1. c.k.pithawalla college of engineering & technology
Transition curve
Prepered by:
Name
Enrollment no:
Patel harikrushna
Paatel harsh d.
Patel jayvadan
Patel kaushal
Patel kerul
Guidance by:
Mital d. dholawala

2. Transition Curve Definition
A transition curve differs from a circular curve in that its radius is always changing.
As one would expect, such curves involve more complex formulae than the curves
with a constant radius and their design is
more complex.

3. Radial Force and Design Speed
When a vehicle moves on a curve, thrre are two forces acting:
Weight of the vehicle (W)
Centrifugal force (p)
A vehicle of mass m, travelling at a constant speed v, along a
curve of radius r, is subjected to a centrifugal force P such that:

4. The centrifugal force (p) is inversely proportional to the radius of the curve.

5. The need for Transition Curves
Circular curves are limited in road designs due to the forces which act on a vehicle
as they travel around a bend. Transition curves are used to introduce those forces
gradually and uniformly thus ensuring the safety of passenger.
Transition curves have much more complex formulae and are more difficult to set
out on site than circular curves as a result of the varying radius.

6. Superelevation
The difference in height between the two sides of the road is known as the superelevation (SE).

7. usually slopes between 2.5 – 7 %
Super elevation:
b=width
W=mg
Normal Force
SE=Super Elevation
usually slopes between 2.5 – 7 % SE

8. Types of Transition Curve
There are two types of curved used to form the transitional section of a composite
or wholly transitional curve. These are:
The clothoid
The cubic parabola

9. The clothoid
For a transition curve the equation rl = K must apply i.e. the radius must reduce in proportion to the length. This is the property of a spiral and one curve which has
This property is the clothoid.

10. The equation of the clothoid can be derived from the above diagram, which shows two points close together (M and N) on a transition curve of length LT:
Φ is the deviation angle between the tangent at M and the straight TI
Δ is the tangential angle to M from T with reference to TI
x is the offset to M from the straight TI at a distance y from T
l is the length from point T to any point M on the curve (not shown)
δl is the length along the curve from M to N
Φl is the angle subtended by the arc δl of radius r

11. Vertical curve

12. Crest Curve G2 G3 G1 Sag Curve
Like the horizontal alignment, the vertical alignment is made up of tangent and curves
In this case the curve is a parabolic curve rather than a circular or spiral curve
Crest Curve G2 G3 G1
Sag Curve

13. Maximum and Minimum Grade
One important design consideration is the determination of the maximum and minimum grade that can be allowed on the tangent section
The minimum grade used is typically 0.5%
The maximum grade is generally a function of the
Design Speed
Terrain (Level, Rolling, Mountainous)
On high speed facilities such as freeways the maximum grade is generally kept to 5% where the terrain allows (3% is desirable since anything larger starts to affect the operations of trucks)
At 30 mph design speed the acceptable maximum is in the range of 7 to 12 %

14. Maximum Grade
Harlech, Gwynedd, UK (G = 34%)

15. Properties of Vertical Curves
BVC G1 G2
EVC PI
L/2
L/2 L
Change in grade: A = G2 - G1
where G is expressed as % (positive /, negative \)
For a crest curve, A is negative
For a sag curve, A is positive

16. Properties of Vertical Curves
BVC G1 G2
EVC PI
L/2
L/2 L
Rate of change of curvature: K = L / |A|
Which is a gentler curve - small K or large K?

17. Properties of Vertical Curves
BVC G1 G2
EVC PI
L/2
L/2 L
Rate of change of grade: r = (g2 - g1) / L
where,
g is expressed as a ratio (positive /, negative \)
L is expressed in feet or meters
Note – K and r are both measuring the same characteristic of the curve
but in different ways

18. Properties of Vertical Curves
BVC G1
Elevation = y G2
EVC PI L
Equation for determining the elevation at any point on the curve
y = y0 + g1x + 1/2 rx2
where,
y0 = elevation at the BVC
g = grade expressed as a ratio
x = horizontal distance from BVC
r = rate of change of grade expressed as ratio

19. Properties of Vertical Curves
Distance BVC to the turning point (high/low point on curve)
xt = -(g1/r)
This can be derived as follows
y = y0 + g1x + 1/2 rx2
dy/dx = g1 + rx
At the turning point, dy/dx = 0
0 = g1 + rxt
Therefore,

20. Properties of Vertical Curves
BVC G1 G2
EVC PI
Example:
G1 = -1% G2 = +2%
Elevation of PI = m
Station of EVC = 25+00
Station of PI = 24+00
Length of curve?
L/2 = Sta. EVC – Sta. PI
L/2 = 2500 m m = 100 m
L = 200 m

