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ENGINEERING SURVEYING

(221 BE)

Horizontal Circular CurvesHorizontal Circular Curves

Sr Dr. Tan Liat ChoonEmail: tanliatchoon@gmail.com

Mobile: 016-4975551

INTRODUCTION

�The centre line of road consists of series of

straight lines interconnected by curves that are

used to change the alignment, direction, or slope of

the roadthe road

�Those curves that change the alignment or

direction are known as Horizontal Curves, and

those that change the slope are Vertical Curves

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DEFINITIONS

�Horizontal Curves: curves used in horizontal

planes to connect two straight tangent sections

�Simple Curve: circular arc connecting two �Simple Curve: circular arc connecting two

tangents. The most common

�Spiral Curve: a curve whose radius decreases

uniformly from infinity at the tangent to that of

the curve it meets

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INTRODUCTION

�Compound Curve: a curve which is composed of two or

more circular arcs of different radii tangent to each other,

with centres on the same side of the alignment

�Broken-Back Curve: the combination of short length of �Broken-Back Curve: the combination of short length of

tangent (less than 100 ft) connecting two circular arcs that

have centres on the same side

�Reverse Curve: Two circular arcs tangent to each other,

with their centres on opposite sides of the alignment

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HORIZONTAL CURVES

�When a highway changes horizontal direction, making the point where it changes direction a point of intersection between two straight lines is not feasible. The change in direction would be too abrupt for the safety of modern, high-speed vehicles. It is therefore necessary to increase a curve between the straight lines. The straight lines of a road are called 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 for all modern 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 that sharp. This means they must be large-radius curves 5

HORIZONTAL CURVES

�In highway work, the curves needed for the location of improvement of small secondary roads may be worked out in the field. Usually, however, the horizontal curves are computed after the route has been selected, the field surveys have been done, and the survey base line and necessary topographic features have been plottedfeatures have been plotted

�In urban work, the curves of streets are designed as an integral part of the preliminary and final layouts, which are usually done on a topographic map. In highway work, the road itself is the end result and the purpose of the design. But in urban work, the streets and their curves are of secondary importance; the best use of the building sites is of primary importance 6

HORIZONTAL CURVES

Simple Horizontal Curve:

�The simple curve is an arc of a circle. �The simple curve is an arc of a circle.

The radius of the circle determines

the sharpness or flatness of the

curve

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HORIZONTAL CURVES

Compound Horizontal Curve:

�Frequently, the terrain will require �Frequently, the terrain will require

the use of the compound curve. This

curve normally consists of two

simple curves joined together and

curving in the same direction

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HORIZONTAL CURVES

Reverse Horizontal Curve:

�A reserve curve consists of two �A reserve curve consists of two

simple curves joined together, but

curving in opposite direction. For

safety reasons, the use of this curve

should be avoided when possible

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HORIZONTAL CURVES

Spiral Horizontal Curve:

�The spiral is a curve that has a varying

radius. It is used on railroads and most radius. It is used on railroads and most

modern highways. Its purpose is to

provide a transition from the tangent to a

simple curve or between simple curves in

a compound curve

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11

INTRODUCTION

12

SIMPLE CURVE LAYOUT

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ELEMENTS OF A HORIZONTAL CURVE

�PI - POINT OF INTERSECTION. The point of intersection is the

point where the backward and forward tangents intersect.

Sometimes, the point of intersection is designed as V (vertex)

�I – INTERSECTING ANGLE. The intersecting angle is the �I – INTERSECTING ANGLE. The intersecting angle is the

deflection angle at the PI. Its value either computed from the

preliminary traverse angles or measured in the field

�A – CENTRAL ANGLE. The central angle is the angle formed by

two radius drawn from the centre of the circle (O) to the PC and

PT. The value of the central angle is equal to the I angle. Some

authorities call both the intersecting angles and central angle

either I or A14

ELEMENTS OF A HORIZONTAL CURVE

�R – RADIUS. The radius of the circle of which the curve is an arc, or

segment. The radius is always perpendicular to backward and forward

tangents

�PC – POINT OF CURVATURE. The point of curvature is the point on the back

tangent where the circular curve begins. It is sometimes designed as BCtangent where the circular curve begins. It is sometimes designed as BC

