Horizontal curve

1. Geometric Design of Highways
Dr. Eman Magdy Ibrahim Youssef
Assistant professor, Civil Engineering Department, Delta Higher Institute of Engineering and Technology
February- 2022
1

2. 2
The roadway horizontal alignment is a
series of horizontal tangents (straight
roadway sections), circular curves, and
spiral transitions.
Horizontal Alignment
The geometric quality of road
when seen from above in “plan”
view.
Plan view and profile
1
5
+
0
0
1
6
+
0
0
1
7
+
0
0
18+00
19+00
20+00
21+00
2
2
+
0
0
2
3
+
0
0
2
4
+
0
0
200
300
400
500
600
700
700
15+00 16+00 17+00 18+00 19+00 20+00 21+00 22+00 23+00 24+00
plan
profile

3. 3
Highway Design Example

4. 4
General Types of Horizontal Alignment
Straight Tangent-Curve
Continuously Curved Road

5. 5
Horizontal Alignment
Horizontal
Alignment
Geometric
Elements of
Horizontal
Curves
Transition
or Spiral
Curves
Sight
Distance
Super
elevation
Design

6. 6
Horizontal Alignment
Purpose:
To provide change in direction to the C.L of a road
Process:
When a vehicle transverse a horizontal curve, the
centrifugal force acts horizontally outwards through
the center of gravity of the vehicle
P = W V2 / g R
The centrifugal force acting on a vehicle passing through a
horizontal curve has two effects:
1- Overturning Effect
2-Transverse Skidding Effect

7. 7
Types of Horizontal Curves
Simple Curve Compound Curve

8. 8
Types of Horizontal Curves
Broken Back Curve Reverse Curve
Spiral

9. 9
Curve with Spiral Transition
Types of Horizontal Curves
Simple Curve

10. 10
Geometry of Circular Curves
PC PT
PI
T
Direction
PI: Point of Intersection
PC: Point of Curvature
PT: Point of Tangency
T: Tangent Length
R: Radius of Curve
L: Length of Curve
Lc: Chord Length
: Deflection Angle
180 -  : Intersection Angle
E: External ordinate
M: Middle ordinate
L
R R


M
E
/2
Lc

11. 11
Degree of Curvature
Arc definition (Da)
Central angle subtended by 100 feet of arc (along curve)
Note: 100'  one full station
100ft
Da
Relationship of Da and R
so
R'
2π
100'
360
Da


a
D
5729.58
R 
R
5729.58
Da 
Metric equivalents by conversion, e.g.
R
1746.38
Da 
2R
Da
360
ft/m)
m)(3.28083
(R
5729.58
Da 
(m)
(ft)
(ft)

12. Design Elements of Horizontal Curves
Deflection Angle
Deflection Angle
( )
2
I
T RTan
= 
Also known as Δ

13. Design Elements of Horizontal Curves
5729.58
D
R
=
Larger D = smaller Radius
100
I
L
D
=

14. Design Elements of Horizontal Curves
E=External Distance
M=Length of Middle
Ordinate

15. Design Elements of Horizontal Curves
L=Length of Long Cord

16. Super Elevation
Purpose:
To provide change in direction to the C.L of a road
Process:
When a vehicle transverse a horizontal curve, the centrifugal force acts horizontally
outwards through the center of gravity of the vehicle
The centrifugal force acting on a vehicle passing through a horizontal curve has two effects:
1-Overturning Effect
2-Transverse Skidding Effect
P/W = V2 /g R
P

17. Overturning Effect
∑M A = P h – w b/2
0.0 = P h – w b/2
P h = w b/2
P/W (Centrifugal Ratio) = b/2h
This means there is a danger of overturning when the Centrifugal
Ratio or V2/ GR attains a value of b/2h

18. Transverse Skidding Effect
P = f RA + f RB
P = f (RA + RB)
P = f W
P/W (Centrifugal Ratio) = f
This means there is a danger of Transverse Skidding when the
Centrifugal Ratio or V2/ GR attains a value of f

19. Horizontal Alignment
• Design based on appropriate relationship between design speed and curvature and
their relationship with side friction and super elevation
• Along circular path, vehicle attempts to maintain its direction (via inertia)
• Turning the front wheels, side friction and super elevation generate an acceleration to
offset inertia
Super elevation “e” & side friction coefficient “f” on horizontal
curves

20. Relationship between speed v, e, f, and curve radius, R
gR
v
ef
f
e 2
01
.
0
1
01
.
0



In practice:
1
01
.
0
1 
 ef and g is calculated:
R
v
R
v
f
e
15
067
.
0
01
.
0
2
2



v : vehicle speed, ft/s
R: radius of curve, ft
e: rate of superelevation, percent
f: side friction factor (lateral ratio)

