This document discusses curves used in transportation routes. It defines different types of curves including simple, compound, and reverse curves. It provides the nomenclature and key elements of simple circular curves, such as tangents, points of intersection and tangency, deflection angle, chord length, arc length, and mid-ordinate. The document also discusses the designation and degree of curves, and describes methods for setting out curves using linear measurements or angular measurements along the curve. Sample problems are provided to demonstrate how to calculate elements of a simple circular curve given radius, deflection angle, and chainage information.
1. The document describes the geometry of simple circular curves used in highway design. It defines key terms like radius (R), deflection angle (Δ), subtangent distance (T), and presents formulas to calculate these values from the radius or each other.
2. Methods for describing the "sharpness" of a curve are discussed, including radius, degree of curve based on chord length (Dc) or arc length (Da). Formulas are provided to calculate radius from these values.
3. The document provides an example problem calculating values for a given curve with a 40 degree deflection angle, including the radius (R), tangent distance (T), arc length (La), and stationing of points of
This document describes different methods of trigonometric leveling to determine the elevation of points. Trigonometric leveling uses vertical angles measured with a theodolite and distances to calculate elevations. There are methods to determine elevations when the base is accessible and inaccessible, and when instrument stations and objects are in the same or different vertical planes. Calculations use trigonometric functions and relationships between angles and distances in triangles formed by the instrument stations and object.
Surveying is the technique of determining the relative position of different features on, above or beneath the surface of the earth by means of direct or indirect measurements and finally representing them on a sheet of paper known as plan or map. Please refer this pdf to learn about Circular Curves.
This document discusses horizontal curves used in transportation infrastructure to gradually change the direction of roads, railways, and other linear structures. It defines simple, compound, and reverse circular curves used for this purpose. Simple curves are arcs of a circle defined by their radius, while compound curves consist of two simple curves joined together curving in the same direction. Reverse curves curve in opposite directions. Spiral curves with varying radii are also used. The document then provides details on calculating elements of simple circular curves like radius, tangent distance, external distance, middle ordinate, and degree of curvature using standard geometric relationships. It concludes with describing the deflection angle method for laying out horizontal curves in the field using one or two theodolites.
The document provides information about theodolites. It begins with an introduction stating that a theodolite is used to measure horizontal and vertical angles more precisely than a magnetic compass. It then discusses the main parts of a theodolite including the horizontal circle, vertical circle, telescope, and levels. The document also covers the history of theodolites from their early origins to modern electronic versions. It describes how to operate a transit vernier theodolite including terms like centering, transiting, swinging the telescope, and changing face. Finally, it discusses the permanent and temporary adjustments needed to ensure accurate theodolite measurements.
Curves are used in transportation routes to gradually change direction between straight segments. There are several types of curves including simple, compound, reverse, and transition curves. A simple circular curve connects two tangents with a single arc, and is defined by its radius or degree. Transition curves provide a gradual transition between tangents and circular curves to avoid abrupt changes in grade or superelevation that could cause vehicles to overturn. There are several methods for laying out circular curves, including using offset distances from the long chord between tangent points or measuring deflection angles from the initial tangent.
This document discusses different methods for balancing a closed traverse survey by distributing corrections to station coordinates. It provides examples of using Bowditch's Rule, the Transit Rule, and the Third Rule to balance a sample traverse with given length, latitude, and departure coordinates. Bowditch's Rule distributes corrections proportionally to leg lengths, while the Transit Rule uses angular precision assumptions and the Third Rule separates corrections between northings/southings and eastings/westings.
The document discusses transition curves in highways. Transition curves are curves that gradually change the horizontal alignment from straight to circular. This is done to introduce centrifugal force, super elevation, extra widening, and aesthetics gradually for driver comfort and safety. There are three main types of transition curves: spiral, cubic parabola, and lemniscate. The length of the transition curve can be calculated based on the rate of change of centrifugal acceleration, rate of introduction of designed super elevation, or empirical formulas based on vehicle speed and radius of the circular curve. The maximum length from these three criteria is used as the final length of the transition curve.
1. The document describes the geometry of simple circular curves used in highway design. It defines key terms like radius (R), deflection angle (Δ), subtangent distance (T), and presents formulas to calculate these values from the radius or each other.
2. Methods for describing the "sharpness" of a curve are discussed, including radius, degree of curve based on chord length (Dc) or arc length (Da). Formulas are provided to calculate radius from these values.
3. The document provides an example problem calculating values for a given curve with a 40 degree deflection angle, including the radius (R), tangent distance (T), arc length (La), and stationing of points of
This document describes different methods of trigonometric leveling to determine the elevation of points. Trigonometric leveling uses vertical angles measured with a theodolite and distances to calculate elevations. There are methods to determine elevations when the base is accessible and inaccessible, and when instrument stations and objects are in the same or different vertical planes. Calculations use trigonometric functions and relationships between angles and distances in triangles formed by the instrument stations and object.
Surveying is the technique of determining the relative position of different features on, above or beneath the surface of the earth by means of direct or indirect measurements and finally representing them on a sheet of paper known as plan or map. Please refer this pdf to learn about Circular Curves.
