The document discusses circular curves and their use in highway and railway alignment. It defines key terms related to circular curves like deflection angle, chord, radius, and introduces different types of horizontal curves - simple circular curves, compound curves, reverse curves, spiral curves, and lemniscate curves. It also discusses vertical curves like valley and summit curves. The document provides formulas to calculate length of tangent, external distance, middle ordinate, length of chord, length of curve, degree of curve, and minimum radius of curvature for circular curves. It includes examples of problems calculating radius, offset distance, and degree of curve given different curve elements.

1. TOPIC:
HIGHWAY & RAILWAY CURVES
SUBJECT: ADVANCE ENGINEERING SURVEY
LECTURE#01 (a)
BY: Engr. Shujaat Abbas

2. Contents
• Circular curves
• Deflection and chord calculation
• Setting out circular curves by
various methods
• Compound curves
• Reverse curves
• Vertical Curves
• Parabolic Curves
• Computation of high or low
point on a vertical curve
• Design consideration
• Spiral curves
• Spiral curves Computations
• Approximate solution for spiral
problems
• Super Elevations

3. What is a curve?
• A curve is an arc which connects two straight lines which are
separated by some angle called deflection angle.
• This situation occurs where the alignment of a road way or rail way
changes its direction because of unavoidable objects or conditions.
• The object may be a hill or a lake or a temple etc. so, for the ease of
movement of vehicle at this point a curve is provided.

4. Types of Curves in Alignment of Highways
In general, there are two types of curves and they are
A. Horizontal curves
B. Vertical curves

5. A. Horizontal Curves
The curve provided in the horizontal plane of earth is called as
horizontal curve. In connects two straight lines which are in same level
but having different directions. Horizontal curves are of different types
as follows
1. Simple circular curve
2. Compound curve
3. Reverse curve
4. Transition curve
5. Spiral
6. Lemniscate

6. 1- Simple Circular Curve
• Simple circular curve is normal horizontal curve which connect two
straight lines with constant radius.

7. 2- Compound Curve
• Compound curve is a combination of two or more simple circular
curves with different radii. In this case both or all the curves lie on the
same side of the common tangent.

8. 3- Reverse Curve
• Reverse curve is formed when two simple circular curves bending in
opposite directions are meet at a point. This points is called as point
of reverse curvature. The center of both the curves lie on the
opposite sides of the common tangent. The radii of both the curves
may be same or different.

9. 4- Transition Curve
• A curve of variable radius is termed as transition curve. It is generally
provided on the sides of circular curve or between the tangent and
circular curve and between two curves of compound curve or reverse
curve etc. Its radius varies from infinity to the radius of provided for
the circular curve.
• Transition curve helps gradual introduction of centrifugal force by
gradual super elevation which provides comfort for the passengers in
the vehicle without sudden jerking.

11. 5- Spiral Curve
• Spiral is a type of transition curve
• Ideal transition curve because of its smooth introduction of
centrifugal acceleration. It is also known as clothoid.

12. 6- Lemniscate
• Lemniscate is a type of transition curve which is used when the
deflection angle is very large. In lemniscate the radius of curve is
more if the length of chord is less.

13. B. Vertical Curves
• The curves provided in vertical plane of earth is called as vertical
curve. This type of curves are provided when the ground is non-
uniform or contains different levels at different points. In general
parabolic curve is preferred as vertical curve in the vertical alignment
of roadway for the ease of movement of vehicles. But based on the
convexity of curve vertical curves are divided into two types
1. Valley curve
2. Summit curve

14. 1- Valley Curve/ Sag Curve
• Valley curve connects falling gradient with rising gradient so, in this
case convexity of curve is generally downwards. It is also called as sag
curve.

15. Summit Curve
• Summit curve connects rising gradient with falling gradient hence, the
curve has its convexity upwards. It is also called as crest curve.

16. Deflection Angle
The amount of angular deviation from a straight line to stay on course
is called deflection

17. Chord
Chord length is the straight line distance between two points on the
curve.
An arc is a segment of a curve between two points
In the graph below, the solid red part is an arc. The solid blue part is a
chord.

18. Setting out circular curves

19. 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.

20. • x = offset distance from tangent to the curve. Note: x is perpendicular to T.
• 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.

21. 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.

22. Formulas for Circular Curves

23. 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= R* tan I/2

24. External distance, E
External distance is the distance from PI to the midpoint of the curve.
From the same right triangle PI-PT-O,

25. 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,

26. 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,

27. 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,
An alternate formula for the length of curve is by ratio and proportion
with its degree of curve.

29. 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.
Arc Basis
In arc definition, the degree of curve is the central angle subtended by one
station of circular arc. This definition is used in highways. Using ratio and
proportion,

31. 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,

32. Minimum Radius of Curvature
• Vehicle traveling 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.

35. For the above formula, v must be in meter per second (m/s) and R in
meter (m). For v in kilometer per hour (kph) and R in meter, the
following convenient formula is being used.

36. Problem 1
The angle of intersection of a circular curve is 45° 30' and its radius is 198.17
m. PC is at Sta. 0 + 700. Compute the right angle offset from Sta. 0 + 736.58 on
the curve to tangent through PC.
Solution

37. Problem 2
The angle of intersection of a circular curve is 36° 30'. Compute the
radius if the external distance is 12.02 m.

38. Problem 3
Given the following elements of a circular curve: middle ordinate = 2 m; length of
long chord = 70 m. Find its degree of curve, use arc basis.