26. Horizontal Curves

7/30/2019 26. Horizontal Curves

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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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Todays Class:

Components of the horizontal alignment

Properties of a simple circular curve


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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 = PIBC = EC - PI)

L = Length of Curvature

(L = ECBC)

M = Middle Ordinate

E = External Distance

C = Chord Length

= 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 unitsbecause D is defined in terms of feet.)


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Length of Curve

By simple ratio: D/ = ?

D/ = 100/L

L = 100 / D

Therefore

L = 100 / D

Or (from R = 5730 / D, substitute for D = 5730/R)

L = R / 57.30

(D is not .)


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Properties of

Circular Curves

Other Formulas

Tangent: T = R tan(/2)

Chord: C = 2R sin(/2)

Mid Ordinate: M = RR cos(/2)

External Distance: E = R sec(/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

c = Lc D / 100

But for spiral

s = Ls D / 200

Central (or Deflection) Angleof Euler Spiral

The total deflection angle for a

spiral/circular curve system is

= c+ 2 s


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Lengthof 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 CalculationSpiral 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 / D or

L = R / 57.30 = 24*1000/57.30 = 418.8 ft

R = 5730 / D >> D = 5.73 degree


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Example CalculationSpiral 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

s = 4 degrees

s = 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

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26. Horizontal Curves