lec 1 _ superelevation

7/29/2019 Lec 1 _ Superelevation

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SuperelevationSuperelevation&&

Spiral CurvesSpiral Curves

SuperelevationSuperelevation&&

Spiral CurvesSpiral Curves

Horizontal Curves• Purpose:

To provide change in direction to the C.Lof a road

• Process:When a vehicle transverse a horizontal

curve, the centrifugal force actshorizontally outwards through thecenter of gravity of the vehicle



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The centrifugal force acting on a vehicle

passing through a horizontal curvehas two effects:

1-Overturning

Effect

2-Transverse

Skidding EffectP = W V2 / g R

P

P/W = V2 /g R

P

h

C. G

A B

b/2b/2

W

11--Overturning EffectOverturning Effect

∑M A = P h – w b/2

0.0 = P h – w b/2

P h = w b/2P/W (Centrifugal Ratio) = b/2h

This means there is a danger ofoverturning when the Centrifugal Ratio

or V2/ GR attains a value of b/2h



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P

h

C. G

A B

b/2b/2

W

2-Transverse Skidding Effect

P = f R A – f RB

P = f (R A + RB)

P = f W

P/W (Centrifugal Ratio) = f

This means there is a danger ofTransverse Skidding when the

Centrifugal Ratio or V2/ GR attains a

RA RB

fRBfRA

Horizontal Alignment

• Design based on appropriate relationshipbetween design speed and curvature andtheir relationship with side friction and

superelevation• Along circular path, vehicle attempts to

maintain its direction (via inertia)

• Turning the front wheels, side frictionand superelevation generate anacceleration to offset inertia



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Superelevation “e”

&side friction coefficient “f”

on horizontal curves

e

f

Relationship between speed v, e, f, and curve radius, R

gR

v

ef

f e2

01.01

01.0=

−

+

In practice:

101.01 ≈− ef and g is calculated:

R

v

R

v f e

15

067.001.0

22

==+

v : vehicle speed, ft/s

R: radius of curve, ft

e : rate of superelevation, percent

f: side friction factor (lateral ratio)



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Radius CalculationRmin = ___V2 ______

15(e + f)

Where:

Rmin is the minimum radius in feet

V = velocity (mph)

e = superelevation

f = friction (15 = gravity and unitconversion)

Radius Calculation• Rmin uses max e and max f (defined by AASHTO,

DOT, and graphed in Green Book) and designspeed

• f is a function of speed, roadway surface,weather condition, tire condition, and based on

comfort – drivers brake, make sudden lanechanges, and change position within a lane whenacceleration around a curve becomes“uncomfortable”

• AASHTO: 0.5 @ 20 mph with new tires and wetpavement to 0.35 @ 60 mph

• f decreases as speed increases (lesstire/pavement contact)



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normally, f is given ( from 0.12 to 0.16)

, e is also knownwhen the location of the designed highway is known.

The rest is to determine

v when R is known, or determine R when v is given.

Application: Minimum radius

)(15 maxmax

2

min f e

V R+

=

Max eMax e•• Controlled byControlled by 44 factors:factors:

– Climate conditions (amount of iceand snow)

– Terrain (flat, rolling, mountainous)– Type of area (rural or urban)

– Frequency of slow moving vehicleswho might be influenced by highsuperelevation rates



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Source: A

Policy on

Geometric Design of

Highways and

Streets (The

Green Book).

Washington,

DC. American

Association of

State Highway

and

Transportation

Officials,

2001 4th Ed.

Radius Calculation (Example)Design radius example: assume a

maximum e of 8% and design speedof 60 mph, what is the minimumradius?

fmax = 0.12 (from Green Book)

Rmin = _____602 _________ _______

15(0.08 + 0.12)

Rmin = 1200 feet



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Radius Calculation (Example)For emax = 4%? (urban situation)

Rmin = _____602 _________ _______

15(0.04 + 0.12)

Rmin = 1,500 feet

Minimum Safe RadiusR = V2/127 (e+f)

Where:

R: Radius in metersV: Speed in Kilometers per hour

e: superelevation, 0.06-0.08

f: Side-friction factor, 0.14 for 80kmph



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Horizontal CurvesSpiral (Transition)

Straight road

section

R = ∞

R = Rn

Spiral Curve

A spiral curve is a curve which has aninfinitely long radius at its junctionwith the tangent end of the curve;

this radius is gradually reduced inlength until it becomes the same asthe radius of the circular curve withwhich it joins.