(beginning of curve) or TC (tangent to curve)

�PT – POINT OF TANGENCY. The point of tangency is the point on the

forward tangent where the curve ends. It is sometimes designated as EC (end

of curve) or CT (curve to tangent)

� POC – POINT OF CURVE. The point of curve is any point along the curve

�L – LENGTH OF CURVE. The length of curve is the distance from the PC to

the PT, measured along the curve15

ELEMENTS OF A HORIZONTAL CURVE

� T – TANGENT DISTANCE. The tangent distance is the distance along

the tangents from the PI to the PC or the PT. These distances are

equal on a simple curve

� LC – LONG CHORD. The long chord is the straight line distance from

the PC to the PTthe PC to the PT

� C – The full chord distance between adjacent stations (full, half,

quarter, or one-tenth stations) along a curve

� E – EXTERNAL DISTANCE. The external distance (also called the

external secant) is the distance from the PI to the midpoint of the

curve. The external distance bisects the interior angle at the PI

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ELEMENTS OF A HORIZONTAL CURVE

�M – MIDDLE ORDINATE. The middle ordinate is the

distance from the midpoint of the curve to the

midpoint of the long chord. The extension of the

middle ordinate bisects the central angle

�D – DEGREE OF CURVE. The degree of curve defines

the sharpness of flatness of the curve

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DEGREE OF CURVES

�Degree of curve deserves special

attention. Curvature may be expressed by

simply stating the length of the radius of

the curve. Stating the radius is common the curve. Stating the radius is common

practice in land surveying and in the design

of urban roads. For highway and railway

work, however, curvature is expressed by

the degree of curve

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DEGREE OF CURVES

�For a 1° curve, D = 1; therefore

R = 5,729.58 feet, or metres, depending

upon the system of units you are using. In

practice, the design engineer usually practice, the design engineer usually

selects the degree of curvature on the

basis of such factors as the design speed

and allowable supper elevation. Then the

radius is calculated

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INTRODUCTION

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DEGREE OF CURVES

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DEGREE OF CURVES

23

SIGHT DISTANCE ON

HORIZONTAL CURVES

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DEFLECTION ANGLES

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CURVE THROUGH FIXED POINT

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COMPOUND CURVES BETWEEN

SUCCESSIVE TANGENTS

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CIRCULAR CURVES

�Portion of a circle

�I – Intersection angleI

R

�R - Radius

�Defines rate of change

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DEGREE OF CURVATURE

�D defines Radius

�Chord Method�R = 50/sin(D/2)

�Arc Method�(360/D)=100/(2πR)

�R = 5729.578/D

�D used to describe curves

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TERMINOLOGY

�PC: Point of Curvature

�PC = PI – T�PI = Point of Intersection

�T = Tangent�T = Tangent

�PT: Point of Tangency

�PT = PC + L�L = Length

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CURVE CALCULATIONS

�L = 100I/D

�T = R * tan(I/2)

�L.C. = 2R* sin(I/2)

�E = R(1/cos(I/2)-1)

�M = R(1-cos(I/2))

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CURVE CALCULATION - EXAMPLE

�Given: D = 2°30’

'83.22915.2

578.5729=

°=R

5.22 °'87.455

2

5.22tan38.2291 =

°⋅=T

13.94170)87.554()50175( +=+−+=PC

'00.9005.2

5.22100 =

°

°⋅=L

13.94179)009()13.94170( +=+++=PT

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CURVE CALCULATION - EXAMPLE

�Given: D = 2°30’

'83.2291=R

'23.8942

5.22sin)83.2291(2.. =

°=CL

2

'04.442

5.22cos183.2291 =

°−=M

'90.441

2

5.22cos

183.2291 =

−

°=E

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CURVE DESIGN

�Select D based on:

�Highway design limitations

�Minimum values for E or M

�Determine stationing for PC and PT

�R = 5729.58/D

�T = R tan(I/2)

�PC = PI –T

�L = 100(I/D)

�PT = PC + L

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CURVE DESIGN EXAMPLE

�Given:

�I = 74°30’

�PI at Sta 256+32.00

�Design requires D < 5°

�E must be > 315’