This document discusses horizontal curves used in transportation infrastructure to gradually change the direction of roads, railways, and other linear structures. It defines simple, compound, and reverse circular curves used for this purpose. Simple curves are arcs of a circle defined by their radius, while compound curves consist of two simple curves joined together curving in the same direction. Reverse curves curve in opposite directions. Spiral curves with varying radii are also used. The document then provides details on calculating elements of simple circular curves like radius, tangent distance, external distance, middle ordinate, and degree of curvature using standard geometric relationships. It concludes with describing the deflection angle method for laying out horizontal curves in the field using one or two theodolites.
The document provides information about theodolites. It begins with an introduction stating that a theodolite is used to measure horizontal and vertical angles more precisely than a magnetic compass. It then discusses the main parts of a theodolite including the horizontal circle, vertical circle, telescope, and levels. The document also covers the history of theodolites from their early origins to modern electronic versions. It describes how to operate a transit vernier theodolite including terms like centering, transiting, swinging the telescope, and changing face. Finally, it discusses the permanent and temporary adjustments needed to ensure accurate theodolite measurements.
Curves are used in transportation routes to gradually change direction between straight segments. There are several types of curves including simple, compound, reverse, and transition curves. A simple circular curve connects two tangents with a single arc, and is defined by its radius or degree. Transition curves provide a gradual transition between tangents and circular curves to avoid abrupt changes in grade or superelevation that could cause vehicles to overturn. There are several methods for laying out circular curves, including using offset distances from the long chord between tangent points or measuring deflection angles from the initial tangent.
This document discusses different methods for balancing a closed traverse survey by distributing corrections to station coordinates. It provides examples of using Bowditch's Rule, the Transit Rule, and the Third Rule to balance a sample traverse with given length, latitude, and departure coordinates. Bowditch's Rule distributes corrections proportionally to leg lengths, while the Transit Rule uses angular precision assumptions and the Third Rule separates corrections between northings/southings and eastings/westings.
The document discusses transition curves in highways. Transition curves are curves that gradually change the horizontal alignment from straight to circular. This is done to introduce centrifugal force, super elevation, extra widening, and aesthetics gradually for driver comfort and safety. There are three main types of transition curves: spiral, cubic parabola, and lemniscate. The length of the transition curve can be calculated based on the rate of change of centrifugal acceleration, rate of introduction of designed super elevation, or empirical formulas based on vehicle speed and radius of the circular curve. The maximum length from these three criteria is used as the final length of the transition curve.
CHAPTER 5 Earth work quantity and mass haul diagram..pdfMohdMusaHashim1
This document discusses earthwork quantities and mass haul diagrams for roadway design. It begins by defining key terms related to earthwork, such as borrow, waste, free haul, and overhaul. It then describes the various activities involved in earthwork, including clearing, excavation, embankment construction, and slope preparation. Methods for calculating cut and fill volumes using end area and prismoidal formulas are provided. The document outlines the steps to develop a mass haul diagram, including plotting cumulative cut and fill volumes versus station. It describes how mass haul diagrams can be used to determine material distribution, waste and borrow needs, and overhaul quantities. Finally, economic analyses for determining the limit of economical haul are presented.
The document discusses setting out curves for highways. It defines different types of curves, including horizontal and vertical curves. Horizontal curves can be circular, transition, or lemniscate curves. Vertical curves connect different gradients and can be circular arcs or parabolic arcs. Common elements of circular curves are defined, such as tangents, points of intersection, and long chords. Methods for setting out simple circular curves include linear methods using offsets from long chords, bisection of arcs, tangents, or chords produced.
1) Curves are gradual bends provided in transportation infrastructure like roads, railways and canals to allow for a smooth change in direction or grade.
2) There are two main types of curves - horizontal curves which provide a gradual change in direction, and vertical curves which provide a gradual change in grade.
3) Curves are needed to safely guide vehicles and traffic when changing directions or grades, to improve visibility, and to prevent erosion of canal banks from water pressure.
Levelling, also known as heighting, is the process of determining relative height differences between points on the Earth's surface. If the height of one point is known relative to a datum, then the heights of other points can be found relative to the same datum. This is done using a leveling instrument, leveling staff, and following standard procedures such as taking backsight and foresight readings. Care must be taken to eliminate errors from things like atmospheric refraction. Results are typically recorded in a level book or form and can be reduced using methods like height of instrument or rise and fall.
Transition curve and Super-elevation
Transition Curve
Objectives of Transition Curve
Properties Of Transition Curve
Types Of Transition Curve
Length Of Transition Curve
Superelevation
Objective of providing superelevation
Advantages of providing superelevation
Superelevation Formula
Numerical
A compound curve consists of two or more circular arcs that join at common tangent points, turning in the same direction between two main tangents. They are rarely used to provide a gradual transition from straight lines to curves but are mainly found in interchange loops and ramps. Reverse curves have an instantaneous change in direction at the point of reverse curve (PRC) and are seldom used in highway alignment due to this, but can be pleasing for other applications like park roads. Both compound and reverse curves problems can be solved using equations that consider the curve geometry as a five-sided polygon traverse.
