Overview: The vertical alignment of a road consists of gradients(straight lines in a vertical plane) and vertical curves. The vertical alignment is usually drawn as a profile, which is a graph with elevation as vertical axis and the horizontal distance along the centre line of the road as the the horizontal axis.

1. CURVES

2. CURVES
Definition:
Curves are provided whenever a road changes its direction from
right to S (vice versa) or changes its alignment from up to down
(vice versa). Curves are a critical! element in the pavement design.
They are provided with a maximum speed limit that should lie
followed very strictly. Following the speed limit becomes essential
as the exceed in speed may lead to the chances of the vehicle
becoming out of control while negotiating a turn and thus increase
the odds of fatal accidents. Also, it is very necessary that
appropriate safety measures be adopted at all horizontal and
vertical curves to make the infrastructure road user friendly and
decrease the risks of hazardous circumstances.
The low cost safety measures that can be adopted at curves
included chevron signs, delineators, pavement markings, flexible
posts, fluorescent strips, road safety barriers, rumble strips etc.
Types of Curves
There are two types of curves provided primarily for the comfort
and ease of the motorists in the road namely:
1) Horizontal Curve
2) Vertical Curve
Horizontal Curves:
Horizontal curves are provided to change the direction or alignment
of a road. Horizontal Curve are circular curves or circular arcs. The
sharpness of a curve increases as the radius is decrease which
makes it risky and dangerous. The main design criterion of a

3. horizontal curve is the provision of an adequate safe stopping sight
distance.
Types of Horizontal Curve:
i) Simple Curve: A simple arc provided in the road to impose a
curve between the two straight lines.
ii) Compound Curve: Combination of two simple curves combined
together to curve in the same direction.
iii) Reverse Curve: Combination of two simple curves combined
together to curve in the same direction.
iv) Transition or Spiral Curve: A curve that has a varying radius. Its
provided with a simple curve and between the simple curves in a
compound curve.
While turning a vehicle is exposed to two forces. The first force
which attracts the vehicle towards the ground is gravity. The
second is centripetal force, which is an external force required to
keep the vehicle on a curved path. At any velocity, the centripetal
force would be greater for a tighter turn (smaller radius) than a
broader one (larger radius). Thus, the vehicle would have to make a
very wide circle in order to negotiate a turn.
This issue is encountered when providing horizontal curves by
designing roads that are tilted at a slight angle thus providing ease
and comfort to the driver while turning. This phenomenon is
defined as super elevation, which is the amount of rise seen on a
given cross-section of a turning road, it is otherwise known as
slope.

4. Vertical Curves:
Vertical curves are provided to change the slope in the road and
may or may not. be symmetrical. They are parabolic and not
circular like horizontal curves. Identifying the proper grade and the
safe passing sight distance is the main design criterion of the
vertical curve, iln crest vertical curve the length should be enough
to provide safe stopping sight distance and in sag vertical curve the
length is important as it influences the factors such as headlight
sight distance, rider comfort and drainage requirements.
Types of Vertical Curve:
i) Sag Curve: Sag Curves are those which change the alignment of
the road from uphill to downhill.
ii) Crest Curve/Summit Curve: Crest Curves are those which
change the alignment of the road from downhill to uphill. In
designing crest vertical curves it is important that the grades be not]
too high which makes it difficult for the motorists to travel upon it.

5. Simple Curves
Terminologies in Simple Curve:
 PC = Point of curvature. It is the beginning of curve.
 PT = Point of tangency. It is the end of curve.
 PI = Point of intersection of the tangents. Also called
vertex
 T = Length of tangent from PC to PI and from PI to PT. It
is known as subtangent.
 R = Radius of simple curve, or simply radius.
 L = Length of chord from PC to PT. Point Q as shown
below is the midpoint of L.
 Lc = Length of curve from PC to PT. Point M in the the
figure is the midpoint of Lc.
 E = External distance, the nearest distance from PI to the
curve.
 m = Middle ordinate, the distance from midpoint of curve
to midpoint of chord.
 I = Deflection angle (also called angle of intersection and
central angle). It is the angle of intersection of the tangents.
The angle subtended by PC and PT at O is also equal to I,
where O is the center of the circular curve from the above
figure.
 x = offset distance from tangent to the curve.
Note: x is perpendicular to T.

6.  θ = offset angle subtended at PC between PI and any point
in the curve
 D = Degree of curve. It is the central angle subtended by a
length of curve equal to one station. In English system, one
station is equal to 100 ft and in SI, one station is equal to
20 m.
 Sub chord = chord distance between two adjacent full
stations.
Sharpness of circular curve: The smaller is the degree of
curve, the flatter is the curve and vice versa. The sharpness
of simple curve is also determined by radius R. Large
radius are flat whereas small radius are sharp.

