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Template for Distance Education Documents Unit CIV2202: Surveying 12.1 Topic 12: Curves CIV2202.12: CURVES Table of Contents PREVIEW .........................................................................................................................3 Introduction ................................................................................................................3 Objectives...................................................................................................................3 Readings .....................................................................................................................3 HORIZONTAL CURVES.................................................................................................3 CIRCULAR HORIZONTAL CURVES ...........................................................................4 Types of Circular Curves............................................................................................4 Simple : one arc .....................................................................................................4 Compound: more than 1 arc .................................................................................4 Reverse ..................................................................................................................4 SETTING OUT METHODS.............................................................................................5 Small radius curves.....................................................................................................5 Offsets from the Tangent Length ...............................................................................5 Offsets from the Long Chord .....................................................................................5 Tangential Angles.......................................................................................................6 Coordinates.................................................................................................................6 Obstructions................................................................................................................7 FACTORS AFFECTING DESIGN OF CIRCULAR CURVES ......................................8 Tangential Angles - Terminology ..............................................................................8 Important Relationships..............................................................................................9 Useful Lengths............................................................................................................9 Through Chainage ......................................................................................................9 Location of Intersection Pt and Tangent Pts.............................................................10 Location of TP's if IP is inaccessible........................................................................11 SETTING OUT- TANGENTIAL ANGLES METHOD ................................................11 EXAMPLE ......................................................................................................................12 Procedure..................................................................................................................13 TRANSITION CURVES ................................................................................................14 DESIGN CONSIDERATIONS................................................................................14 BASIC EQUATION........................................................................................................15 VERTICAL CURVES.....................................................................................................18 Design Considerations..............................................................................................18 Gradients ..................................................................................................................18 Sign Convention .......................................................................................................19 Design Criteria..........................................................................................................19 Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.2 Topic 12: Curves Type of Curve...........................................................................................................19 Assumptions .............................................................................................................20 Equation for the VC..................................................................................................20 Design and Setting Out.............................................................................................21 Other Factors ............................................................................................................21 Plotting and Setting Out ...........................................................................................21 Setting Out................................................................................................................22 Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.3 Topic 12: Curves PREVIEW Introduction Calculating curves for design or setout demands a thorough understanding of their basic geometry. The descriptive geometry learnt at school must be recalled or revised. The theorems relating to circles are particularly important. Objectives After completing this topic you should be able to : • understand the various methods for setting out a horizontal, circular curve, and be able to perform the calculations for the Tangential Angles Method. • understand the basis for Transition Curves. • calculate the elevations and displacements for a Vertical Curve, and understand the underlying assumptions. Readings Read Muskett, Chapter 10 REQUIRED HORIZONTAL CURVES Horizontal curves are used for deflecting roads, channels, pipelines, railways etc. There are 2 types: 1. Circular Curves (with constant radius), and 2. Non-Circular Transition Curves (of varying radius, which will be considered later) Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.4 Topic 12: Curves CIRCULAR HORIZONTAL CURVES Types of Circular Curves Simple : one arc Compound: more than 1 arc Reverse Centres on opposite sides of a common tangent. Useful for joining 2 straights that are nearly parallel, eg. Mulgrave freeway inbound approaching Warrigal Rd. Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.5 Topic 12: Curves SETTING OUT METHODS Small radius curves Small radius curves (eg. kerb at road intersections) can be set out using a tape swung from the peg which marks the centre of the curve. However, large radius curves (eg. R > 25 m) or obstructions (eg. centre point is inside the building on the corner), require less direct methods for setting out curves. Offsets from the Tangent Length Tabulate x and y values as shown, and using two tapes, locate the pegs. This could be messy where x and y are large, or on steeply sloping ground. Offsets from the Long Chord Tabulate x and y to locate the pegs on the curve. Again, this method is likely to be difficult if x and y are large, or in steeply sloping terrain. Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.6 Topic 12: Curves Tangential Angles This is the only method that we'll consider in detail (see a later section). It requires a theodolite (at one tangent point) which is used to turn off the angle to each of the pegs, and a tape which is used to measure from the previous peg to the next peg. This method can cope with rough terrain as long as a sighting is possible (or the theodolite can be moved to a new position if necessary). Coordinates Points are determined relative to control points (i) bearings or distances from 2 or more control points This method can produce a weak position fix depending on the intersection geometry. Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.7 Topic 12: Curves (ii) bearings and distance from one control point (increasingly used with EDM and computer based methods). The set out should be checked by measurements from another point. Obstructions Coordinate (and other) methods can be used to set out on either side of an obstruction, then later used to peg the missing points. Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.8 Topic 12: Curves FACTORS AFFECTING DESIGN OF CIRCULAR CURVES Roads, railways - vehicle speed, existing road/rail reserve Pipelines - min. radius for joint pipes. Tangential Angles - Terminology I = Intersection Pt. Θ = Deflection Angle, Radius Angle (TCU) T, U = Tangent Points, TP's. R = Radius of Curvature C = Centre of Curvature TU = Long Chord α = example of a Tangential Angle PI = External Distance (secant) IT = IU = Tangent Length Department of Civil Engineering, Monash University Edition 6/2002 Unit CIV2202: Surveying 12.9 Topic 12: Curves Important Relationships 1. Triangle ITU is isosceles 2. The tangential angle a, for any point K, is half the angle subtended at the centre. 3. The tangential angle to any point on the curve is equal to the sum of the tangential angles from each chord up to that point. This relationship yields a method of setting out the curve using a theodolite at T and a tape. We will return to this method soon. Useful Lengths
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