21. Radius Calculation
Rmin = ___V2______
15(e + f)
Where:
Rmin is the minimum radius in feet
V = velocity (mph)
e = super elevation
f = friction (15 = gravity and unit
conversion)
Radius Calculation
 Rmin uses max e and max f (defined by AASHTO, DOT, and
graphed in Green Book) and design speed
 f is a function of speed, roadway surface, weather condition,
tire condition, and based on comfort – drivers brake, make
sudden lane changes, and change position within a lane
when acceleration around a curve becomes “uncomfortable”
 AASHTO: 0.5 @ 20 mph with new tires and wet pavement
to 0.35 @ 60 mph
 f decreases as speed increases (less tire/pavement contact)

22. normally, f is given ( from 0.12 to 0.16), e is also known when 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 max
max
2
min
f
e
V
R


Radius Calculation

23. Max SUPERELEVATION (e)
Controlled by 4 factors:
• Climate conditions (amount of
ice and snow)
• Terrain (flat, rolling,
mountainous)
• Type of area (rural or urban)
• Frequency of slow moving
vehicles who might be
influenced by high super
elevation rates
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.

24. Radius Calculation
Example: assume a maximum e of 8% and design speed of 60 mph, what is the
minimum radius?
fmax = 0.12 (from Green Book)
Rmin = _____602_______________
15(0.08 + 0.12)
Rmin = 1200 feet
For emax = 4%? (urban situation)
Rmin = _____602
15(0.04 + 0.12)
Rmin = 1,500 feet

25. Minimum Safe Radius
R = V2/127 (e+f)
Where:
R: Radius in meters
V: Speed in Kilometers per hour
e: super elevation, 0.06-0.08
f: Side-friction factor, 0.14 for 80 kph

26. Horizontal Curves Spiral (Transition)
A spiral curve is a curve which has an infinitely long radius at its junction with the tangent end
of the curve; this radius is gradually reduced in length until it becomes the same as the radius
of the circular curve with which it joins.
Curve with Spiral Transition

27. Advantages of Spirals Curve
a. Provides natural, easy to follow, path for drivers (less encroachment, promotes more
uniform speeds), lateral force increases and decreases gradually
b. Provides location for superelevation runoff (not part on tangent/curve)
c. Provides transition in width when horizontal curve is widened
d. Aesthetic
Source: Iowa DOT Design Manual

28. Minimum Length of Spirals
Larger 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 centripetal acceleration (ft/s3)
(use 1ft/s3 to 3 ft/s3 for highway)

29. Super elevation Design
Desirable super elevation:
for R > Rmin
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 the desirable super elevation, but the
equation above offers a simple way to do it. The other methods are presented in the next
few overheads.
2
max
d
V
e f
gR
 

30. Attainment of Super elevation - General
1. Tangent to super elevation
2. Must be done gradually over a distance without appreciable reduction in
speed or safety and with comfort
3. Change in pavement slope should be consistent over a distance
4. Methods
a. Rotate pavement about centerline
b. Rotate about inner edge of pavement
c. Rotate about outside edge of pavement

31. Super elevation Transition Section
• Tangent Run out Section
• Length of roadway needed to accomplish
a change in outside-lane cross slope from
normal cross slope rate to zero
For rotation about centerline
• Super elevation Runoff Section
• Length of roadway needed to accomplish
a change in outside-lane cross slope from
0 to full super elevation or vice versa
• For undivided highways with cross-
section rotated about centerline

32. Method 1: (Centerline)
c
c
s
s
C = w *0.02
S = w * e
1 : 200
L1 = 200 c Ls = 200 s or 1.6 v3 /R
c

33. Method 2: (Inside Edge)
c c
s
s
C = w *0.02
S = w * e
c
c

34. Method 3:(Outside Edge)
c
c s
s
C = w *0.02
S = w * e
c
c
c

35. Which Method?
• In overall sense, the method of rotation about the centerline (Method 1) is
usually the most adaptable
• Method 2 is usually used when drainage is a critical component in the
design
• In the end, an infinite number of profile arrangements are possible; they
depend on drainage, aesthetic, topography among others

36. Widening on Horizontal Curves
1- Mechanical Widening
Wm = n l2/2 R
l = length of wheel base (m)
n = Number of lanes
R = radius of the curve
2- Psychological Widening
Wps = V/9.5 √ R
V = Design speed (Km/hr)

37. Sight Distance on Horizontal Curve
Minimum sight distance (for safety) should be equal to the safe stopping distance

38. Example
• Consider
• Curve with R = 1909.86 ft
• Sight obstruction (e.g. building) 12 ft from
curve (M = 12 ft)
• Question
• Recall: car going 60 mph needs SSD of 475 ft
• Does curve have enough SSD for a car going
60 mph?
Sight Distance on Horizontal Curve
M = 12'
LC = ?
R = 1909.86'
 
427.5'
)
35'34"
sin(6
1909.86
2
2
Δ
2Rsin
LC






 
2
cos
1
R
M 


1909.86'
12'
1909.86'
R
M
R
2
cos






35'34"
6
2




• Available sight distance = 428';
Required SSD60 = 475'
• Not enough sight distance for 60 mph
• Post lower speed limit or redesign
curve
 
2
2Rsin
LC 


39. ANY QUESTIONS