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CURVE STAKING

� Deflection Angles

� Transit at PC, sight PI

� Turn angle δ to sight on Pt

along curvealong curve

� Angle enclosed = ∆

� Length from PC to Pt = l

� Chord from PC to point = c

200,

2,

100

DlD

l ⋅=∴

∆=⋅=∆ δδ

)sin(22

sin2 δRRc =

∆=

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CURVE STAKING EXAMPLE

13.94170,'302 +=°= PCD

"24'040200

5.287.5

,'87.500171

°=°⋅

=

=+

δ

l

'86.105)"24'191sin()83.2291(2

"24'191200

)5.2(87.105

00172

00172

=°=

°=°

=

+

+

c

δ

200

'87.5)"24'40sin()83.2291(2

,83.2291

=°=

=

c

R

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CURVE STAKING

If chaining along the curve, each station has the same c:

'99.99)'151sin()83.2291(2

'151200

)5.2(100

100

100

=°=

°=°

=

c

δ

'99.99)'151sin()83.2291(2100 =°=c

With the total station, find δ and c, use stake-out

'34.405)"24'045sin()83.2291(2

"24'045200

)5.2(87.405

00175

00175

=°=

°=°

=

+

+

c

δ

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MOVING UP ON THE CURVE

� Say you can’t see past Sta 177+00.

�Move transit to that Sta,

sight back on PC.

�Plunge scope, turn 7° 34’ 24”

to sight on a tangent line.

�Turn 1°15’ to sight on

Sta 178+00.

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CIRCULAR CURVES NOTATIONS

�Definitions:

�Point of intersection (vertex) PI, back and forward

tangents.

�Point of Curvature PC, beginning of the curve�Point of Curvature PC, beginning of the curve

�Point of Tangency PT, end of the Curve

�Tangent Distance T: Distance from PC, or PT to PI

�Long Chord LC: the line connecting PC and PT

�Length of the Curve L: distance for PC to PT:�measured along the curve, arc definition

�measured along the 100 chords, chord definition40

CIRCULAR CURVES NOTATIONS

�Definitions:

�External Distance E: The length from PI to curve

midpoint

�Middle ordinate M: the radial distance between the �Middle ordinate M: the radial distance between the

midpoints of the long chord and curve

�POC: any point on the curve

�POT: any point on tangent

�Intersection Angle I: the change of direction of the two

tangents, equal to the central angle subtended by the

curve

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DEGREE OF CIRCULAR CURVE

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DEGREE OF CIRCULAR CURVE

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CIRCULAR CURVES NOTATIONS

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CIRCULAR CURVES FORMULAS

45

CIRCULAR CURVE STATIONING

46

CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES

WITH A TOTAL STATION OR AN EDM

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CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES

WITH A TOTAL STATION OR AN EDM

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CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES

WITH A TOTAL STATION OR AN EDM

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CIRCULAR CURVE LAYOUT BY COORDINATES

WITH A TOTAL STATION

�Given: Coordinates and station of PI, a point

from which the curve could be observed, a

direction (azimuth) from that point, AZPI-PC , and

curve infocurve info

�Required: coordinates of curve points (stations

or parts of stations) and the data to lay them out

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CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES

WITH A TOTAL STATION OR AN EDM

� Solution: - from XPI, YPI, T, AZPI-PC, compute XPC, YPC

�compute the length of chords and the deflection angles

�use the deflection angles and AZPI-PC, compute the azimuth of

each chordeach chord

�knowing the azimuth and the length of each chord, compute the

coordinates of curve points

�for each curve point, knowing it’s coordinates and the total

station point, compute the azimuth and the length of the line

connecting them

�at the total station point, subtract the given direction from the

azimuth to each curve point, get the orientation angle

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CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES

WITH A TOTAL STATION OR AN EDM

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SPECIAL CIRCULAR CURVE PROBLEMS

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INTERSECTION OF A CIRCULAR CURVE

AND A STRAIGHT LINE

�Form the line and the circle

equations, solve them equations, solve them

simultaneously to get the

intersection point

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INTERSECTION OF TWO

CIRCULAR CURVES

�Simultaneously solve the two

circle equationscircle equations

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T H A N K YO UT H A N K YO U

&&

Q U E S T I O N & A N S W E RQ U E S T I O N & A N S W E R

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