriageway centreline (axis of rotation).

On divided rural roads where the median isrelatively narrow, i.e. less than 5m, the twocarriageways may be rotated about the centrelineof the median. Where the median is wide, the axisof rotation is usually along the median edge(particularly in flat country).

With the two-lane, two-way road case and thenarrow median divided road case, the applicationof superelevation can result in the profiles at theedges of the carriageways being in opposite

vertical directions to each other, or one of theprofiles being flat. Where this causes any drainageor appearance problems, the application of thesuperelevation will need to be modified. Thesubjective nature of both the rotation and relativegrade controls means that there is some scope tomodify the application of superelevation.

Any appearance problems are likely to be morenoticeable if an edge line is defined by a kerb,guard rail or bridge barrier.

11.4.8 Positioning ofSuperelevationDevelopment

Transitioned Curves

For curves with transitions, it is normal practice tomatch superelevation runoff with the transition.That is, the point of 0% crossfall corresponds tothe start of the transition (for a vehicle enteringthe curve) and the full superelevation for the curve(e%) is attained at the end of the transition. Thesuperelevation development is extended backfrom the start at the same rotation rate to the pointof normal crossfall on the approach tangent.Figure 11.1 uses a superelevation diagram tofurther explain this practice.

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Notes to Table 11.6 (contd):3. The radius corresponding to 5% superelevation is the desirable minimum for most design situations, being that

which corresponds to a desirable maximum superelevation of 5% and the desirable maximum coefficient of sidefriction.

4. The shaded rows in the table represent cases that should be avoided where possible.5. Plan Transition Type: S denotes a Spiral transition is required for the application of superelevation; U denotes

that the curve is untransitioned with the superelevation runoff being applied over the given length in accordancewith Figure 11.1.

6. Transition Criteria Satisfied: R denotes rate of rotation criterion; g1 denotes relative grade for 1 lane; g2 denotesrelative grade for 2 lanes; g3 denotes relative grade for 3 or more lanes. See Table 11.4. For the 40, 50, 60 and70km/h design speeds, the g1 criterion yields longer transition lengths than the rotation criterion (0.035 rad/s) sothat the rounded transition lengths end up being sufficient for a rotation of 0.025 rad/s. 0.025 rad/s is used forall curve design speeds > 70 km/h.

7. If a required transition criterion is not satisfied, use a plan transition length 20m longer than shown in the table.8. Desirable minimum curve length is for appearance. It includes the length of any spiral transitions. 9. Other combinations of superelevation and coefficient of side friction are possible for a given radius and

curve design speed. Other engineering requirements are drainage of the pavement surface, constraint onchange in superelevation between adjoining curves, temporary connections etc. However, the combinationshould be supported by some engineering requirement. In particular, the need to reuse existing pavementmay mandate a different superelevation with the coefficient of side friction being checked against desirableand/or absolute maximum limits and for consistency with any adjacent horizontal curves.

11-14 May 2002

The theoretical basis for the uniform developmentof the superelevation along the length of thetransition is that when combined with the uniformincrease in curvature provided by the clothoidspiral transition, there will be a uniformattainment of the side friction demand that will beused on the horizontal curve.

Untransitioned Curves

Normal practice for positioning thesuperelevation development for curves withouttransitions is as follows:

• Tangent to Curve and Vice Versa: Positionedsuch that 50% of the superelevation runofflength (from 0% crossfall to e% for the curve)is located on the tangent and 50% is located onthe curve. This positioning is considered tocorrespond with the transition path that isdescribed by the majority of vehicles enteringor leaving the curve. However, in constrainedsituations such as mountainous areas or urbanroads, shorter than desirable arc lengths mayforce the positioning to be 70% on the tangentand 30% on the curve (see Figure 11.1).

Development of the superelevation entirely onthe tangent for an untransitioned curve isnormally not preferred because thesuperelevation development will notcorrespond sufficiently with the transition paththat is described by the majority of vehicles.Therefore when vehicles approach the curve,there will be a section where a vehicle will stillbe travelling straight and the crossfall issufficient to affect tyre slip angles. This willcause drivers to steer slightly in the directionopposite to the pending curve.

Development of the superelevation entirely onthe tangents is not precluded however. Therecan be exceptional circumstances that favourthis practice. The most common will be forexiting from a horizontal curve on a steepdowngrade. Application of any curve wideningwill have to be treated separately in such cases.

• Reverse Curves with no intervening straight:Zero crossfall is provided at the commontangent point. The full superelevation runoff is

then located on each curve. See also Section11.6.

• Compound Curves: Where compound curvesare provided, the full superelevation on thesharper curve must be retained to the commontangent point. Therefore, the superelevationdevelopment length is provided on the largerradius curve.

11.5 Lengths of HorizontalCurves

With deflection angles less than 1 degree, a curveis not required.

At small angles greater than 1 degree, theappearance of a kink must be avoided. This isachieved by the use of long curves as shown inTable 11.7.

Table 11.7 Curves for Small Deflection Angles

Angle Min.Lth. Radius Metres Secantof Arc* Calculated Rounded Metresmetres Value Value

5° 150 1719 1800 1.7154° 180 2578 2600 1.5853° 210 4011 4000 1.3712° 240 6875 7000 1.0661° 270 15470 15000 0.572Less than 1° - No curve required

* Minimum length of arc to be 150 m for a 5° angle plusan additional 30 m for each degree less than 5°.

The lengths of curves given in Table 11.7 aremore applicable to important and high speedroads than to minor roads. The lengths do notapply in difficult locations such as mountainousterrain.

Besides appearance requirements, horizontalcurves must be long enough to accommodate theapplication of superelevation. The desirableminimum lengths given in Tables 11.5 and 11.6are based on:


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Figure 11.1 Standard Methods of Applying Curve Widening and Superelevation

11-16 May 2002

• 30m minimum length of arc with fullsuperelevation (see 11.4.5);

• transition curve lengths.

There is scope to use shorter lengths when thecurve is untransitioned because half of thesuperelevation runoff is applied on the tangent.

