GEOMATICS ENGINEERING

CHAPTER 5

Road Curve Design

1

Horizontal and vertical curve

• The center line of a road consists of series of straight lines

interconnected by curves that are used to change the

alignment, direction, or slope of the road.

• Curves that change the alignment or direction are

known as horizontal curves, and those that change the

slope are vertical curves.

2

Tangents and Curves

Tangents Curves

3

Horizontal curves

• Curves used in horizontal planes to connect two straight tangent sections are called as horizontal curves.

• When a highway changes horizontal direction, making the point of intersection. Change in direction would be too abrupt for the safety of modern, high-speed vehicles.

• It is therefore necessary to introduce a curve between the straight lines.

• The straight lines of a road are called tangents because the lines are tangent to the curves used to change direction.

• In practically all modem highways, the curves are circular curves; that is, curves that form circular arcs. The smaller the radius of a circular curve, the sharper the curve.

• For modern, high speed highways, the curves must be flat, rather than sharp. That means they must be large radius curves.

4

TYPES OF HORIZONTAL CURVES

• There are five types of horizontal curves.

• 1. SIMPLE. The simple curve is an arc of a circle connecting two tangents. The radius of the circle determines the sharpness or flatness of the curve.

• 2. COMPOUND. This curve normally consists of two simple curves of different radii joined together and curving in the same direction.

• 3. BROKEN BACK. The combination of short length of tangent (less than 100 ft) connecting two circular arcs that have center on same side.

• 4. REVERSE. It consists of two circular arcs tangent to each other, with their centers on opposite sides of the alignment.

• 5. SPIRAL. The spiral is a curve that has a varying radius. It is used on railroads and most modem highways. Its purpose is to provide a transition from the tangent to a simple curve or between simple curves in a compound curve.

5

TYPES OF HORIZONTAL CURVES

6

Degree of circular curve • Two definitions are used for the degree of curve.

Degree of Curve (Arc Definition) • The arc definition is most frequently used in high- way design.

• This definition states that the degree of curve is the central angle formed by two radii that extend from the center of a circle to the ends of an arc measuring 100 meters long.

• Therefore, if you take a sharp curve, mark off a portion so that the distance along the arc is exactly 100 m, and determine that the central angle is 12°, then you have a curve for which the degree of curvature is 12°; it is referred to as a 12° curve.

D

R

R

100 ft (D/360) = (100/2πR)

R = 5729.58/D = 5730/D

7

Degree of circular curve

Degree of Curve (Chord

Definition)

• The chord definition is used in railway

practice and in some highway work.

• This definition states that the degree of

curve is the central angle formed by two

radii drawn from the center of the circle

to the ends of a chord 100 meters long.

• If you take a flat curve, mark a 100-m

chord, and determine the central angle

to be 0°30’, then you have a 30-minute

curve.

8

ELEMENTS OF A HORIZONTAL CURVE • Point of Intersection (PI): the point at which the two

tangents to the curve intersect

• Delta Angle: the angle between the tangents is also equal to the angle at the center of the curve

• Back Tangent: for a survey progressing to the right, it is the straight line that connects the PC to the PI

• Forward Tangent: for a survey progressing to the right, it is the straight line that connects the PI to the PT

• Point of Curvature (PC): the beginning point of the curve

• Point of Tangency (PT): the end point of the curve

• Tangent Distance (T): the distance from the PC to PI or from the PI to PT

• External Distance (E): the distance from the PI to the middle point of the curve

• Middle Ordinate (M): the distance from the middle point of the curve to the middle of the chord joining the PC and PT

• Long Chord (LC): the distance along the line joining the PC and the PT

• Length of Curve: the difference in stationing along the curve between the PC and the PT

9

ELEMENTS OF A HORIZONTAL CURVE

10

2cos

4tan

EM

TE

RLor

or

or

Δ is in radian

Circular Curve Stationing

• Initial route survey consists of establishing the PIs according to plan, layout the tangents, and establishing continuous stationing along them, from the start of the project, through each PI, to the end of the job.

• Starting point of any project is assigned a station value, and all other points along the reference line are related to it.

• If starting point is also the end point of a previous adjacent project, its station value may be retained.

• Otherwise an arbitrary value such as 10+00 for metric stationing is assigned.

• In Metric stationing, full stations are generally 1 km apart. Or Sometimes full station = Full length of tape or chain i.e. 30 m

• Staking at the closer spacing is usually done in urban situations, on sharp curves.

• Stakes are placed farther apart in relatively flat or gently rolling rural areas.

• After the tangents have been staked and stationed, the Δ angle is measured at each PI, and curves computed and staked.

• The station locations of points on any curve are based upon the stationing of the curves PI.

