- APSEd

# Transition Curves: Concept, Types and Lenght Calculations

A transition curve is a horizontal curve having a variable radius i.e., an infinite radius at the tangential point to the selected radius of the primary horizontal curve. A horizontal curve provides a transition between two tangent strips of roads and transition curves help make the transition smoother. Concepts, types and length calculations of a transition curve are discussed further in this blog.

## Horizontal Curves in a Highway

Before getting into the transition curve it is important to understand horizontal curves. As said, horizontal curves connect two tangent strips of roads i.e., it provides a transition between them.

If horizontal curves are not present then two tangential strips of roads may never meet or could be made to forcibly meet with sharp cuts. These sharp cuts could prove fatal to the vehicles. Also having sharp cuts forces the vehicle to reduce their speed less than the design speed which is not an intention of the highway. Therefore, horizontal curves are absolutely necessary.

Even after providing a horizontal curve, it is important to have a smooth transition between the straight road and the horizontal curve. A transition curve is used to achieve this as discussed further.

## Transition Curves in Highways

As said, a transition curve is a horizontal curve having a variable radius i.e., infinity at the tangential point to the selected radius of the horizontal curve. The infinite radius at the tangential point implies that the transition curve is straight at the point where it offsets the straight road.

The main purpose of a transition curve is to provide a smooth transmission between the straight road and the main horizontal curve. By having a variable radius, a gradual increase in centrifugal force is achieved thus helping in having a smooth transition without any sudden jerk. A transition curve connecting a straight road and a horizontal curve on either side of a turn is shown below.

### Transition Curves Concept

Centrifugal force is a force that acts on a body moving along a curved path. It acts outwards from the centre and tries to push the body away from the centre. Centrifugal force is expressed as ** m * (V^2) / R,** where m is the mass of the body, V is the velocity of the body and R is the radius of the curved path. Centrifugal force is the main cause of the jerk due to the sudden transition experienced while transitioning a horizontal curve.

To avoid this sudden jerk there should be a gradual increase in the centrifugal force. As the transition curve has an infinite radius at starting tangential point the centrifugal force at that point is zero. Thereafter the radius of the transition curve increases gradually which in turn increases the centrifugal force gradually without any jerk. At the start of the main horizontal curve, the radius of the transition curve will be equal to the radius of the horizontal curve and the centrifugal force reaches its max value thereby completing the transition from a straight road to a horizontal curve in a gradual manner.

### Types of Transition Curves

There are three types of transition curves based on their shape as listed below.

Spiral/Clothoid

Bernoulli's Lemniscate

Cubic parabola

## Transition Curve Lenght Calculation

Lenght of the transition curve is an important factor as it directly influences the rate of attainment of centrifugal force. ** Lenght of a transition curve is inversely proportional to the radius of the horizontal curve. **There are three ways by which a transition curve's length can be fixed as discussed below.

**1. Rate of Change of Centrifugal Acceleration**

This method is based on the allowable rate of change of centrifugal acceleration. The centrifugal acceleration is expressed as, ** (V^2)/R**. The rate of change of centrifugal force is given as,

c = (V^2)/(R*t),

where t = V/L (length of transition curves)

Therefore, Lenght of transition curve (L) = (V^3)/(c*R)

c is empirically given as** c = 80/(75 + v), **

where,

c is in m/(s^3)

v is design speed in kmph

c is usually taken between 0.5 to 0.8 m/(s^3)

**2. Rate of Attainment of Superelevation**

This method is based on the rate of attainment of superelevation. There are two subdivisions within this method as mentioned below.

**Inner edge rotation**

Lenght of transition curve (L) = N*e*B

where,

N - the rate of change of superelevation (usually 150)

e - superelevation rate

B - width of carriage including widening

**Centre line rotation**

Lenght of transition curve (L) = (N*e*B)/2

where,

N - the rate of change of superelevation (usually 150)

e - superelevation rate

B - width of carriage including widening

**3. Empirical Formula**

Thi method is a purely empirical method based on road terrain. Details of the same is shared in the table below.

Terrain | Cross slope | Length of transition curve |

Plain | 0 to 10% | 2.7*(V^2)/R |

Rolling | 10 to 25% | 2.7*(V^2)/R |

Mountain | 25 to 60% | (V^2)/R |