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SC

ST

Curve with SpiralTransitionCircular Curve

Tangent

Tangent to Spiral

Spiral to Tangent

Spiral

TS

Spiral to Curve

CS

Curve to Spiral

Location ofTransition Sections



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Spiral Curve Transitions

• Vehicles follow as transition path asthey enter or leave a horizontalcurve

• Combination of high speed and sharpcurvature can result in lateral shifts

in position and encroachment onadjoining lanes

Spirals

1. Advantagesa. Provides natural, easy to follow, path

for drivers (less encroachment,promotes more uniform speeds), lateral

force increases and decreasesgraduallyb. Provides location for superelevation

runoff (not part on tangent/curve)c. Provides transition in width when

horizontal curve is widenedd. Aesthetic



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Source: Iowa DOT Design Manual

Minimum Length of SpiralsLarger of L = 3.15 V3 L = 1.6 V3

RC R

Where:

L = minimum length of spiral (ft)

V = speed (mph)

R = curve radius (ft)

C = rate of increase in centripetalacceleration (ft/s3)

(use 1ft/s3 -> 3 ft/s3 for highway)



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Length of SpiralsMore practical = assume L = to length of

superelevation runoff

The radius of a spiral (by definition) variesinversely with distance from the TS frominfinite (at TS) to circular curve radius atSC.

Maximum Length ofSpirals

• Safety problems may occur whenspiral curves are too long – driversunderestimate sharpness ofapproaching curve (driverexpectancy)



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Superelevation DesignDesirable superelevation:

for R > R min

Where,

V= design speed in ft/s or m/s

g = gravity (9.81 m/s2 or 32.2 ft/s2)

R = radius in ft or m

Various methods are available for determining thedesirable superelevation, but the equation aboveoffers a simple way to do it. The other methods arepresented in the next few overheads.

2

m a xd

V e f

g R= −

Attainment of Superelevation -

General1. Tangent to superelevation

2. Must be done gradually over a distance withoutappreciable reduction in speed or safety andwith comfort

3. Change in pavement slope should be consistentover a distance

4. Methods

a. Rotate pavement about centerline

b. Rotate about inner edge of pavement

c. Rotate about outside edge of pavement



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SuperelevationTransition Section

• Tangent Runout Section

• Superelevation Runoff Section

Tangent Runout Section

• Length of roadway needed toaccomplish a change in outside-lanecross slope from normal cross slope

rate to zero

For rotation about centerline



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Superelevation Runoff

Section

• Length of roadway needed toaccomplish a change in outside-lanecross slope from 0 to fullsuperelevation or vice versa

• For undivided highways with cross-

section rotated about centerline

Method 1Centerline

c cc

s

s

C = w *0.02

S = w * e

1 : 200

L1 = 200 c

Ls = 200 s or 1.6 v 3 /R



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Method 2Inside Edge

c c

s

s

C = w *0.02

S = w * e

cc

Method 3Outside Edge

c

c s

sC = w *0.02

S = w * e

c

c

c



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Which Method?• In overall sense, the method of rotation

about the centerline (Method 1) is usuallythe most adaptable

• Method 2 is usually used when drainage is acritical component in the design

• In the end, an infinite number of profilearrangements are possible; they depend ondrainage, aesthetic, topography amongothers



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Median width

Pivot points

Example where pivot points are important

Bad design

Good design

15 ft to 60 ft

Source: CalTrans Design Manual online,

http://www.dot.ca.gov/hq/oppd/hdm/pdf/chp0200.pdf

http://www.dot.ca.gov/hq/oppd/hdm/pdf/chp

http://www.dot.ca.gov/hq/oppd/hdm/pdf/chp



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Attainment Length Location

Without a horizontal transition curve (spiralor compound), superelevation must beattained over a length that includes thetangent and the curve

Typical: distribution of runoff is 2/3 ontangent and 1/3 on curve if no spiral

Widening on Horizontal Curves

1- Mechanical Widening

Wm = n l2/2 R

l = length of wheel base (m)

n = Number of lanesR = radius of the curve

2- Psychological Widening

Wps = V/9.5 √ R

V = Design speed (Km/hr)



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Front axle

Rear axle

Sight Distance

on

Horizontal CurveMinimum sight distance (for safety) should be equal to the safe

stopping distance

R R

HSOPC PT

Sight Obstruction

Line of sight

Centerline of inside lane

Highway Centerline

sight



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Stopping Sight Distance &

Horizontal Sightline Offset (HSO)

Exhibit 3-53, p 225.



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Sight Distance onHorizontal Curves• Issue

• Standards

– Set by American Association of StateHighway and Transportation Officials(AASHTO)

Example of Using SSD• Consider

– Curve with R = 1909.86 ft

– Sight obstruction (e.g. building) 12 ftfrom curve (M = 12 ft)

• Question– Recall: car going 60 mph needs SSD of

475 ft

– Does curve have enough SSD for a cargoing 60 mph?



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M = 12'LC = ?

R = 1909.86'

427.5'

)35'34"sin(61909.862

2 Δ2RsinLC

=

°××=

=

( )2

cos1RM ∆−=

1909.86'

12'1909.86'

R

MR

2cos

−=

−=∆∴

35'34"62

°=∆∴

• Available sight distance = 428'; RequiredSSD60 = 475'

• Not enough sight distance for 60 mph– Post lower speed limit or redesign curve

( )2

2RsinLC ∆=

lec 1 _ superelevation

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