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Superelevation & Spiral Curves

Horizontal Curves
Purpose:
To provide change in direction to the C.L of a road

Process: When a vehicle transverse a horizontal curve, the centrifugal force acts horizontally outwards through the center of gravity of the vehicle

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The centrifugal force acting on a vehicle passing through a horizontal curve has two effects:
P

1-Overturning Effect 2-Transverse P = W V2 / g R Skidding Effect

P/W = V2 /g R

1-Overturning Effect
P

C. G

M A = P h w b/2 0.0 = P h w b/2

A
b/2 b/2

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

This means there is a danger of overturning when the Centrifugal Ratio or V2/ GR attains a value of b/2h

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2-Transverse Skidding Effect

P C. G

P = f RA f RB P = f (RA + RB)

fRA
A
b/2 b/2

fRB
B

P= fW P/W (Centrifugal Ratio) = f

RA

RB

This means there is a danger of Transverse Skidding when the Centrifugal Ratio or V2/ GR attains a value of f

Horizontal Alignment
Design based on appropriate relationship between design speed and curvature and their relationship with side friction and superelevation Along circular path, vehicle attempts to maintain its direction (via inertia) Turning the front wheels, side friction and superelevation generate an acceleration to offset inertia

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Superelevation e & side friction coefficient f on horizontal curves

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

0.01e + f v2 = 1 0.01ef gR
In practice:

1 0.01ef 1

and g is calculated:

0.067v 2 v2 0.01e + f = = R 15 R
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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Rmin = ___V2______ 15(e + f) Where: Rmin is the minimum radius in feet V = velocity (mph) e = superelevation f = friction (15 = gravity and unit conversion)

Radius Calculation

Rmin uses max e and max f (defined by AASHTO, DOT, and graphed in Green Book) and design speed f is a function of speed, roadway surface, weather condition, tire condition, and based on comfort drivers brake, make sudden lane changes, and change position within a lane when acceleration around a curve becomes uncomfortable AASHTO: 0.5 @ 20 mph with new tires and wet pavement to 0.35 @ 60 mph f decreases as speed increases (less tire/pavement contact)

Radius Calculation

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normally, f is given ( from 0.12 to 0.16) , e is also known when 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

Rmin

V2 = 15(emax + f max )

Max e
Controlled by 4 factors: Climate conditions (amount of ice and snow) Terrain (flat, rolling, mountainous) Type of area (rural or urban) Frequency of slow moving vehicles who might be influenced by high superelevation 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 speed of 60 mph, what is the minimum radius? 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 Radius

R = V2/127 (e+f) Where: R: Radius in meters V: Speed in Kilometers per hour e: superelevation, 0.06-0.08 f: Side-friction factor, 0.14 for 80 kmph

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Horizontal Curves Spiral (Transition)

R = Rn

R=

Straight road section

Spiral Curve
A spiral curve is a curve which has an infinitely long radius at its junction with the tangent end of the curve; this radius is gradually reduced in length until it becomes the same as the radius of the circular curve with which it joins.

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Curve with Spiral Transition

Circular Curve Tangent Spiral

SC TS Spiral to Curve

CS Curve to Spiral ST

Spiral to Tangent Tangent to Spiral

Location of Transition Sections

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Spiral Curve Transitions

Vehicles follow as transition path as they enter or leave a horizontal curve Combination of high speed and sharp curvature can result in lateral shifts in position and encroachment on adjoining lanes

Spirals
1. Advantages
a. Provides natural, easy to follow, path for drivers (less encroachment, promotes more uniform speeds), lateral force increases and decreases gradually b. Provides location for superelevation runoff (not part on tangent/curve) c. Provides transition in width when horizontal curve is widened d. Aesthetic

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Source: Iowa DOT Design Manual

Minimum Length of Spirals

Larger of L = 3.15 V3 RC L = 1.6 V3 R

Where: L = minimum length of spiral (ft) V = speed (mph) R = curve radius (ft) C = rate of increase in centripetal acceleration (ft/s3) (use 1ft/s3 -> 3 ft/s3 for highway)

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Length of Spirals
More practical = assume L = to length of superelevation runoff The radius of a spiral (by definition) varies inversely with distance from the TS from infinite (at TS) to circular curve radius at SC.

Maximum Length of Spirals

Safety problems may occur when spiral curves are too long drivers underestimate sharpness of approaching curve (driver expectancy)

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Superelevation Design
Desirable superelevation:
ed = V2 f m ax gR

for R > Rmin

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 the desirable superelevation, but the equation above offers a simple way to do it. The other methods are presented in the next few overheads.

Attainment of Superelevation General

1. Tangent to superelevation 2. Must be done gradually over a distance without appreciable reduction in speed or safety and with comfort 3. Change in pavement slope should be consistent over 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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Superelevation Transition Section

Tangent Runout Section Superelevation Runoff Section

Tangent Runout Section

Length of roadway needed to accomplish a change in outside-lane cross slope from normal cross slope rate to zero

For rotation about centerline

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Superelevation Runoff Section

Length of roadway needed to accomplish a change in outside-lane cross slope from 0 to full superelevation or vice versa For undivided highways with crosssection rotated about centerline

Method 1
Centerline
Ls = 200 s or 1.6 v3 /R L1 = 200 c C = w *0.02 c S = w * e 1 : 200 c c s s

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Inside Edge

Method 2

C = w *0.02 c S = w * e c c c

s s

Outside Edge

Method 3

c c

c c

s s

C = w *0.02

S = w * e

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Which Method?
In overall sense, the method of rotation about the centerline (Method 1) is usually the most adaptable Method 2 is usually used when drainage is a critical component in the design In the end, an infinite number of profile arrangements are possible; they depend on drainage, aesthetic, topography among others

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Example where pivot points are important Bad design

Pivot points Good design

Median width 15 ft to 60 ft

Source: CalTrans Design Manual online, http://www.dot.ca.gov/hq/oppd/hdm/pdf/chp0200.pdf

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Attainment Length Location

Without a horizontal transition curve (spiral or compound), superelevation must be attained over a length that includes the tangent and the curve Typical: distribution of runoff is 2/3 on tangent 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 lanes R = 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 Curve

Minimum sight distance (for safety) should be equal to the safe stopping distance
Highway Centerline sight

PC
Line of sight

HSO

PT

Sight Obstruction Centerline of inside lane

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Stopping Sight Distance & Horizontal Sightline Offset (HSO) Exhibit 3-53, p 225.

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Sight Distance on Horizontal Curves

Issue

Standards
Set by American Association of State Highway and Transportation Officials (AASHTO)

Example of Using SSD

Consider
Curve with R = 1909.86 ft Sight obstruction (e.g. building) 12 ft from curve (M = 12 ft)

Question
Recall: car going 60 mph needs SSD of 475 ft Does curve have enough SSD for a car going 60 mph?

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M = R 1 cos

2 LC = 2Rsin 2
cos = 2 R M R 1909.86' 12' = 1909.86'
= 635'34" 2

( )

M = 12' LC = ?

R = 1909.86'

LC = 2Rsin = 427.5'

( 2)

= 2 1909.86 sin(635'34" )

Available sight distance = 428'; Required SSD60 = 475' Not enough sight distance for 60 mph
Post lower speed limit or redesign curve