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7/21/2019 Horizontal Curves basics
http://slidepdf.com/reader/full/horizontal-curves-basics 1/22
Geometric Design of Highways
Highway Alignment is a three-dimensional problem – Design & Construction would be difficult in 3-D so highway
design is split into three 2-D problems
– Horizontal alignment, vertical alignment, cross-section
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Austin, TX
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Near Cincinnati, OH
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Components of Highway Design
Plan View
Profile View
Horizontal Alignment
Vertical Alignment
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Horizontal Alignment
Today‟s Class:
• Components of the horizontal alignment
• Properties of a simple circular curve
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Horizontal Alignment
Tangents Curves
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Tangents & Curves
Tangent
Curve
Tangent to
Circular Curve
Tangent to
Spiral Curve to
Circular Curve
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Layout of a Simple Horizontal Curve
R = Radius of Circular Curve
BC = Beginning of Curve
(or PC = Point of Curvature)
EC = End of Curve
(or PT = Point of Tangency)
PI = Point of Intersection
T = Tangent Length
(T = PI – BC = EC - PI)
L = Length of Curvature
(L = EC – BC)
M = Middle Ordinate
E = External Distance
C = Chord Length
Δ = Deflection Angle
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Properties of Circular CurvesDegree of Curvature• Traditionally, the “steepness” of the curvature is defined by either the radius
(R) or the degree of curvature (D)
• In highway work we use the ARC definition
• Degree of curvature = angle subtended by an arc of length 100 feet
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Degree of CurvatureEquation for D
Degree of curvature = angle subtended by an arc of length 100 feet
By simple ratio: D/360 = 100/2*Pi*R
Therefore
R = 5730 / D
(Degree of curvature is not used with metric units because D is defined in terms of feet.)
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Length of Curve
By simple ratio: D/ Δ = ?
D/ Δ = 100/L
L = 100 Δ / D
Therefore
L = 100 Δ / D
Or (from R = 5730 / D, substitute for D = 5730/R)
L = Δ R / 57.30
(D is not Δ .)
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Properties of
Circular Curves
Other Formulas…
Tangent: T = R tan(Δ/2)
Chord: C = 2R sin(Δ/2)
Mid Ordinate: M = R – R cos(Δ/2)
External Distance: E = R sec(Δ/2) - R
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Spiral CurveA transition curve is sometimes used in horizontal alignment design
It is used to provide a gradual transition between tangent sections and circular curve sections.
Different types of transition curve may be used but the most common is the Euler Spiral.
Properties of Euler Spiral
(reference: Surveying: Principles and Applications, Kavanagh and Bird, Prentice Hall]
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Degree of Curvature of a spiral at any point is proportional to its length at that point
The spiral curve is defined by „k‟ the rate of increase in degree of curvature per
station (100 ft)
In other words,
k = 100 D/ Ls
Characteristics of Euler Spiral
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Degree of Curvature of a spiral at any point is proportional to its length at that point
The spiral curve is defined by „k‟ the rate of increase in degree of curvature per
station (100 ft)
In other words,
k = 100 D/ Ls
Characteristics of Euler Spiral
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As with circular curve the central angle is also important for spiralRecall for circular curve
Δc = Lc D / 100
But for spiral
Δs = Ls D / 200
Central (or Deflection) Angle of Euler Spiral
The total deflection angle for a
spiral/circular curve system is
Δ = Δc + 2 Δs
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Length of Euler Spiral
Note: The total length of curve (circular plus spirals) is longer
than the original circular curve by one spiral leg
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Example Calculation – Spiral and Circular Curve
The central angle for a curve is 24degrees - the radius of the circular curve
selected for the location is 1000 ft.
Determine the length of the curve (with no spiral)
L = 100 Δ / D or
L = Δ R / 57.30 = 24*1000/57.30 = 418.8 ft
R = 5730 / D >> D = 5.73 degree
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Example Calculation – Spiral and Circular Curve
The central angle for a curve is 24degrees - the radius of the circular curve
selected for the location is 1000 ft.
If a spiral with central angle of 4
degrees is selected for use,
determine the
i) k for the spiral,
ii) ii) length of each spiral leg,
iii) iii) total length of curve
Δs = 4 degrees
Δs = Ls D / 200 >> 4 = Ls * 5.73/200 >>
Ls = 139.6 ft
k = 100 D/ Ls = 100 * 5.73/ 139.76 = 4.1 degree/100 feet
Total Length of curve = length with no spiral + Ls = 418.8+139.76 = 558.4 feet