11.6 Reverse Curves

Reverse Curves are horizontal curves turning inopposite directions that adjoin (have commontangent points) or have a short length of tangentbetween the curves. Desirably, reverse curvesshould not be used unless there is sufficientdistance between the curves to introduce the fullsuperelevation required for each of the two curveswithout exceeding the standard rate of change ofcrossfall for the particular design speed.

The following cases of reverse curves areacceptable as shown in Figure 11.2A:

• Case 1, reverse transitioned curves with acommon point of tangency - these canaccommodate the required change insuperelevation with the point of zero crossfalloccurring at the tangent point. See Case 1 inFigure 11.2A. This case requires no change inthe method of applying any curve widening.

• Case 2, reverse transitioned curves with asufficient length of intervening tangent to havea section of the tangent with normal crossfall.On a two-lane, two-way road with the controlline down the center of the road, a spacinggreater than about 0.7V metres (V in km/h)results in a normally crowned length of tangent.See Case 2 in Figure 11.2A. It has beencommon practice to try to achieve a minimumof 30 metres of normally crowned tangent. It isoften possible to increase the length of tangentbetween the two curves by slightly increasingthe spacing between the centres of the twocircular arcs. This case requires no change inthe method of applying any curve widening.

• Case 3A, reverse circular curves with acommon point of tangency are possible foroperating speeds less than or equal to 80 km/h

if the superelevation on each curve does notexceed 3% and a nominal rotation rate of 0.025rad/s is used. This allows the superelevation tochange from zero crossfall at the commontangent point to 3% in 25 metres (or less) inline with established practice of attaining fullsuperelevation 25 metres into an untransitionedcurve. See Case 3A in Figure 11.2A. However,the need to switch curve widening from oneside of the road to the other will require thetreatment shown in Figure 11.5. For loweroperating speeds, the acceptable rate ofpavement rotation (0.035 rad/s or 0.04 rad/s)may allow more than 3% superelevation to beachieved over the 25m length. Case 3Arepresents acceptable compromising of thelocation of the superelevation runoff.

• Case 3B, reverse circular curves spaced suchthat the length of intervening tangent is lessthan the length in case 4A or case 4B(depending upon the operating speed). As forcase 3A, this case is based on attaining fullsuperelevation within a maximum distance of25 metres into each curve. The pavement isrotated directly between the two curves asshown in Case 3B in Figure 11.2A. Dependingupon the spacing between the two curves, thiscase is slightly less restrictive than Case 3A.The maximum superelevation that is achievableon each curve depends upon the maximum rateof rotation for the operating speed. If themaximum achievable superelevation for acurve is less than that needed for the operatingspeed, this case is unacceptable. The need toswitch curve widening from one side of theroad to the other may require the treatmentshown in Figure 11.5. Case 3B representsacceptable compromising of the location of thesuperelevation runoff.

• Case 4A, reverse circular curves spaced atabout (e1 + e2)V/0.18 metres (where V is theoperating speed in km/h, e1 and e2 are theabsolute value of the superelevation on eachcurve in m/m). The rate of rotation of thepavement is nominally 0.025 rad/s. This casewill accommodate the standard positioning ofthe superelevation development for eachcircular curve (see 11.4.8 and Figure 11.1). Thepavement will be rotated directly between the


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two curves as shown for Case 4 in Figure11.2A. This case will also provide scope forswitching any curve widening from one side ofthe road to the other in the normal manner (seeFigure 11.1). However, the treatment shown inFigure 11.5 may simplify construction and havebetter appearance.

• Case 4B, reverse circular curves spaced atabout (e1 + e2)V/0.25 metres (where V is theoperating speed in km/h, e1 and e2 are theabsolute value of the superelevation on eachcurve in m/m). The rate of rotation of thepavement is nominally 0.035 rad/s because theoperating speed is less than or equal to 70km/h.This case will accommodate the standardpositioning of the superelevation developmentfor each circular curve (see 11.4.8 and Figure11.1). The pavement will be rotated directlybetween the two curves as shown for Case 4 inFigure 11.2A. This case will also provide scopefor switching any curve widening from one sideof the road to the other in the normal manner(see Figure 11.1). However, the treatmentshown in Figure 11.5 may simplifyconstruction and have better appearance.

• Case 5, reverse circular curves with asufficient length of intervening tangent to havea section of the tangent with normal crossfall.On a two-lane, two-way road with the controlline down the centre of the road, a spacinggreater than about 1.5V to 2V metres (V inkm/h) results in a normally crowned length oftangent. See Case 5 in Figure 11.2A. It has beencommon practice to try to achieve a minimumof 30 m of normally crowned tangent. It is oftenpossible to increase the length of tangentbetween the two curves by slightly increasingthe spacing between the centres of the twocircular arcs. This case requires no change inthe method of applying any curve widening.

• Acceptable cases involving a transitioned curvethat is reversed with a circular curve can beinferred from the above cases. For the circularcurve, the superelevation runoff length (zerocrossfall to full superelevation for the curve)corresponds to the assumed transition path thatis described by vehicles.

Most other spacing of reverse curves will requiresome compromising of the superelevationdevelopment method and/or location. Case 6 inFigure 11.2A shows the most common situation.Case 6 occurs when there is a sufficient length oftangent between the two curves to have anacceptable rate of rotation of the pavement butthere is no room to have a length of tangent withnormal crossfall. The figure shows that thestandard superelevation development method andlocation can be accommodated (see 11.4.8 andFigure 11.1). In fact, the treatment of each trafficlane is no different from that for Cases 2 and 5above. However, it is usually preferable todirectly rotate the pavement between the twocurves as also shown in the figure. Thisalternative treatment simplifies setting out andconstruction. It also results in a lower rate ofrotation of the pavement so that drainage of thepavement surface must be checked. If a circularcurve is involved, there is scope to have moresuperelevation development occur on the tangentif this helps drainage of the pavement surface.

Case 6 requires no change in the method ofapplying any curve widening. However, if thepavement is directly rotated between the twocurves and each curve requires curve widening,the curve widening may be switched inconjunction with the pavement rotation.

Where the spacing between reverse curves willnot even permit acceptable compromising of thedevelopment of superelevation, the spacingbetween the curves will need to be adjusted. Inmany cases, this can be achieved by slightlyadjusting the spacing between the centres of thetwo circular arcs and having a differentorientation of the intervening tangent.

