ENGINEERING SURVEYING (221 BE)

Horizontal Circular Curves

Sr Tan Liat Choon

Email: 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 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 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 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 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 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 radius of the circle determines the sharpness or flatness of the curve

7

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

8

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

9

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

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 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

14

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

15

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

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 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 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 angle

R

I

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/(2R) R = 5729.578/D

D used to describe curves

29

TERMINOLOGY

PC: Point of Curvature

PC = PI – T PI = Point of Intersection 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

'87.4552

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’

'83.2291R

'23.8942

5.22sin)83.2291(2..

CL

'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 to sight on Pt

along curve Angle enclosed = Length from PC to Pt = l Chord from PC to point = c

200,

2,

100

DlD

l

)sin(22

sin2 RRc

36

CURVE STAKING EXAMPLE

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

"24'191200

)5.2(87.105

00172

00172

c

13.94170,'302 PCD

"24'040200

5.287.5

,'87.500171

l

'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

With the total station, find and c,

use stake-out

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

"24'045200

)5.2(87.405

00175

00175

c

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 115’ 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 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

40

CIRCULAR CURVES NOTATIONS Definitions: 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

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 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 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 simultaneously to get the intersection point

54

INTERSECTION OF TWO CIRCULAR CURVES

Simultaneously solve the two circle equations

55

T H A N K YO U &

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

56