Horizontal Transition Curves for Highways and Its Calculation

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What is transition curve and when it is needed?

Transition curve is a curve in plan which is provided to change the horizontal alignment from straight to circular curve gradually means the radius of transition curve varies between infinity to R or R to infinity.

Objectives for providing transition curves

1. For the gradual introduction Centrifugal force
2. To introduce super elevation gradually
3. To introduce extra widening gradually
4. To provide comfort for the driver that is to enable smooth vehicle operation on road.
5. To enhance aesthetics of highways.

Types of transition curves

• Spiral or clothoid
• Cubic parabola
• Lemniscate

IRC recommends Spiral or clothoid as the ideal transition curve due to following reasons:

1. It satisfies that rate of change of centrifugal acceleration is constant i.e., L_s.R = constant. Where L_s = length of transition curve R = radius of curve.
2. The calculation and field implementation of spiral curve is simple and easy.
3. It enhances aesthetics also.

Determining length of transition curve

The length of transition curve can be calculated by 3 conditions.

• Based on rate of change of acceleration
• Based on rate of change of super elevation and extra widening
• Based in IRC empirical formula

Based on rate of change of acceleration

Radius of curve is infinity at the tangent point and hence centrifugal acceleration is zero. Similarly at the straight end radius of curve has minimum value means centrifugal acceleration is maximum. So, the rate of change of centrifugal acceleration should be adopted such that the design should not cause any discomfort to the drivers. Let Ls be the length of transition curve and a vehicle is moving with a speed of V m/s.

Force P = (mV²/R)

Since it is similar to F= ma

P = m (V²/R)

Therefore, centrifugal acceleration = V²/R Let “C” be the coefficient of rate of change of centrifugal acceleration.

C = (V²/R). (1/t)

Where t= time taken to travel the transition curve of length Ls, with a speed of V

t = Ls/V

C = (V²/R). (V/Ls)

Ls = (V³/CR)

According to IRC, C = 80/(75+V) and C should be (0.5<C<0.8).

Based on rate of change of superelevation and extra widening

Let 1 in N is the allowable rate of introduction of super elevation and E is the raise of the outer edge with respect to inner edge. W is the normal width of pavement in meters. We is the extra width of pavement in meters. And e is the rate of superelevation.

E = (W+We).e

Therefore length of transition curve, Ls = (W+We).e.N If the pavement outer edge is raised and inner edge is depressed with respect to center of pavement then,

Ls = [(W+We).e.N]/2

Typical range of introduction of super elevation is as follows according to IRC

Type of terrain	Rate of super elevation 1 in N
For plain and rolling terrains	1 in 150
For built up areas	1 in 100
For hilly and steep terrains	1 in 60

Based on IRC empirical formula

IRC given some direct formulae for finding the length of transition curve.

• For plain and ruling terrain:

Ls = 2.7 (V²/R)

• For mountainous and steep terrains

Ls = V²/R

Hence these are the three criteria to determine the length of transition curve. The maximum of above three conditions will be considered as the length of transition curve.