Where the operating speed of each curve isdifferent, the speed (V) to be used when designingthe superelevation development between thecurves depends on the following conditions:

• For cases 2 and 5 above, the curves are spacedsufficiently for the superelevation developmentto be treated separately for each curve andbased on the operating speed for each curve.With Case 1 and where a transitioned curve isinvolved in Case 6, the transitions and

May 2002 11-17


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superelevation development will normally bebased on the operating speeds of the respectivecurves.

• For the other cases, the spacing is close enoughfor one curve to influence the speed towards theend of the other curve. Where a decrease inspeed is involved, drivers will have to startslowing towards the end of the leading curve.Given the deceleration lengths (see Table13.16) and curve perception requirements for aspeed change that is in accordance with Section6.6 or Section 14.10, it is appropriate to basethe superelevation development on the mean ofthe operating speeds for the two curves. On aone-way road where a speed increase isinvolved, drivers are not likely to increasespeed until they are well into the second curve(see Figure 6A.2). Therefore, it is appropriateto base the superelevation development on theoperating speed for the leading curve.

Figure 11.2A Superelevation Development between Reverse Curves (Case 1)

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May 2002 11-19


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Figure 11.2A Superelevation Development between Reverse Curves (Cases 2 and 3A)

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Figure 11.2A Superelevation Development between Reverse Curves (Cases 3B and 4)

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Figure 11.2A Superelevation Development between Reverse Curves (Cases 5 and 6)

11.7 Broken Back Curves

Broken back curves (also called similar curves)are horizontal curves turning in the same directionjoined by a short length of straight or tworelatively small unidirectional curves connectedby a large radius curve. Broken back curvesshould be avoided if possible.

There are two general cases of broken back curve:

• Where the length of straight is less than about0.6V metres (based on about 2 seconds traveltime with V = operating speed in km/h), theseparation of the curves is usually small enoughso that there is no visual complication orproblems with superelevation. Such curvaturemay be tolerated in urban areas if there is aneed to maintain existing pavement andkerbing. However, it is often possible andpreferable to substitute a single curvedepending on the difference in the curve radii.

If a single curve cannot be substituted, the shortlength of straight and the difference in curveradii will cause the same problems formotorcyclists that compound curves cause.

• Where the length of straight is greater thanabout 0.6V and less than 2V to 4V, appearanceis compromised by there not being sufficientseparation of the curves. Within this range ofspacing, it is often possible and preferable tosubstitute a single curve depending on thedifference in the curve radii. The length 2Vmetres may be taken as the absolute minimumwith 4V metres the desirable minimum. Even adistance of 4V may be insufficient if bothcurves are visible at the same time over a longdistance. A length of straight greater than about3V will allow normal crossfall on the straight tobe regained for about 4 seconds of travel.

Figure 11.2B shows how superelevation may betreated when a broken back curve can not beavoided. All potential cases of broken back curveshould be checked for appearance by perspectiveviews in each direction. These can be readilygenerated by road design modelling systems.

Figure 11.2B Superelevation Development between Similar Curves

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11.8 Compound Curves

Compound curves are horizontal curves ofdifferent radii turning in the same direction with acommon tangent point. When radii less than about1000m are involved, compound curves may causeoperational problems due to problems withdrivers not perceiving the change in curvature anddrivers not anticipating a change in side frictiondemand.

Although not conclusive, some literature fromthroughout the world suggests that a smallerradius curve immediately following a largerradius curve (both turning in the same direction)gives drivers inadequate perception of the smallerradius curve which leads to a higher single vehicleaccident rate. This is a particular problem wherelimited visibility of the smaller radius curveexists; for example, where vegetation or a cut-faceon the inside of the larger radius curve limitsvisibility and driver perception of the smallerradius curve. Furthermore, compound curvesinvolve an undesirable change in side frictiondemand for motorcycles. Generally, this geometryshould be avoided.

Where compound curves can not be avoided:

• There should be no more than two curves ofdiminishing radii. And the radius of the smallercurve should be at least 2/3 of the radius of thelarger curve.

• Any change in design speed between the twocircular elements must be within the limitsgiven in Table 6.4.

• The change in side friction demand must meetthe criteria given in 11.4.2.

• Diminishing radii in the downhill directionshould be avoided on steep downgrades.

• On a one-way roadway a smaller curvepreceding a larger curve is preferable.

11.9 Transition Curves

11.9.1 General

Any motor vehicle describes a transition path as itchanges from a straight to a circular horizontalcurve and vice versa, or between the elements ofa compound curve. The way drivers, in general,steer a vehicle at speed results in a path thatprovides a reasonably uniform attainment ofcentripetal acceleration. For most curves, theaverage car driver can achieve a suitabletransition path within the limits of normal lanewidth. However, with particular combinations ofhigh speed, heavy vehicles and a large differencein curvature between successive geometricelements, the resultant vehicle transition path canresult in encroachment into adjoining lanes.Trucks have more problems because of theirgreater width, longer wheel base and heavier, lessresponsive steering. Also, trucks have a greaterswept path width on curves as explained inSection 11.10 on Curve Widening.

In cases where the combination of lane width,vehicle speed and curve radius do not allowsufficient space for a vehicle to describe a suitabletransition path, it is necessary to provide atransition curve between the tangent and thecircular arc. The need for such transition curveswas learned from the early days of railwaybuilding when problems were encountered withpassenger comfort and track wear due to thesudden application of curvature withuntransitioned curves. However, the fact that roadvehicles are not rigidly confined to a specific pathtogether with the characteristics of road vehiclesteering mean that shorter transition lengths aremore appropriate than those used for railways.This is why it is current road design practice tobase transition lengths on superelevation runofflength (see 11.4.6 and 11.9.2 for furtherexplanation) instead of a comfort criterion thatwas once used.

Transition curves provide the followingadvantages:

• A properly designed transition curve allows thevehicle's centripetal acceleration to increase ordecrease gradually as the vehicle enters or

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11-24 May 2002

leaves a circular curve. This transition curveminimizes encroachment on adjoining trafficlanes.