11

Example # 1

• Assume that a metric curve will be used at a PI where I =

8024’. Assume also that the station of the PI is 6 +

427.467, and that terrain conditioned require a minimum

radius of 900m. Calculate the PC and PT stationing, and

other defining elements of the curve. In this example 1 full

station = 1000 m.

• Sometimes, it is required to calculate curve data at 10 m, 20 m, or 30

m intervals or increments.

12

ftxEM

mxTE

mxxRLC

stationPT

L

stationPC

T

stationPI

mxRT

mxradianinRL

416.2)2

'248cos(423.2

2cos

423.2)4

'248tan(092.66

4tan

829.131)2

'248sin(9002

2sin2

319.4936

947.131

372.3616

092.66

464.4276

092.66)2

'248tan(900

2tan

947.131)180

'248(900)(

0

0

0

0

0

Solution

(continued):

The arc distance from the PC to the next station 6 + 380 is (6380 – 6361.372) = 18.628

m. the arc distance for the final stationing is 6493.319 – 6480 = 13.319 m . All other

stations have 20 m stationing intervals.

13

What is degree of this

curve?

Example # 2 • Highway curve with R = 700.0 m, Δ = 13010’, and PI station = 5 +

784.850 m

• Calculate the PC and PT stationing, Long chord, Curve Length,

External and Middle Ordinate and degree of curve.

14

Vertical curves

• Curves exist in vertical plane are called as vertical curves.

• Vertical curves need to be inserted at peaks and troughs

on highways to smooth out gradient changes and provide

adequate sight lines.

• Generally, the simple parabola curve is used as the rate of

change of gradient.

• Function of each curve is to provide gradual change in

grade from the initial (back) tangent to the grade of

second (forward)tangent.

15

Profiles Curve a: Crest Vertical Curve (concave downward)

Curve b: Sag Vertical Curve (concave upward)

Tangents: Constant Grade (Slope)

16

17

BVC: Beginning of Vertical Curve

or PVC

V: Vertex or PVI

EVC: End of Vertical Curve

or PVT

G1 : percent grade of back

tangent (or g1)

G2 : percent grade of forward

tangent (or g2)

L: curve length (horizontal

distance) in m

x: horizontal distance from any

point on the curve to the BVC

r: rate of change of grade

Elements of Vertical Curve

Related Equations r = (g2 – g1)/L

where: g2 & g1 are grades in fraction and L in m

Yx = YBVC + g1x + (r/2)x2

where:

YBVC – elevation of the BVC in m, g1 in fraction

and x in m

Station of BVC = Station of Vertex (V) – L/2

Station of EVC = Station of BVC + L

Elevation of BVC (YBVC) = Elevation of Vertex (V) ± g1(L/2)

Elevation of EVC (YEVC) = Elevation of Vertex (V) ± g2(L/2)

18

‘-’ sign for crest curve, and

‘+’ sign for sag curve.

g1 and g2 are in fraction

Example: Equal-Tangent Vertical Curve A +3.5% grade meets a -1.5% grade at station 60 +15 and elevation 250 m. An equal tangent

parabolic curve 300 m long has been selected to join the two tangents. Compute and tabulate

the curve for stakeout at full stations.

Assume 1 full station = 30 m.

g1 = 3.5%

g2 =1.5%

L = 300 m

Vertex station (V) = 60+15

=60*30+15 = 1815m

Vertex elevation (V) = 250 m

19

V

55 + 15

20

(55 + 15)

ElevEVC = 250 -1.5*150/100 = 247.75 m

Yx = YBVC + g1x + (r/2)x2

High and Low Points on Vertical Curves

Sag Curves:

Low Point defines location of catch basin for drainage.

Crest Curves:

High Point defines limits of drainage area for roadways.

21

mincurveoflengthL

ggA

A

LK

12

1./ gKxptlowhigh

Example: High Point on a Crest Vertical Curve

22

stationsormx

mK

A

70.21060*5.3

0.600.5

300

%0.55.35.1

Example • A grade of -3.5% meets another grade of +0.50%. The

elevation of the point of intersection is 267 m and

stationing is 780 m. Field coordinates require that the

vertical curve should pass through a point of elevation

268 m at stationing 780 m. compute a suitable equal

tangent vertical curve and full station (30 m) elevations

23

24

y = 268 m

x = L/2

yBVC = 267 +(3.5/100)*(L/2)

yBVC = 267 +(3.5L/200)

g1x = -3.5L/200

r/2 = 0.5+3.5/200L

rx2/2 = (0.5+3.5/200L)*(L/2)2

25

Yx = YBVC + g1x + (r/2)x2

g1 in fraction