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ENGINEERING SURVEYING
(221 BE)
Horizontal Circular CurvesHorizontal Circular Curves
Sr Dr. Tan Liat ChoonEmail: [email protected]
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
2
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
3
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
4
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
7
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
8
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
9
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
10
11
INTRODUCTION
12
SIMPLE CURVE LAYOUT
13
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
16
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
17
18
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
19
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
20
INTRODUCTION
21
DEGREE OF CURVES
22
DEGREE OF CURVES
23
SIGHT DISTANCE ON
HORIZONTAL CURVES
24
DEFLECTION ANGLES
25
CURVE THROUGH FIXED POINT
26
COMPOUND CURVES BETWEEN
SUCCESSIVE TANGENTS
27
CIRCULAR CURVES
�Portion of a circle
�I – Intersection angleI
R
�R - Radius
�Defines rate of change
28
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
29
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
30
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))
31
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
32
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
33
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
34
CURVE DESIGN EXAMPLE
�Given:
�I = 74°30’
�PI at Sta 256+32.00
�Design requires D < 5°
�E must be > 315’
35
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 =
∆=
36
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
37
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
38
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.
39
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
41
DEGREE OF CIRCULAR CURVE
42
DEGREE OF CIRCULAR CURVE
43
CIRCULAR CURVES NOTATIONS
44
CIRCULAR CURVES FORMULAS
45
CIRCULAR CURVE STATIONING
46
CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES
WITH A TOTAL STATION OR AN EDM
47
CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES
WITH A TOTAL STATION OR AN EDM
48
CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES
WITH A TOTAL STATION OR AN EDM
49
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
50
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
51
CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES
WITH A TOTAL STATION OR AN EDM
52
SPECIAL CIRCULAR CURVE PROBLEMS
53
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
54
INTERSECTION OF TWO
CIRCULAR CURVES
�Simultaneously solve the two
circle equationscircle equations
55
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
56