lec 1 _ superelevation
TRANSCRIPT
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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
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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 ∆=