• The transition curve length provides aconvenient desirable arrangement forsuperelevation runoff. The change in the crossfall can be effected along the length of thetransition curve in a manner closely fitting theradius-speed relation for the vehicle traversingit.

• A transition facilitates the change in widthwhere the pavement section is to be widenedaround a circular curve. Use of transitionsprovides flexibility in the widening on sharpcurves.

• The appearance of the highway or street isenhanced by the application of transitions. Onmultilane divided roads where the curveoperating speeds are consistent with theoperating speeds on the straights (typically,greater than 100 km/h), there is scope toimprove the appearance by using longertransitions.

• Transitioned curves simplify the application ofcurve widening on closely spaced curves inconstrained mountainous situations becausethey avoid having to switch the widening fromone side of the road to the other.

The use of longer transitions than those based onnormal superelevation runoff length should beavoided when curve operating speeds are suchthat drivers have to reduce speed for the curve(see 11.12). Long transitions should only beconsidered in high speed curvilinear alignments(see Chapter 10) that are designed for a uniformspeed of travel.

Effect on Braking

When drivers brake on curves, a combination offorces apply on the tyres, effectively reducing themaximum force that can be developed for brakingor cornering. Articulated trucks also haveproblems with braking on curves because of thetendency of these vehicles to jackknife. Ontransitioned curves where the operating speed issuch that drivers are expected to reduce speed

prior to the curve, there is a likelihood of brakingon the transition if a long transition is used. Thisis because the change in curvature is not sufficientto identify the approximate start of the transition.

Effect on Overtaking

Transitions can reduce the length of straightsbetween curves, effectively reducing the length ofpossible overtaking opportunities.

Effect on Design

Transitions involve additional work both in designand construction. Transitions that are longer thanthe superelevation runoff length for the curvedesign speed, but still have the superelevationrunoff matched with the longer transition length,are more likely to have problems with drainage ofthe pavement surface. If this occurs, it will benecessary to compromise the application of thesuperelevation.

11.9.2 Use of Transitions

It is normal practice for horizontal curves to betransitioned, with the transition length based onthe superelevation runoff length for therecommended combination of speed, radius andsuperelevation in Tables 11.5 and 11.6 (see11.4.6). This length criterion provides theadvantages given in Section 11.9.1 whileminimising the negative effects on driverperception, braking and overtaking that areassociated with long transitions. It is also normalpractice to round the transition length upwards ifnecessary to the next standard length of: 40, 60,80, 100, 120 and 140 m. The rounding is primarilyin the interests of uniformity and to avoidattributing undue precision to the calculatedlength. However, in constrained situations,adjacent curves may dictate the use of a transitionlength rounded upwards if necessary to the nextmultiple of 5 m.

If circumstances favour a lower superelevationvalue than that shown in Tables 11.5 and 11.6 (seeNote 9 for each table), it is not appropriate to usea shorter transition for the reduced superelevation.This is because the recommended superelevation


11

values in Tables 11.5 and 11.6 yield transitionlengths for the various radius/speed combinationsthat adequately match driver behaviour. It is not acase of transition length being solely related towhatever superelevation is applied (and hence,superelevation runoff length).

Transition curves involve a shift of the circular arcin order to accommodate the transition (seeAppendix 11A). When this shift is less than 0.25to 0.3 m, the transition can be omitted becausethere is sufficient room for a vehicle to describe atransition path. The test for omitting the transitionshould be based on the length originallycalculated for the superelevation runoff (from 0%crossfall to the rounded superelevation value forthe curve) before any rounding of the length isapplied.

Due to the characteristics of vehicle steeringgeometry, there is a transient state (or distance)where it is possible for the front (steered) wheels ofa vehicle travelling at slow speed to follow acircular path while the steering angle is changedfrom straight ahead to the maximum angle neededto describe the turn. This is a major reason whytransitions are not needed for intersection turns andmost curves with a design speed below 60 km/h.

Tables 11.5 and 11.6 reflect the above criteria withthe recommended transition lengths for variouscombinations of radius and speed. The tables alsoshow when a transition should be omitted togetherwith the superelevation runoff length that is thenused. The superelevation runoff lengths foruntransitioned curves show the normal roundedvalues. However, in constrained situations,adjacent curves may dictate the use ofsuperelevation runoff lengths that are roundedupwards if necessary to the next multiple of 5 minstead of the normal 10 m. For untransitionedcurves, the superelevation is applied as describedin Section 11.4.8.

11.9.3 Types of Transitions

The standard form of transition curve is theclothoid spiral. Other curves have been used in thepast such as the cubic parabola, combination ofcubic parabola and cubic spiral, lemniscate and a

suitably large radius compound curve. All of thesealternatives were advocated on the basis that theyclosely matched the shape of the clothoid spiralbut were simpler to calculate. The disadvantage ofthe first three alternatives was that there werepoints of discontinuity which were conveniently'smoothed out' during construction. Furthermore,for simplifying calculations, arc lengths wereassumed to equal the tangent length of thetransition curve. The combination of cubicparabola and cubic spiral was widely used fortransition curves on the Queensland road networkprior to the adoption of the clothoid spiral. Manyoverlay and widening schemes will encounter thisform of transition. But it will be possible to fit acontrol line that contains clothoid spirals instead.

The clothoid spiral was adopted because thesimplifications that were assumed for thealternative forms could not be convenientlyincorporated into computer programs that came tobe used for design calculations. Furthermore,electronic calculators and portable computersmade field calculations feasible.

With a defining property of the clothoid spiralbeing that curvature (reciprocal of the radius)varies uniformly along the spiral (rather than thetangent), other convenient properties can bederived:

• If vehicles travel at a uniform speed and followthe transition curve closely enough, they willexperience a uniform change in centripetalacceleration as they enter and exit the circularsection of the curve.

• In turn, by matching superelevation runoff tothe transition, the uniform rate of application ofthe superelevation then results in a uniformattainment of side friction force.

The properties of transition curves that arerelevant to road design and the derivation ofequations for use in design calculations are givenin Appendix 11A.

