Vertical curves are provided at the intersections of different grades to smooth on the vertical profile The
disadvantages of not providing vertical curves are:
(i) Accident due to inadequate visibility on a high and blind culvert over which roads profile has not
boon is properly designed as a vertical curve.
(ii) Discomfort and damage due to humps and trough, although these may not be a source of accidents
but they make traveling very uncomfortable
(iii) It is much cheaper and easier to construct good curves in the first instance than to set the right road
curves later on The minimum length of the vertical curve as per the IRC recommendations are given in Table
| Maximum grade
change (%) not
Requiring a Vertical curve
| Maximum Length of
the vertical curve in m
(for higher grade change)
The vertical curves are of two types;
(i) Summit curve (ii) Valley curve
A summit curve is a vertical curve with convexity upward or concavity downward. This occurs when an
ascending gradient intersects a descending gradient or when an ascending gradient meets another ascending
gradient or an ascending gradient meets a horizontal ar when descending gradient meat an another Summit
curve descending gradient as shown in Figure
Square parabola is the best shape for summit curve due to good riding qualities, Simplicity of calculation work and uniform rate of change of grade. An ideal shape for a summit curve is circular because the sight distance available throughout the curve is constant.
Generally, there is no problem of discomfort on the summit curve because gravity force acts download and a centrifugal force acts upwards, part of pressure in the tyre is released Length of summit curve depends only
on sight distance (SSD.ISD and OSD).
Deviation Angle of Summit Curve
The deviation angle of a curve is expressed by the algebraic difference of the grade angles. If n, and n, are
the grade angles for the two curves, then assigning proper signs (ie, + for ascending and for descending), the
deviation angle N for Figure (a)
N = Angle EOT2 = (+n1)-(-n2) = n1+n2
Thus deviation angle of a summit curve is the angle that measures the change of direction in the path
of motion at the intersection of two grade lines.
Length of Summit Curve
The length of the summit curve depends only on sight distance i.e. SSD, ISD, and OSD. The length of a
summit curve is a function of
(i) the deviation angle, N (ii) the required sight distance
In determining the length of the summit curve, two cases have to be considered
Case-1: When the length of the curve exceeds the required sight distance i.e., L>S.
Case-2: When the length of the curve is less than the required sight distance, i.e., L<S.
The length of the summit curve is given as,
Valley Curve or Sag Curve
A valley curve is a vertical curve with concavity upward or convexity downward as shown in Figure
This is formed when a descending gradient intersects an ascending gradient when a descending gradient meets another descending gradient or when a descending gradient joins a horizontal path or when an ascending gradient meets another ascending gradient as shown in Figure
A valley curve is usually made up of two transition curves of equal length without having a circular curve in
between. Cubic parabola is generally preferred for valley curves to gradually introduce the centrifugal force. In
the valley curve, centrifugal force and gravity force both act downwards, hence creating discomfort for passengers.
Deviation Angle of Valley Curve
As in the case of summit curves, in this case, also deviation angle is the algebraic difference of the two grade angles
NOTE: There are no restrictions to sight distance at valley curve during daw time but at night only source of visibility is with the help of headlights in case of inadequate street lights.
Length of Valley Curve
Length of valley curve is designed on the basis of two criteria:
(i) Based on Comfort Condition: In this criteria, the rate of change of centrifugal acceleration is limited to a comfortable zone of about 0.6 m/sec, then the length of the transition curve is given by
If C= 0.6 m/s³ then Lv = 0.38 (Nv³)½
(ii) Based on Headlight Sight Distance:
Case-1: When the length of a valley curve is more than the head light sight distance. (L>HSD/SSD) Leth is the height of head light above the road surface and ‘S is the head light sight distance which should be at least equal to stopping sight distance.
if h=0.75 m and ∝ =1°
Case-2: When the length of the valley curve is less than the head light sight distance (L. <HSD/SSD)
In this case, the length of the valley curve is given by