Due to change in grade in the vertical alignment of the highway, it is necessary to introduce a vertical curve at the intersections of different grades to smoothen out the vertical profile and for this reason ease off the modifications in gradients for the fast-shifting vehicles.
The curve in a vertical alignment that’s produced while one-of-a-kind gradients meet is called vertical curves. It is provided to secure safety, look, and visibility. The maximum, not unusual place practice has been to apply parabolic curves in summit curves. This is due to the benefit of setting it out on the sector and the cushty transition from one gradient to another. Furthermore, using parabolic curves offers exquisite driving comfort. In the case of valley curves, using a cubic parabola is favored because it carefully approximates the appropriate transition requirements.

Classification Of Curves

a) Summit Curve (Crest curve with convexity upward)

Summit curves are the ones curves that have convexity upwards. They are formed below the 4 following conditions:

  1. When a superb gradient meets any other moderate superb gradient
  2. When a superb gradient meets a degree 0 gradient
  3. When a superb gradient meets with a poor gradient
  4. When a poor gradient meets any other steeper poor gradient

During the layout of the vertical summit curve the consolation, appearance, and safety of the driving force must be taken into consideration. The sight distance has to be taken into consideration withinside the layout. All the sorts of sight distance must be taken into consideration throughout the layout as a way as possible. During motion in a summit curve, there may be much less soreness to the passengers due to the fact the centrifugal pressure evolved through the motion of the car on a summit curve act upwards that is contrary to the route wherein its weight acts. This relieves the burden at the springs of the car so strain evolved could be much less.

Summit curves with convexity upwards are formed in any one of the cases illustrated in the figures. The deviation angles between the two interacting gradients are equal to the algebraic difference between them.  N=n_1-(-n_2) = (n_1+n_2)

1) Length of summit curve for SSD


When L>SSD

 L= \frac{NS^2}{(\sqrt{2H})\: +\: \sqrt{2h}^2)}


 L= \frac{NS^2}{4.4}

L = Length of summit curve in meter,
S = SSD (m)
N = Deviation angle= Algebraic difference of Grade
H =1.2m =height of eye level of the driving force above street way surface
H = 0.15m = Height of item above the pavement surface

When L<SSD

 L= \frac{(\sqrt{2H}\: +\: \sqrt{2h})^2}{NS^2}

or  L= 2S-\frac{4.4}{N}

2) Length of summit curve for OSD or ISD Conditions:

 L= \frac{NS^2}{(\sqrt{2H})\: +\: \sqrt{2h}^2)}

or  L= \frac{NS^2}{9.6}

Where h=1.2
When L>SSD

 L= \frac{(\sqrt{2H}\: +\: \sqrt{2h})^2}{N}

or  L= 2S-\frac{9.6}{N}

b) Valley Curves (Sag Curve with Concavity Upward)

Valley (Sag) curves are the curves that have convexity downwards. They are fashioned below the 4 following conditions:

  1. When a poor gradient meets any other slight poor gradient
  2. When a poor gradient meets a degree 0 gradient
  3. When a poor gradient meets with a fine gradient
  4. When a fine gradient meets any other steeper fine gradient

As compared to the design of the summit curve, the valley curve requires For extra consideration. During the daytime, the visibility in valley curves isn’t hindered however all through night time the simplest supply of visibility will become headlight withinside the absence of street lights. And in valley curves, the centrifugal force generated through the car shifting alongside a valley curve acts downwards together with the load of the vehicle/car and this provides to the pressure caused withinside the spring of the automobile which reasons jerking of the vehicle/car and pain to the passengers. Thus, the maximum essential matters to don’t forget all through valley curve design are:

  • Impact and jerking free movement of motors at design speed
  • Availability of preventing sight distance below headlight of cars all through night time driving

Valley curves or sag curves are shaped in anybody of the instances illustrated in fig. In all of the instances, the most feasible deviation attitude is received while a descending gradient meets with an ascending gradient.

Valley Curves (Sag Curve with Concavity Upward)

Length of Summit curve for headlight sight distance

When L>SSD

 L= \frac{NS}{2h_1 +2Stan\alpha }

Gives  L= \frac{NS}{1.5+0.035S }

L= Total length of valley curve, S=SSD (m),
N= Deviation angle,
 \alpha = Beam\: Angle\: Say\: 1^0

h1= Average height of head light=0.75m

When L<SSD

 L= 2S - \frac{2h_1 +2Stan \alpha}{N }

 L= 2S - \frac{1.5 + 0.035S}{N }

The lowest point on the valley curve from the tangent point of the first gradient

 x_0= L\sqrt{\frac{n_1}{2N}}

N = deviation angle
n1 = gradient
L = length of the curve

Properties of a Vertical Curve

  1. The distinction in elevation between the BVC and some extent on the g1 grade line at a distance X units (feet or meters) is g1X (g1 is expressed as a decimal).
  2. The tangent offset among the grade line and additionally, the curve is given via way of means of ax2, anyplace x is that the horizontal distance from the BVC; (that is, tangent offsets are proportional to the squares of the horizontal distances).
  3. The elevation of the curve at distance X from the BVC is given (on a crest curve) by 
    BVC + g1x – ax2
    (the signs would be reversed during a sag curve).
  4. The grade lines (g1 and g2) encounter midway between the BVC and also the EVC. That is, BVC to V = 1/2L = V to EVC.
  5. Offsets from the 2-grade lines are symmetrical in relation to the PVI.
  6. The curve lies midway between the PVI and also the center of the chord; that’s, Cm = mV.

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