ENGINEERING SURVEYING (221 BE)

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1 ENGINEERING SURVEYING (221 BE) Horizontal Circular Curves Sr Tan Liat Choon Mobile:

2 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

3 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

4 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

5 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, highspeed 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

6 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

7 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

8 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

9 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

10 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

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12 INTRODUCTION 12

13 SIMPLE CURVE LAYOUT 13

14 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

15 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

16 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

17 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

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19 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

20 DEGREE OF CURVES For a 1 curve, D = 1; therefore R = 5, 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

21 INTRODUCTION 21

22 DEGREE OF CURVES 22

23 DEGREE OF CURVES 23

24 SIGHT DISTANCE ON HORIZONTAL CURVES 24

25 DEFLECTION ANGLES 25

26 CURVE THROUGH FIXED POINT 26

27 COMPOUND CURVES BETWEEN SUCCESSIVE TANGENTS 27

28 CIRCULAR CURVES I Intersection angle Portion of a circle I R - Radius Defines rate of change R 28

29 DEGREE OF CURVATURE D defines Radius Chord Method R = 50/sin(D/2) Arc Method (360/D)=100/(2 R) R = /D D used to describe curves 29

30 TERMINOLOGY PC: Point of Curvature PC = PI T PI = Point of Intersection T = Tangent PT: Point of Tangency PT = PC + L L = Length 30

31 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

32 CURVE CALCULATION - EXAMPLE Given: D = R ' 22.5 T tan ' 2 PC ( ) ( ) L ' 2.5 PT ( ) (9 00)

33 CURVE CALCULATION - EXAMPLE Given: D = 2 30 R ' 22.5 L. C. 2( )sin ' M cos 44.04' 2 1 E ' 22.5 cos 2 33

34 CURVE DESIGN Select D based on: Highway design limitations Minimum values for E or M Determine stationing for PC and PT R = /D T = R tan(i/2) PC = PI T L = 100(I/D) PT = PC + L 34

35 CURVE DESIGN EXAMPLE Given: I = PI at Sta Design requires D < 5 E must be >

36 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 l D, 100, 2 c 2Rsin 2Rsin( ) 2 l D

37 CURVE STAKING EXAMPLE D 2 30', PC l ', '24" 200 R , c 2( )sin(0 4'24") 5.87' c (2.5 ) 1 19'24" 200 2( ) sin(1 19'24") ' 37

38 CURVE STAKING If chaining along the curve, each station has the same c: 100(2.5 ) ' 200 c 2( ) sin(1 15') ' With the total station, find and c, use stake-out c (2.5 ) 5 04'24" 200 2( )sin(5 04'24") ' 38

39 MOVING UP ON THE CURVE Say you can t see past Sta Move transit to that Sta, sight back on PC. Plunge scope, turn to sight on a tangent line. Turn 1 15 to sight on Sta

40 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

41 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

42 DEGREE OF CIRCULAR CURVE 42

43 DEGREE OF CIRCULAR CURVE 43

44 CIRCULAR CURVES NOTATIONS 44

45 CIRCULAR CURVES FORMULAS 45

46 CIRCULAR CURVE STATIONING 46

47 CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES WITH A TOTAL STATION OR AN EDM 47

48 CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES WITH A TOTAL STATION OR AN EDM 48

49 CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES WITH A TOTAL STATION OR AN EDM 49

50 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, AZ PI-PC, and curve info Required: coordinates of curve points (stations or parts of stations) and the data to lay them out 50

51 CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES WITH A TOTAL STATION OR AN EDM Solution: - from X PI, YPI, T, AZPI-PC, compute XPC, YPC compute the length of chords and the deflection angles use the deflection angles and AZ PI-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

52 CIRCULAR CURVES LAYOUT BY DEFLECTION ANGLES WITH A TOTAL STATION OR AN EDM 52

53 SPECIAL CIRCULAR CURVE PROBLEMS 53

54 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

55 INTERSECTION OF TWO CIRCULAR CURVES Simultaneously solve the two circle equations 55

56 T H A N K YO U & Q U E S T I O N & A N S W E R 56

Route Surveying. Topic Outline

Route Surveying. Topic Outline Route Surveying CE 305 Intro To Geomatics By Darrell R. Dean, Jr., P.S., Ph.D. Topic Outline Horizontal alignment Types of Horizontal Curves Degree of Curve Geometric elements of curve Station ti number