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ENGINEERING SURVEYING 221 BE
Horizontal Circular Curves
Sr Tan Liat ChoonEmail: [email protected]
Mobile: 016-4975551
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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 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 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 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 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
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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
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
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Horizontal Curves
Simple Horizontal Curve:
• 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 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 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 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
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Introduction
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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 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 A
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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 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 curve
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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 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 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 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
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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 angle
R
I
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
• 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’
'83.22915.2578.5729
=°
=R
'87.45525.22tan38.2291 =
°⋅=T
13.94170)87.554()50175( +=+−+=PC
'00.9005.25.22100 =°°
⋅=L
13.94179)009()13.94170( +=+++=PT
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Curve Calculation - Example• Given: D = 2 30’
'83.2291=R
'23.89425.22sin)83.2291(2.. =
°=CL
'04.4425.22cos183.2291 =
°−=M
'90.441
25.22cos
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 > 315’35
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Curve Staking
• Deflection Angles
– Transit at PC, sight PI– Turn angle δ to sight on Pt
along curve– Angle enclosed = ∆– Length from PC to Pt = l– Chord
from PC to point = c
200,
2,
100DlDl ⋅=∴∆=⋅=∆ δδ
)sin(22
sin2 δRRc =
∆=
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Curve Staking Example
'86.105)"24'191sin()83.2291(2
"24'191200
)5.2(87.105
00172
00172
=°=
°=°
=
+
+
c
δ
13.94170,'302 +=°= PCD
"24'040200
5.287.5,'87.500171
°=°⋅
=
=+
δ
l
'87.5)"24'40sin()83.2291(2,83.2291
=°==c
R
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Curve StakingIf chaining along the curve, each station has the
same c:
'99.99)'151sin()83.2291(2
'151200
)5.2(100
100
100
=°=
°=°
=
c
δ
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 NotationsDefinitions:
• Point of intersection (vertex) PI, back and forward
tangents.
• 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
definition
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Circular Curves NotationsDefinitions:
• External Distance E: The length from PI to curve midpoint
• 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
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Circular Curve Stationing
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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 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 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
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
simultaneously to get the intersection point
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Intersection of Two Circular Curves
Simultaneously solve the two circle equations
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T H A N K YO U&
Q U E S T I O N & A N S W E R
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ENGINEERING SURVEYING �221
BEIntroductionDefinitionsIntroductionHorizontal CurvesHorizontal
CurvesHorizontal CurvesHorizontal CurvesHorizontal CurvesHorizontal
CurvesSlide Number 11IntroductionSimple Curve LayoutElements of a
Horizontal CurveElements of a Horizontal CurveElements of a
Horizontal CurveElements of a Horizontal CurveSlide Number 18Degree
Of CurvesDegree Of CurvesIntroductionDegree Of CurvesDegree Of
CurvesSight distance on Horizontal CurvesDeflection AnglesCurve
Through Fixed PointCompound Curves Between �Successive
TangentsCircular CurvesDegree of CurvatureTerminologyCurve
CalculationsCurve Calculation - ExampleCurve Calculation -
ExampleCurve DesignCurve Design ExampleCurve StakingCurve Staking
ExampleCurve StakingMoving Up on the CurveCircular Curves
NotationsCircular Curves NotationsDegree of Circular CurveDegree of
Circular CurveCircular Curves NotationsCircular Curves
FormulasCircular Curve StationingCircular Curves Layout by
Deflection Angles with a Total Station or an EDMCircular Curves
Layout by Deflection Angles with a Total Station or an EDMCircular
Curves Layout by Deflection Angles with a Total Station or an
EDMCircular Curve Layout by Coordinates with a Total
StationCircular Curves Layout by Deflection Angles with a Total
Station or an EDMCircular Curves Layout by Deflection Angles with a
Total Station or an EDMSpecial Circular Curve ProblemsIntersection
of a Circular Curve and a straight LineIntersection of Two Circular
CurvesTHANK YOU�&�QUESTION & ANSWER
