horizontal curves basics

7/21/2019 Horizontal Curves basics

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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



Austin, TX



Near Cincinnati, OH



Components of Highway Design

Plan View

Profile View

Horizontal Alignment

Vertical Alignment




Today‟s Class:

• Components of the horizontal alignment

• Properties of a simple circular curve




Tangents Curves



Tangents & Curves

Tangent

Curve

Tangent to

Circular Curve

Tangent to

Spiral Curve to

Circular Curve



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



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



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.)



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 .)



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



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]



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



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



Length of Euler Spiral

Note: The total length of curve (circular plus spirals) is longer

than the original circular curve by one spiral leg



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



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

horizontal curves basics

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