May 2002 11-25


11

11-26 May 2002

11.10 Curve Widening

11.10.1 General

On smaller radius horizontal curves, traffic lanesmay need to be widened in order to maintain thelateral clearances that apply to vehicles on straightsections of road. This is due to the followingvehicle and driver characteristics:

• The rear wheels of a vehicle offtrack withrespect to the front wheels on a curve and thisalso causes the front overhang of a vehicle tohave a lateral component. The offtracking canbe either what is commonly called low speedofftracking (or more correctly, offtrackingwhen vehicles have a low side friction demand)or high speed offtracking (or more correctly,offtracking when vehicles have a high sidefriction demand). With the former form ofofftracking, the rear wheels offtrack towardsthe centre of the curve; with the latter form, therear wheels offtrack outwards from the centreof the curve.

• Vehicles deviate more from the centreline of alane when on a curve.

The amount of widening per lane depends on:

• The radius of the curve

• Type (size) of vehicle operating on the road.

• Some allowance for steering variation bydifferent drivers.

However, there is a lower practical limit towidening due to construction feasibility. For atwo-lane road, curve widening is omitted whenthe total widening is less than 0.5m.

Curve widening will be based on the designvehicle for the section of road. This means that aB-Double will be used on B-Double routes andthe appropriate type of road train will be used onroad train routes. For other roads, the designprime mover & semi-trailer will normally be usedgiven the incidence of these vehicles on statehighways and many other main roads (see Cox1998). On roads where the incidence of primemover & semi-trailer operation is small (less than

about 3% of AADT), or where for some otherreason the single unit truck is used as the designvehicle, the curve widening may be based onsingle unit truck operation; for example, the singleunit truck is commonly used as the design vehiclefor turns between minor streets and urban sub-arterial roads.

With one possible exception, curve widening isalways based on vehicles of the same typemeeting or passing each other; even when theincidence of design vehicles such as road trains islow. This is primarily in the interests of safetywith the lane marking accommodating the sweptpath of the design vehicle in each lane.Furthermore, by applying meeting rate equations(see Taneka and Troutbeck 1996) it can be shownthat a flow rate of three design vehicles per hourin each direction will result in an average of onemeeting per day on a given curve.

The possible exception relates to curve wideningon a climbing lane or descending lane (as distinctfrom overtaking lanes) on a Multi-CombinationVehicle route. In these cases it is acceptable forthe curve widening on the climbing lane ordescending lane (that is, the outer lane) to suit theMulti-Combination Vehicle and for the curvewidening on the adjacent through lane to suit aSingle Unit Truck/Bus. In such cases, the lanemarking must reflect the design lane widths.

Figure 11.3 shows the road width componentswhen two vehicles meet on a horizontal curve.These components are:

• Horizontal clearances. These are the same as onstraights.

• Swept path width of each vehicle; being greateron the curve due to offtracking of the rearwheels and the consequent horizontalcomponent of the front overhang.

Note that there is no additional steering allowancecomponent for difficulty of driving on curves.This has been the Austroads practice since 1979and has been based on the following assumptions:

• There is less steering variation with the designvehicle since it is a large commercial vehiclethat is driven by a professional driver.


11

• The swept path width of the design vehicleaccommodates the swept path width of smallervehicles. Plus it provides room for steeringvariation (and driver skill variation) with thesmaller vehicles.

• The now common use of full width or partwidth paved and sealed shoulders compensatesfor not having a steering allowance componentfor the design vehicle.

Furthermore, the swept path component is basedon low speed/low side friction demandofftracking even though normal operation onsome curves may involve high speed / high sidefriction demand offtracking that will be less thanthe low speed offtracking. The greater swept pathallowance in such cases is then assumed toprovide for probable increased steering variationand possible offtracking due to axle misalignmentand grade.

Table 11.8 shows the recommended curvewidening per lane for a range of curves and designvehicles. The need for widening ceases when thewidening is less than 0.25m (to fit the minimumpractical widening for a two lane road of 0.5m).Note that the table also indicates a lower limitwhere the curve should be designed with the aidof turning templates. This is because the angle ofturn starts to affect swept path width and theapplication of the curve widening will have to bechecked against vehicle swept path.

Due to the size of the current design vehicles, thecurve widening results in wide traffic lanes forcurves with a radius less than about 100 metres.However, at traffic flows greater than about 1000vehicles per hour, cars tend to form two laneswithin traffic lanes greater than about 4.6m wide.The width at which two lane operation starts tooccur is also dependent upon the horizontalcurvature. And this width approaches 10m in thecase of a circulating roadway of a roundabout(Austroads 1988, 1993).

Therefore, in practice, potential problems withwide traffic lanes due to curve widening are onlylikely to occur in urban areas. However, due toright of way constraints in urban areas, it willoften not be possible to provide full curve

widening for road train operation or B-doubleoperation or even semi-trailer operation. Figure11.4 shows how large vehicles have to beaccommodated in these cases. It can also be seenin the figure that roadway capacity is adverselyaffected.

Table 11.8 Curve Widening per Lane for CurrentDesign Vehicles, m

Radius SU PM & B- Type 1 Type 2m Truck Semi- double Road Road

or Bus trailer Train Train30.00 Use40.00 1.0350.00 0.82 Turning60.00 0.71 1.2770.00 0.59 1.03 1.3180.00 0.52 0.91 1.16 1.62 Templates90.00 0.46 0.81 1.03 1.44

100.00 0.41 0.71 0.90 1.26 1.80120.00 0.36 0.63 0.80 1.13 1.61140.00 0.32 0.56 0.71 1.00 1.43160.00 0.28 0.49 0.62 0.87 1.25180.00 0.24 0.42 0.53 0.74 1.07200.00 0.35 0.45 0.62 0.89250.00 0.29 0.37 0.51 0.74300.00 0.23 0.30 0.41 0.59350.00 0.26 0.35 0.51400.00 0.22 0.30 0.44450.00 0.27 0.39500.00 0.25 0.35600.00 0.21 0.30700.00 0.25800.00 0.22

Need for using Turning Templates determined byvariation in widening due to angle of turn. Need forcurve widening ceases when widening per lane< 0.25m.The curve widening for a given carriageway will be thewidening/lane x no. of lanes. The resultant total width ofthe traffic lanes may then be rounded to the nearestmultiple of 0.25m.Widening per lane is not dependent on the lane widthon the straight. The widening is intended to maintainthe horizontal clearances used on the straight.

