1. 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

2. Austin, TX

3. Near Cincinnati, OH

6. Components of Highway Design
Horizontal Alignment
Plan View
Vertical Alignment
Profile View

7. Horizontal Alignment Today’s Class:
Components of the horizontal alignment
Properties of a simple circular curve

8. Horizontal Alignment
Tangents
Curves

10. Tangents & Curves Tangent Curve Tangent to Circular Curve Tangent to
Spiral Curve to
Circular Curve

11. 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

12. Properties of Circular Curves
Degree 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

13. Degree of Curvature Equation 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.)

14. Length of Curve Or (from R = 5730 / D, substitute for D = 5730/R)
By simple ratio: D/ Δ = ?
D/ Δ = 100/L
L = 100 Δ / D
Therefore
Or (from R = 5730 / D, substitute for D = 5730/R)
L = Δ R / 57.30
(D is not Δ .)

15. 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

16. Spiral Curve
A 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]

17. Characteristics of Euler Spiral
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

18. Characteristics of Euler Spiral
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

19. Central (or Deflection) Angle of Euler Spiral
As with circular curve the central angle is also important for spiral
Recall for circular curve
Δc = Lc D / 100
But for spiral
Δs = Ls D / 200
The total deflection angle for a spiral/circular curve system is
Δ = Δc + 2 Δs

20. Length of Euler Spiral
Note: The total length of curve (circular plus spirals) is longer than the original circular curve by one spiral leg

21. Example Calculation – Spiral and Circular Curve
The central angle for a curve is 24 degrees - 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 / = 24*1000/57.30 = ft
R = 5730 / D >> D = 5.73 degree

22. Example Calculation – Spiral and Circular Curve
The central angle for a curve is 24 degrees - 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
k for the spiral,
ii) length of each spiral leg,
iii) total length of curve
Δs = 4 degrees
Δs = Ls D / 200 >> 4 = Ls * 5.73/200 >> Ls = ft
k = 100 D/ Ls = 100 * 5.73/ = 4.1 degree/100 feet
Total Length of curve = length with no spiral + Ls = = feet