Object: To set out the simple curve by the deflection angle method.
Theory:
Highway Curves:
- During the survey of the alignment of a project involving roads or railways, the direction of a line of track may change due to unavoidable circumstances i.e. it may change towards right or left or up or bottom.
- In such cases, to ensure the vehicle runs easily and smoothly along a track, the two straight lines (the original line and the changed line) are connected by an arc which is known as the “Curve” or more correctly “Highway curve”.
- Thus a curve may be defined as “the arc provided where it is necessary to change the direction of the motion from right to left and vice versa or from up to down and vice versa”.
Objectives:
The main objectives of providing highway curves are:
- Two connect two lines (tangents) in different directions.
- To maintain the speed of vehicles.
- For safe turnings to avoid accidents etc.
Types of Highway Curves:
There are following two main types of highway curves.
Horizontal Curves:
- The curve is provided in the horizontal plane i.e. from right to left and vice versa.
Vertical Curves:
- The curve is provided in the vertical plane i.e. from up to down and vice versa.
Highway Curves
We shall discuss here only a simple curve.
Definition of Simple Curve:
- It is the curve that consists of a single arc of a circle to connect two straight lines.
- This curve is tangential to both lines.
- It has a constant radius.
Definitions and Notations for Simple Curve:
Back Tangent:
- The tangent AT1 previous to the curve is called the back tangent / original straight line or first tangent.
Forward Tangent:
- The tangent T2B following the curve is called the forward tangent or second tangent or deflected line.
Point Of Intersection:
- If the two tangents are produced, they will meet at a point, called the point of intersection PI or vertex V.
Point Of Curve:
- It is the beginning of the curve where the alignment changes from a tangent to the curve.
Point Of Tangency:
- It is the end of the curve where the alignment changes from a curve to a tangent.
Intersection Angle or External Deflection Angle:
- The angle AVB between the tangent lines AV and VB produced is called the intersection angle.
Deflection Angle:
- The angle V’VB i.e. the angle by which the forward tangent deflects from the rear tangent is called deflection angle.
Tangent Distance:
- It is the distance from the point of intersection to the tangent point i.e. between PC to P. I or vice versa
Apex Distance:
- It is the distance from the midpoint of the curve to the PI
Length of Curve:
- It is the total length of the curve from PC to PI.
Long Chord:
- It is the chord joining PC to PT.
Mid Ordinate:
- It is the distance or ordinate from the midpoint of the long chord to the midpoint of the curve.
Normal Chord:
- A chord between two successive regular stations on a curve.
Sub-Chord:
- It is any chord shorter than the normal chord.
Parameters or Elements of Simple Curve:
- Length Of Curve: L=⊼ R⏀/180˚
- Deflection Angle: ⏀=180◦-I
- Radius: 1719/D D=Degree of curve
- Tangent Length: T=R x tan (⏀/2)
- Length Of Long Chord: LC=2RSin (⏀/2)
- Apex Distance: E=R Sec [(⏀/2)-1]
- Mid Ordinate: M=R [1- Cos (⏀/2)]
- Chainages:
- Chainage of first tangent point (PC) = Chainage of intersection point (PI) – back tangent( T1)
- Chainage of second tangent point (PT) = Chainage of first tangent point + curve length.
Setting out of Simple Curve by Deflection Angle or Rankin’s Method:
- Rankin’s method is based on the principle that the deflection angle to any point on a circular curve is measured by one-half the angle subtended by the arc from PC to that point.
- The curve is set by the deflection angles (tangential angles) with the help of theodolite or total station.
- This method is used for large radius curves and high-speed roads.
Methods:
There are two methods of setting out simple curves by deflection angle:
- By one theodolite method
- By two theodolite method
Procedure:
Office Work:
- In both methods, office work is common.
- First of all, calculate all the setting out parameters of a simple curve.
- Divide the length of the curve into several small sub-chords at regular peg intervals.
- Calculate initial sub-chord, final sub-chord, and number of full sub-chords.
- Calculate small deflection angles for small sub-chords by using the formula:
δ= 90 x L/⊼R
- Apply arithmetical check.
- Prepare setting out table.
Field Procedure of Setting out Curve by Deflection Angle by One Theodolite:
- Let AB and BC be the two tangents intersecting at the point B.
- The points T1 and T2 are marked by intersecting pegs on the ground.
- In this method, one surveyor and three helpers are needed.
- The surveyor stands with the theodolite and one of the three helpers will hold the staff and the other two will hold the tape.
- The theodolite is centered over T1 and properly leveled.
- By setting the horizontal angle at 0˚ and fixing the upper clamp, direct the theodolite to bisect the ranging rod at the intersection point B.
- Set the first deflection angle thus, the line of sight is directed along chord T1P.
- Now, the zero end of the tape is held as T1 and the distance T1P1 is measured equal to the length of the initial sub-chord in such a way that the ranging rod at P1 is also bisected by the telescope. Thus, the first point P1 is fixed.
- Set the second deflection angle on the scale so that the line of sight is directed along T1P2.
- With the zero end of the tape pinned at P1 swing the other end around P1 until the arrow held at the other end is bisected by the line of sight, thus locating the second point on the curve (P2).
- Repeat the process until the last point T2 is reached.
Check:
- The last point so located must coincide with the point of tangency (T2) fixed independently by measurements from the point of intersection.
- If the discrepancy is small, the last few pegs may be adjusted. If it is more, the whole curve should be reset.
Field Procedure of Setting out Curve by Deflection Angle by Two-Theodolite Method:
- This method is employed in railway curve setting, as it gives the correct location of points.
- In this method, no chain or tape is required to fix the points on the curve.
- It is mostly suitable when the ground surface is not favorable for chaining along the curve due to undulations,
- First of all, all the necessary data for setting out the curve is calculated in the usual manner, and a setting out table is prepared.
- The tangent points T1 and T2 are marked on the ground by inserting pegs.
- This method consists of two theodolites. A theodolite is centered over T2 and leveled properly.
- Set the horizontal angle at 0˚ and the upper clamp is tightened.
- Direct the line of sight of the instrument at T2 towards T1 when the reading is zero, till the ranging rod at T1 is perfectly bisected.
- The ranging rod at T1 is taken off and another theodolite is centered over this point T1 and leveled.
- The horizontal angle is set at 0˚ and the upper clamp is tightened.
- The line of sight of theodolite is directed towards B and the ranging rod at B is perfectly bisected.
- Set the reading of each of the instruments to the deflection angle for the first point P1. The line of sight of both the theodolites is thus directed towards P1 along T1P1 and T2P1 respectively.
- Move a ranging rod or an arrow in such a way that it is bisected simultaneously by the cross-hairs of both instruments. Thus, the point is fixed.
- To fix point P2, set the reading of both the instruments to the second deflection angle and bisect the ranging rod.
- This process is repeated until all the deflection angles are set out and all the points are marked.
- Finally, when the total deflection angle (δn) is set out in both instruments, the line of sight of the theodolite at T1 should bisect T2 and that of the theodolite at T2 should bisect B.
- This method is expensive since two instruments and two surveyors are required.
- However, it is most accurate since each point is fixed independently of the others. An error in setting out one point is not carried right through the curve as in the method of tangential angles.