May 2002 11-27


11

11-28 May 2002


11

Figure 11.3 Lane Width Components on a Horizontal Curve

Figure 11.4 Examples of Heavy Vehicle Operation on Multi-lane Roads with Insufficient Curve Widening

11.10.2 Application of CurveWidening

Except for small radius curves where it isnecessary to design curve widening and theapplication of curve widening with the aid ofturning templates (see Table 11.8), it has alwaysbeen AASHTO and Austroads practice to applycurve widening by tapering over the length of theroadway used for superelevation runoff. This isthe length where the outer lane goes from level tofull superelevation. In the case of a transitionedcurve, the superelevation runoff corresponds withthe plan transition. In the case of an untransitionedcurve, it is usually applied equidistant about thetangent point of the horizontal curve. Thisadequately matches where drivers make their owntransition when entering or leaving anuntransitioned horizontal curve.

The basis for tying the application of curvewidening to the superelevation runoff is largelyone of convenience. Also, the changing curvatureover the transition path has some correspondenceto the change in swept path width, and hence thechange in lane width. Most importantly, this hasbeen found to work in practice. Thecorrespondence to the change in swept path widthhas been checked for current prime mover &semi-trailer, B-double and road train operation(see Cox 1998).

For transitioned curves, it is normal practice toapply half of the curve widening to each side ofthe road (see Figure 11.1). However this meansthat the shift associated with the transition (shift =L²/24/R approximately) must be greater than thecurve widening that is applied to the outer side ofthe curve so that:

• the design vehicle will make use of thewidening; and

• for appearance.

This will usually only be a problem when thecurve widening has to suit a road train. And agreater proportion of the total widening will thenhave to be applied on the inside of the curve. Thepainted centreline will then be offset from thecontrol line in order to provide equal lane widths.

For untransitioned curves, it is normal practice toapply all of the curve widening to the inside of thecurve (see Figure 11.1). The painted centreline isthen offset from the control line in order toprovide equal lane widths. This practice aidsdrivers in making their own transition. Driverssteer slightly closer to the centre of the curvewhen making their own transition.

With closely spaced reverse untransitioned curves(Cases 3A and 3B, and possibly Case 4, in Section11.6), it will be necessary to modify how thecurve widening is applied between the two curves.This is necessary because the widening still has tobe fully applied by the normal 15m to 25mdistance into the curve in order to match vehicletracking characteristics. But when the curves areclosely spaced, it is necessary to start applying thecurve widening for the second curve at a pointback on the first curve. Again, this is to matchvehicle tracking characteristics.

Figure 11.5 shows how the modified practice isderived from the standard practice shown in Figure11.1. With both the standard and modifiedpractices, the full curve widening is achieved at thepoint where full superelevation is attained (denotedas distance L/2 into the curve). With both thestandard and modified practices, the curvewidening starts to be applied at a point that isdistance L/2 prior to the curve tangent point (orlonger distance if necessary). Figure 11.5 alsoshows an alternative treatment. This will often bepreferred when it improves appearance andsimplifies construction. The alternative treatmentalso gives drivers more latitude with where theyposition their vehicle. However, the range ofvehicle paths will still adequately match thechanging superelevation between the curves. Theapplication of the curve widening should always bechecked against vehicle swept paths in these cases.

May 2002 11-29


11

11.11 Radius to Meet SightDistanceRequirements

Figure 11.5 shows the relationship betweenhorizontal sight distance, curve radius and lateralclearance to an obstruction. It is valid when thesight distance at the appropriate design speed isnot greater than the length of the curve. Thisrelationship assumes that for a car, the driver’seye position and the object are in the centre of theinside lane (in terms of radius for the direction oftravel).

For a truck, the object is assumed to be in thecentre of the inside lane (in terms of radius for thedirection of travel). But the driver’s eye positionhas to be offset from the centre of the lane tobetter reflect the operation of these vehicles. For aleft hand curve, the driver’s eye position isassumed to be 1.25m inwards from the right handedge of the traffic lane. For a right hand curve ona one-way carriageway, the driver’s eye positionis assumed to be 2.85m from the left-hand edge ofthe traffic lane. The object is assumed to be in thecentre of the lane.

When the design sight distance is greater than thelength of curve, the line of sight may be checkedon a layout drawing or digital model by using theparameters shown in Figure 11.6.

11.12 Curve PerceptionIssues

Chapter 6 explains how operating speeds tend to beuniform on rural roads in high speed environments(100km/h and greater) when all of the geometricelements have been designed to a standard equal toor greater than the speed environment. This mayalso apply to some urban roads and motorwayswhere the speed environment is less than 100km/h.Chapter 6 also explains how operating speeds tendto vary on rural roads and some urban roads inlower speed environments (less than 100km/h). Inthese cases, drivers will slow down, if necessary,for a curve and increase speed where possible onstraights and larger radius horizontal curves. Theoperating speeds on long straights and large radiuscurves will match the speed environment (bydefinition). Chapter 6 further explains that theremay also be cases on roads in high speed

11-30 May 2002


11

Figure 11.5 Application of Curve Widening on Closely Spaced Reverse Untransitioned Curves

May 2002 11-31


11

Figure 11.6 Horizontal Stopping Sight Distance for Type F Restriction to Visibility(Reaction Time = 2.5 seconds)

Where:R = radius in metresS = sight distanceO = offset in metresNote: Use this formula only when S ≤ the length ofcircular curve. Angles in degrees.

Eye position is in centre of lane for cars;1.15m from right-hand edge of lane fortrucks turning left; 2.85m from left-handedge of lane for trucks turning right.To determine offset for truck:1. Use stopping sight distance for truck and

radius of centre of lane to calculate offset.2. Subtract 0.30 from the value obtained in

step 1 for truck turning left.3. Add 0.55 to the value obtained in step 1

for a truck turning right.

(11.10) )]R

OR([cos65.28

RS

(11.9) )]R

S65.28cos(1[RO

1 −=

−=

−

Eye height (see Chapter 9)

11-32 May 2002

environments where operating speeds have to varydue to some local constraint or short constrainedsection of road that could not be designed to astandard equal to the speed environment.

