Test: Vertical Alignment - Civil Engineering (CE) MCQ

The design of valley curves, summit curves and gradient are included in the vertical alignment of highway.

• The ideal vertical curve is a 2° Parabola
• The most preferred curve for vertical alignment is parabolic.

Important Points

• Square parabola is generally preferred due to the best riding quality, simplicity of calculation, and uniform rate of change of gradient.
• In a circular curve, sight distance is available throughout the curve is constant.
• Cubic parabola is generally preferred in the valley curve.

The ruling gradient value required for plain and rolling terrain type of road is 1 in 30 or 3.3%.

Vertical Curves

• Vertical curves are used in highway and street vertical alignment to provide a gradual change between two adjacent grade lines.

Length of a vertical curve
= (Total grade/Rate of change of grade per chain length) × Length of chain
Vertical Curves are of two types:

• Summit Curve and
• Valley Curve

Important Points

• The ideal shape for the summit curve is circular because the sight distance available throughout its length remains constant. However, if the deviation angle is very small, the behavior of parabola and circular is the same. Hence, a parabola is preferred for summit curves.
• For Valley curves, a cubic parabola is preferred as the introduction of centrifugal force on this curve is gradual and hence, there is no problem of discomfort. However, IRC suggests a Square parabola for valley curves due to the simplicity of calculation. 

The maximum grade compensation is 75/R = 75/75 = 1%.

Vertical curve: 
Vertical alignment is the elevation of the centerline of the road. It may be in the form of a circular arc or a parabola
The vertical curve is divided into two categories

Summit curve:

• It is a crest curve with convexity upward.
• When an ascending gradient meets with descending gradients the deviation angle will be maximum.

The deviation angle between the two interacting gradients is equal o the algebraic difference between them.
N = n₁ – (-n₂) = n₁ + n₂
As Square parabola is the best shape of the summit curve due to good riding qualities, simplicity of calculation, and uniform rate of change of grade.

Valley curve/Sag curve:
The maximum possible deviation angle is obtained when a descending gradient(downhill) meets with an ascending (uphill) gradient.
Valley curves or sag curves are vertical curves with convexity downwards.
The overtaking sight distance is not included in the valley curve.
The cubic parabola is generally preferred.
The headlight sight distance available at the valley curve should be at least equal to the stopping sight distance.

Important Points
Transition curve:
The curve provided on the road or on the rails at the junction of a straight line and a circular curve, whose radius of curvature changes from infinite at the straight line to a fixed value at the circular portion is known as the transition curve.

• It is provided to change the radius gradually so that the effect of curvature can be reduced on the passengers as well as on the vehicle.
• It is also used to introduce super-elevation gradually.

Different types of transition curves are adopted by different organizations according to their specific needs and arrangement, such as;

• Euler's spiral
• Cubical spiral
• Cubic parabola
• Lemniscate curve

The vertical alignment influences sight distance, vehicle operation cost, vehicle speed and comfort while travelling at high speed.

The length of summit curve is based on stopping sight distance, it is further divided into two cases when L>SSD and L<SSD.

The length of the summit curve is given by
L = NS²/4.4
L = 1*150*150/ (4.4*30)
L = 170m.

The equation of summit curve for L< OSD or L<ISD is given by NS²/8H and the height of the eye of driver is generally assumed as 1.2m.