7. Formulas for Circular Curves: The formulas we are
about to present need not be memorized. All we need is
geometry plus names of all elements in simple curve. Note
that we are only dealing with circular arc, it is in our great
advantage if we deal it at geometry level rather than
memorize these formulas.
Circular curve
Length of tangent, T: Length of tangent (also referred to as
subtangent) is the distance from PC to PI. It is the same distance
from PI to PT. From the right triangle PI-PT-O,
Tan
𝐼
2
=
𝑇
𝑅
T = R𝑡𝑎𝑛
𝐼
2

8. External distance, E: External distance is the distance from PI to
the midpoint of the curve. From the same right triangle PI-PT-O,
Cos
I
2
=
R
R + E
R + E =
R
Cos
I
2
E = Rsec
I
2
− 𝑅
Middle ordinate, m: Middle ordinate is the distance from the
midpoint of the curve to the midpoint of the chord. From right
triangle O-Q-PT,
Cos
I
2
=
R − m
R
Rcos
I
2
= 𝑅 − 𝑚
m = R − Rcos
I
2
Length of long chord, L: Length of long chord or simply length of
chord is the distance from PC to PT. Again, from right triangle O-
Q-PT,
Sin
I
2
=
L
2⁄
R
Rsin
I
2
=
L
2
L = 2Rsin
I
2

9. Length of curve, Lc: Length of curve from PC to PT is the road
distance between ends of the simple curve. By ratio and proportion,
L 𝑐
7
=
2πR
360°
L 𝑐 =
πRI
180°
L 𝑐
I
=
1 Station
D
L 𝑐 =
1 Station x I
D
SI units: 1 station = 20m
L 𝑐 =
20I
D
English System: 1 station = 100ft
L 𝑐 =
100I
D
If given the stationing of PC and PT, then
L 𝑐 = stationing of PT − stationing of PC
Degree of curve, D: The degree of curve is the central angle
subtended by an arc (arc basis) or chord (chord basis) of one
station. It will define the sharpness of the curve. In English system,
1 station is equal to 100 ft. In SI, 1 station is equal to 20 m. It is
important to note that 100 ft is equal to 30.48 m not 20 m.

10. Arc Basis: In arc definition, the degree of curve is the central angle
angle subtended by one station of circular arc. This definition is
used in highways. Using ratio and proportion,
1 Station
D
=
2πR
360°
SI units (1 station=20m)
20
D
=
2πR
360°
English System (1 station=100ft):
100
D
=
2πR
360°
Chord Basis: Chord definition is used in railway design. The
degree of curve is the central angle subtended by one station length
of chord. From the dotted right triangle below,

11. Sin
D
2
=
Half station
D
SI units (half station=10m):
Sin
D
2
=
10
R
English System (half station=50ft):
Sin
D
2
=
50
R
Minimum Radius of Curvature: Vehicle travelling on a
horizontal curve may either skid or overturn off the road due to
centrifugal force. Side friction f and super elevation e are the
factors that will stabilize this force. The super elevation e = tan θ
and the friction factor f = tan ϕ. The minimum radius of curve so
that the vehicle can round the curve without skidding is determined
as follows.

12. From the force polygon shown in the right
Tan(θ + ∅) =
CF
W
Tan(θ + ∅) =
Wv2
gR⁄
W
Tan(θ + ∅) =
Wv2
WgR
Tan(θ + ∅) =
v2
gR
The quantity v2
gR⁄ is called impact factor.
Impact Factor:
if =
v2
gR
Back to the equation tan (θ + ∅)=
v2
gR
tan(θ + ∅) =
v2
gR
tanθ + tan∅
1 − tanθtan∅
=
v2
gR
Recall that tanθ = e and tan∅ = f
e + f
1 − ef
=
v2
gR
But ef = 0, thus
e+f =
v2
gR

13. Radius of curvature with R in meter and v in meter per second
R =
v2
g(e + f)
For the above formula, v must be in meter per second (m/s) and R
in meter (m). For v in kilometre per hour (kph) and R in meter, the
following convenient formula is being used.
R =
(v
km
hr
)
2
(
1000m
km
x
1hr
3600sec
)2
g(e + f)
R =
𝑣2
(
1
3.6
)2
g(e + f)
R =
𝑣2
(3.62)g(e + f)
R =
v2
(3.62)(9.80)(e + f)
Radius of curvature with R in meter and v in kilometre per
hour
R
𝑣2
127(e + f)
Using the above formula, R must be in meter (m) and v in
kilometre per hour (kph).