Valley curves are a type of vertical curves used in the vertical alignment of highways. The length of a valley curve is calculated based on two criteria i.e., headlight distance (or stopping sight distance) and passenger comfort criteria.

## Valley Curves Concept

As said, valley curves are a type of vertical curves used in the vertical alignment of highways. It is also called a sag curve.

#### Where is a Valley Curve Used?

A valley curve has upward concavity. Such a situation is possible only when a descending gradient meets the ascending gradient of a highway. Therefore, valley curves are located at places where a descending portion of the highway meets the ascending section of the highway.

#### The Shape of a Valley Curve

Generally, the cubic parabola is adopted for a valley curve. But, for small angles of deviations, all types of transition curves like a spiral, lemniscate, and cubic parabola are suitable.

## Design Criteria of Valley Curves

### Comfort Criteria

While traversing a valley curve, the centrifugal force acts downwards and exerts pressure on tyre. The tyre is now subjected to both the weight of the car and centrifugal force due to the valley curve. This causes discomfort to the passengers. Therefore, the length of a valley curve is designed so as to minimize the discomfort to passengers.

The comfort criteria can be satisfied only by transition curves which help in gradually increasing and decreasing the centrifugal force as the car traverses the valley curve. Therefore, the length of the valley curve as per comfort criteria is calculated as the length of two transition curves for descending and ascending sections of the highway.

### Headlight Sight Distance Criteria

The headlight sight distance is the amount of highway that should be visible at any time in night conditions in order for the vehicle to stop before the obstacle. Numerically it is the same as the stopping sight distance.

In a valley, during the daytime, there is no problem in satisfying the sight distance. But during the nighttime, it becomes a problem. Therefore headlight sight distance is also considered a criterion.

## Valley Curve Length Formula

### Comfort Criteria

As said, the comfort criteria can be satisfied only by transition curves. Therefore, the length of the valley curve is calculated as the length of two transition curves for descending and ascending sections of the highway. The formula is given as,

L = 2 * Lt1 = 2 * ((V^3)*N/C)^(1/2)

where,

C = rate of change of centrifugal acceleration. Generally fixed at 0.6 m/s^3

N = |(±n1) - (±n2)|

n1 - descending gradient which is in -ve

n2 - ascending gradient which is in +ve

V - design speed of the highway in m/s

### Headlight Sight Distance Criteria

Length of valley curve as per headlight sight distance is derived using basic trigonometry. The formula is given as,

For L > SSD

L = (N*S^2) / (1.5 + 0.035*S)

For L < SSD

L = 2S - ((1.5 + 0.035*S) / N)

where,

S - headlight sight distance (stopping sight distance)

N = |(±n1) - (±n2)|

n1 - descending gradient which is in -ve

n2 - ascending gradient which is in +ve

V - design speed of the highway in m/s

Initially, we don't know whether the length of the curve is greater or lesser than the sight distance. Therefore, it is calculated on a trial and error basis. The formulas are better illustrated below.

## Example Problem

Question: For a portion of the highway, descending gradient 1 in 25 meets ascending gradient 1 in 20. The design speed is given as 90 kmph. Calculate the length of the valley curve (L) based on comfort and headlight distance criteria. Take S value as 153.66 m. Assume L is greater than SSD.

Solution:

N = |(±n1) - (±n2)|

N = |-1/25 - 1/20| = 0.09

Comfort Criteria:

L = 2 * ((V^3)*N/C)^(1/2)

L = 2 * (((0.278*90)^3) * 0.09 / 0.5)^(1/2)

L = 106.066 m

Headlight Distance Criteria:

For L > SSD

L = (N*S^2) / (1.5 + 0.035*S)

L = (0.09*153.66^2) / (1.5 + 0.035*153.66)

L = 308.96 m

Therefore, the length of the transition curve as per the comfort criterion is 106.66m and as per headlight sight distance is 308.96m. Usually, a higher value is adopted for design.

## Practise Problem

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