The foregoing explanation of vehicle operatingspeeds means that many horizontal curves aredesigned on the basis of the speed that drivers areprepared to slow down to for that curve. Whenapproaching a curve, drivers regulate their speedfrom the apparent curvature of the road ahead.The amount of superelevation on a curve does notnormally influence curve entry speeds(Kanellaidis, 1999) - probably because drivers donot perceive the superelevation.

In practice, there is some variation in curve entryspeeds. In turn, this gives rise to three importantcurve perception issues:

• Horizontal Curve Perception Distance.When vehicles have to slow down for ahorizontal curve, drivers must see a sufficientamount of the curve in order to perceive itscurvature, react and slow down appropriatelyfor the curve. Normally, sufficient sightdistance for a horizontal curve is providedthrough the practice of not having a horizontalcurve start over a crest and through the normalprinciples of alignment coordination (seeChapter 10). However, there are times wherethis can not be avoided. Chapter 9 sets out thecriteria that should be applied in order to checkthat sufficient visibility is provided for anycurve where vehicles have to slow down for it.In some cases, provision of horizontal curveperception distance may require a larger crestvertical curve than is required for stoppingsight distance.

• Horizontal Curves in the Range of 300m to440m. New South Wales experience has foundthat horizontal curves with a radius in the rangeof 300m to 440m should be avoided foroperating speeds greater than 70 km/h. Thecurves are deceptive to the driver in that theyappear to be able to be travelled at higherspeeds than is actually possible. As a result,drivers may not slow down appropriately forthem. Therefore, these curves should only beused when the predicted curve operating speed

meets the following conditions:- The curve operating speed is at least 5km/h

below the limiting curve speed standard (see6.1.5);

- The decrease in operating speed from thepreceding horizontal alignment element isless than or equal to 5km/h; or

- If the decrease in operating speed from thepreceding horizontal alignment element isgreater than 5km/h but less than or equal to10km/h, then the decrease in limiting curvespeed between the elements must also beless than or equal to 10km/h (see 6.6.3).

Closely spaced preceding curves help achievethese conditions. In practice, this restriction onthe use of curves in the range 300m to 440mlimits the range of curves suitable for 80km/h,and particularly, 90km/h.

• Length of Transition Curves. The use oflonger transitions than those based on thenormal superelevation runoff length should beavoided when curve operating speeds are suchthat drivers have to reduce speed for the curve.In these circumstances, longer transitions maycause drivers to perceive a higher standard ofcurvature than there is, with consequentincreased speed and friction demand on thecircular section of the curve. Overseas studieshave found that there have been higher accidentrates on some curves with a combination oflong transition (typically with more than twicethe length based on superelevation runoff) andsmall to medium radius.

11.13 Curves on SteepDown Grades

On steep downgrades there is more chance ofsome drivers tending to overdrive horizontalcurves. Therefore, on steep downgrades, theminimum horizontal curve radius for the designspeed should be increased in accordance with thefollowing empirical formula:

Rmin on grade = Rabs min [1+(G-3)/10] (11.11)

where


11

Rabs min = absolute minimum radius for the designspeed from Table 11.5 or 11.6 (m). That is, theradius based on absolute maximum side frictioncoefficient and maximum superelevation.

G = grade (%).

Radii are in metres.

The resulting minimum radius will commonly beof a size where Table 11.5 or 11.6 recommends alower value of superelevation than the maximumshown in Table 11.2. However, it will usually bebetter to use the maximum superelevation shownin Table 11.2. The higher superelevation providesan additional margin for drivers who tend tooverdrive the curves.

Where it is not possible to increase the minimumradius in accordance with equation 11.11, it willbe necessary to try to provide either:

• Preceding curves that limit the speed ofvehicles. Using such curves will reinforce thelow speed environment that is needed on steepdowngrades (see 6.2.1).

• Using curves that require less than the desirablemaximum value of side friction. The use ofmaximum (or close to maximum)superelevation values may also be necessary.

References

AASHO (1954): A Policy on Geometric Design ofRural Highways. American Association of StateHighway Officials. Washington, D.C.

AASHTO (1994): A Policy on Geometric Designof Highways and Streets. American Association ofState Highway and Transportation Officials.Washington, D.C.

Australian Road Research Board (1993):Unsealed Roads Manual - Guidelines to GoodPractice. Melbourne.

Austroads (1989): Rural Road Design – A guideto the Geometric Design of Rural Roads. Sydney.

Austroads (1988): Guide to Traffic EngineeringPractice. Part 5 - Intersections at Grade. Sydney.

Austroads (1993): Guide to Traffic EngineeringPractice. Part 6 - Roundabouts. Sydney.

Austroads (1999): Guide to Traffic EngineeringPractice. Part 15 - Motor Cycle Safety. Sydney.

Austroads (2000): May 2000 draft of the Guide tothe Geometric Design of Major Urban Roads.Sydney.

Cox, R.L. (1972): Clothoid (Euler) Spiral Theoryas Applied to Transition Curves. QueenslandDepartment of Main Roads.

Cox, R.L. (1998): A Review of Geometric RoadDesign Standards based on Vehicle Swept Path.Technology Transfer Forum July 1998.Queensland Department of Main Roads.

Kanellaidis, G. (1999): Road CurveSuperelevation Design: Current Practices andProposed Approach. Vol 8, No 2, Jinee 1999,Road and Transport Research.

McLean, J., Tziotis, M. and Gunitallake, T.(2001): Final Draft, Designing for Trucks - When,Where and How? Austroads, Sydney.

Rahmann, W.M. (1981): Design of HorizontalCurves Using the Revised NAASRA Method -Side Friction Factors. Road Design References.Main Roads Department Queensland.

Taneka, D. and Troutbeck, R. (1996): Equationsfor Meeting and Overtaking Rates. PhysicalInfrastructure Centre Research Report 96-11.Queensland University of Technology.

Transportation Association of Canada (1999):Geometric Design Guide for Canadian Roads.Ottawa.