21. Properties of Vertical Curves
BVC G1 G2
EVC PI
Example:
G1 = -1% G2 = +2%
Elevation of PI = m
Station of EVC = 25+00
Station of PI = 24+00
r - value?
r = (g2 - g1)/L
r = ( [-0.01])/200 m
r = / meter

22. Properties of Vertical Curves
BVC
EVC G2 G1 PI
Example:
G1 = -1% G2 = +2%
Elevation of PI = m
Station of EVC = 25+00
Station of PI = 24+00
Station of low point?
x = -(g1/r)
x = -([-0.01] / [ /m])
x = m
Station = [23+00] m
Station 23+67

23. Properties of Vertical Curves
BVC G1 G2
EVC PI
Example:
G1 = -1% G2 = +2%
Elevation of PI = m
Station of EVC = 25+00
Station of PI = 24+00
Elevation at low point?
y = y0 + g1x + 1/2 rx2
y0 = Elev. BVC
Elev. BVC = Elev. PI - g1L/2
Elev. BVC = 125 m - [-0.01][100 m]
Elev. BVC = 126 m

24. Properties of Vertical Curves
BVC
EVC G2 G1 PI
Example:
G1 = -1% G2 = +2%
Elevation of PI = m
Station of EVC = 25+00
Station of PI = 24+00
Elevation at low point?
y = y0 + g1x + 1/2 rx2
y = 126 m + [-0.01][66.67 m] + 1/2 [ /m][66.67 m]2
y = m

25. Properties of Vertical Curves
BVC
EVC G2 G1 PI
Elevation at station 23+50?
y = 126 m + [-0.01][50 m] /2 [ /m][50 m]2
y = m
Elevation at station 24+50?
y = 126 m + [-0.01][150 m] /2 [ /m][150 m]2
y = m
Example:
G1 = -1% G2 = +2%
Elevation of PI = m
Station of EVC = 25+00
Station of PI = 24+00

26. Design of Vertical Curves

27. Design of Vertical Curves
The first step in the design is to determine the minimum length (or minimum K) for a given design speed.
Factors affecting the minimum length include
Sufficient sight distance
Driver comfort
Appearance

28. Design of Vertical Curves
Crest Vertical Curve
If sight distance requirements are satisfied then safety, comfort, and appearance requirements are also satisfied.
h1 = height of driver’s eyes, in ft
h2 = height of object, in ft

29. Design of Vertical Curves
Sag Vertical Curve
Stopping sight distance not an issue for sag vertical curves
Instead the design controls are one of the following
Headlight sight distance
Rider comfort
Drainage
Appearance

30. Design of Vertical Curves
Crest Vertical Curve
Equation relating sight distance to minimum length
From AASHTO:
h1 ≈ 3.5 ft
h2 ≈ 0.5 ft (stopping sight distance)
h3 ≈ 4.25 ft (passing sight distance)

31. Cubic parabola transition curve

32. Cubic parabola in railway applications
Cubic parabola is used in transition curves of the railway. The cubic
parabola function is y=kx^3 (1)

33. The “main” elements in railway transition curve are: The radius of curvature
at the end of transition, the length L of the curve, the length l of its
projection on x axis and the coefficient k. Two of these elements must be
given in order for the curve to be defined, the other two can be calculated
as described in this article.

34. The 'secondary elements” of the curve are: the coordinates Rx and Ry of the
center of curvature at the end of the transition, the shift f between the
curve R and the x axis, the angle τ of the tangent to the curve at the end of
the transition and and Yl, the y coordinate of the curve at length l. When the
“main” elements are known, the “secondary” elements can easily be calculated.

35. Fig. 1. Elements of transition curve

36. The Transition Curves (Spiral Curves)
The transition curve (spiral) is a curve that has a varying radius. It is used on
railroads and most modem highways. It has the following purposes:
1- Provide a gradual transition from the tangent (r=∞ )to a simple circular curve
with radius R
2- Allows for gradual application of superelevation.

37. decrease gradually as the vehicle enters and leaves the circular curve
Advantages of the spiral curves:
1- Provide a natural easy to follow path such that the lateral force increase and
decrease gradually as the vehicle enters and leaves the circular curve
2- The length of the transition curve provides a suitable location for the
superelevation runoff.
3- The spiral curve facilitate the transition in width where the travelled way is
widened on circular curve.
4- The appearance of the highway is enhanced by the application of spiral curves.

38. When transition curves are not provided, drivers tend to
create their own transition curves by moving laterally
within their travel lane and sometimes the adjoining
lane, which is risky not only for them but also for other
road users.

39. Maximum length of spiral.
International experience indicates that there is a need to
limit the length of spiral transition curves. Safety problems have been found to occur
on spiral curves that are long (relative to the length of the circular curve). Such
problems occur when the spiral is so long as to mislead the driver about the sharpness
of the approaching curve.