Relationship to OtherChapters

• Close relationship with Chapter 6 - SpeedParameters;

• MUST be read in conjunction with Chapter 10(coordination) and Chapter 12 (verticalalignment);

May 2002 11-33


11

11-34 May 2002

• Sight distance requirements are defined inChapter 9;

• Elements of Chapters 4, 6, 13, 14, 15, 16, 20and 22 require information from this Chapter.


11

Appendix 11A: ClothoidSpiral Transition Curves

11A.1 Properties of the ClothoidSpiral

Basic properties of the Clothoid (or Euler Spiral)are given below. More detailed information,including the derivation of all the equations, isavailable in Cox, 1972.

Radians - Length Relationships

The clothoid spiral is a curve that is defined by theproperty that curvature (which is 1/R) variesuniformly along the length of spiral. Therefore,the curvature at the end of the spiral isproportional to the length:

or RL = Constant

where L = length of clothoidR = radius of clothoid at end

It follows that for any intermediate point P in aclothoid of length L, the following expressionapplies:

(11A.1)

where Li =length of clothoid to point PRi = instantaneous radius at point P

Since RL is constant, any particular transitioncurve can be defined by an “RL value”.

Deflection Angles

From equation 11A.1 it can be derived that theangle between the straight and a line tangential tothe clothoid at any point is proportional to thesquare of the spiral length to that point from thebeginning (or origin) of the spiral.

At the end of the transition curve, this angle,called the spiral angle (θ) is given by

For any intermediate point on the clothoid, thefollowing expression applies:

(11A.2)

where θi = angle between straight and tangent topoint P.

Figure 11A.1 Clothoid Spiral Transition Curve

Coordinates of a Point on the Spiral

A clothoid spiral at any general location andorientation does not lend itself to the directcalculation of the Cartestian coordinates for anypoint on the spiral. In practice, the mostconvenient means for setting out points on aclothoid spiral is through the direct or indirect useof distances along the tangent through the spiralorigin and corresponding offsets from the tangent.This means that the spiral origin becomes theorigin of a local coordinate system and the tangentthrough the origin becomes the X axis. Thispractice further suits road construction practicesince the tangent line also has to be set out forother purposes. Even so, the calculation ofcoordinates in the local coordinate system is morecomplex than for many other curves but areacceptable through the use of computers orcalculators. The equations for the X and Ycoordinates (and other properties) are in the formof infinite series. The infinite series, however,rapidly converge, and usually only the first twoterms of each expression, and occasionally thefirst three, are significant for the standardtransition curves used in road design.

radiansRL2L2

ii =θ∴

2ii )LL(=

θθ

radiansR2L=θ

i

iLL

RR =

LR1 ∝

May 2002 11-35


11

11-36 May 2002

From equation 11A.2 it can be derived that:

Figure 11A.2

11A.2 Curve Components

Standard curve component descriptions andnotation are shown in Figure 11A.3 and Table11A.1.

Figure 11A.3

...)RL2(!5.11

L)RL2(!3.7

L)RL2(3

L5

11i

3

7i

3i −+−=

...))!7(15)!5(11)!3(73

(Ly7i

5i

3ii

i +θ−θ+θ−θ=

...)RL2(!613

L

)RL2(!49L

)RL2(!25LL

6

13i

4

9i

2

5i

i

+×

−

−×

+×

−=

...))!6(13)!4(9)!2(5

1(Lx6i

4i

2i

i +θ−θ+θ−=


11

Table 11A.1

Q Alignment number SLT Long Tangent of spiralBgBT Bearing Back Tangent SST Short Tangent of spiralBgAT Bearing Ahead Tangent SEC Secant lengthR Radius of circular curve TC Tangent to CircleT Tangent length of circular curve CT Circle to TangentTT Total Tangent length CC Circle to Circle∆ Intersection Angle at point of intersection (in degrees) SS Spiral to SpiralL Spiral Length TS Tangent to SpiralS Shift Distance SC Spiral to Circleθ Spiral Angle CS Circle to Spiralβ Intersection Angle of circular curve (Vertex angle) ST Spiral to TangentARC Total arc length IP Intersection Point

(circular curve plus spirals) SP Secant PointFor transitioned curves with unequal spiral lengths, the abbreviations TT, L, S, θ, SLT, SST, are to be suffixed bythe letter A (ahead) or B (back), e.g. LA=Spiral Length Ahead.

11A.3 Formulae for TransitionCurves

Spiral Angle

Offset from tangent to SCwhere θ isexpressed inradians

Projection of transition curve on tangent

where θ isexpressed inradians

Offset from tangent to transition curve at anyintermediate point P

where

Projection of portion of transition curve from anypoint P to tangent

where

...LR3456

LLR40

LL 44

9i

22

5i

i −+−=

radiansRL2L2

ii =θ

...)21610

1(Lx4i

2i

i −θ+θ−=

55

11i

33

7i

3i

LR42240L

LR336L

RL6L +−=

radiansRL2L2

ii =θ

...)1320423

(Ly5i

3ii

i −θ+θ−θ=

...R3456

LR40

LL 4

5

2

3−+−=

...)21610

1(Lx42

s −θ+θ−=

...R42240

LR336

LR6

L5

6

3

42−+−=

...)1320423

(LY53

s −θ+θ−θ=

radiansR2L=θ

May 2002 11-37


11

Figure 11A.4

11-38 May 2002

Shift distance (S)

This distance is the offset from the tangent to thepoint on the circle at which the radius is normal toit.


Displacement of the tangent point along thetangent due to the introduction of the transitioncurve (k)


For a transitioned curve with equal length spirals:

Total arc length (ARC)where ∆ is expressedin radians.

Total tangent length (TT)

Secant length

R2

sec)SR(SEC −∆+=

k2

tan)cosRY( s +∆θ+=

k2

tan)SR(TT +∆+=

LRARC +∆=

...R34560

LR240

L2L

4

5

2

3−+−=

...)2160602

1(L42

−θ+θ−=

θ−= sinRxk s

...R506880

LR2688

LR24

L5

6

3

42−+−=

...)1584033612

(L53

−θ+θ−θ=

)cos1(RYS s θ−−=


11

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RPDM_Chapter11