Tacheometric surveying is a method of surveying that determines horizontal and vertical distances optically rather than through direct measurement with a tape or chain. It uses an instrument called a tacheometer fitted with a stadia diaphragm to rapidly measure distances. The key principles are that the ratio of perpendicular to base is constant in similar triangles, allowing horizontal distance and elevation to be calculated from observed angles and staff intercept readings. Common tacheometric systems include fixed hair stadia, subtense stadia, and tangential methods. Distance and elevation formulas are derived for horizontal, inclined, and depressed line of sights depending on staff orientation. Tacheometric surveying is well-suited for difficult terrain where direct measurement is challenging
Tacheometric surveying uses a tacheometer to determine horizontal and vertical distances through angular measurements. A tacheometer is a theodolite fitted with stadia hairs and an anallatic lens. The tacheometric formula relates the staff intercept, focal length, stadia interval and additive constant to calculate horizontal distances. Methods include stadia, fixed/movable hair, and non-stadia techniques. Determining the tacheometer constant involves measuring distances and staff intervals at stations to solve equations. Errors arise from incorrect stadia intervals or graduations. Tacheometric surveying provides distances in rough terrain but with less precision than other methods.
The document provides information about stress distribution in soil due to self-weight and surface loads. It discusses Boussinesq's formula for calculating vertical stress in soil due to a concentrated surface load. The formula shows that vertical stress is directly proportional to the load, inversely proportional to depth squared, and depends on the ratio of radius to depth. A table of coefficient values used in the formula for different ratios of radius to depth is also provided.
Horizontal curves are used in highway design to provide a gradual transition between two intersecting roadways. A simple curve is an arc of a circle, with the radius determining the sharpness of the curve. Key elements of a simple curve include the radius, tangent distance, intersection angle, and stationing of the point of curvature and point of tangency. Compound curves consist of two simple curves joined together curving in the same direction. Reversed curves have two simple curves joined and curving in opposite directions, connected at the point of reversed curve.
This document discusses vertical curves used in transportation design. Vertical curves provide a smooth transition between different road or rail grades. They are designed using parabolic equations to maintain a constant rate of change in slope. The key points are:
- Vertical curves connect two different grades using a parabolic shape.
- Their design ensures a constant rate of change in slope for driver comfort.
- The general parabolic equation and methods for computing curve elements like high/low points and elevations at different points are presented.
1. The document presents information from a slideshow on tacheometric surveying. It discusses various methods of tacheometric surveying including fixed hair, movable hair, tangential, and subtense bar methods.
2. Formulas are provided for calculating horizontal distance, vertical distance, and elevation of points using these different tacheometric surveying methods under various sighting conditions such as inclined or depressed lines of sight.
3. The document also discusses tacheometric constants, anallatic lenses, and procedures for conducting field work in a tacheometric survey including selecting instrument stations, taking observations of vertical angles and staff readings, and shifting to subsequent stations.
This document describes the elements and setting out procedure of a compound curve. A compound curve consists of two circular arcs connected by a common tangent. It provides equations to calculate the seven elements (radii, tangents, deflection angles) of the curve given four known values. The setting out procedure involves locating the points of tangency and common tangent using deflection angles, then setting out each arc using Rankine's method starting from the first and second points of tangency.
This document discusses the theory of errors in survey measurements. It begins with an introduction to the types of errors that can occur, including mistakes, systematic errors, and accidental errors. It then provides definitions for key terms like direct observation, indirect observation, conditioned quantity, true value, most probable value, true error, and residual error. The document goes on to explain statistical formulas used to calculate the probable error of single observations, the probable error of the mean, standard deviation, standard error, and precision. It also covers the law of accidental errors and laws of weights. Finally, it discusses the theory of least squares and provides examples of determining probable errors and finding the most probable value of quantities.
This document discusses methods for measuring horizontal distances in chain surveying. It describes direct measurement using tapes, chains, and EDM instruments. Common sources of error in tape measurements include temperature fluctuations, sag, tension applied, and incorrect standardized length. Corrections are made to account for these errors. Perpendicular and oblique offsets from chain lines to features are also measured. Instruments like open cross staffs, adjustable cross staffs, and optical/prism squares are used to lay out right angles when measuring offsets. Proper procedures are outlined for collecting distance and offset measurements in the field.
This document provides information about errors in measurements and their propagation. It defines different types of errors like gross errors, systematic errors, and random errors. It discusses concepts like most probable value, standard deviation, variance, standard error of mean, most probable error, confidence limits, and weight. It distinguishes between precision and accuracy. It also describes the propagation of errors, where the error in a computed quantity is evaluated as a function of the errors in the measured quantities used to compute it.
s2 5 tutorial traversing - compass rule and transit ruleEst
This document contains 7 questions related to surveying techniques. Each question provides field data like angles, distances, and coordinates from closed traverse surveys. The questions ask to calculate angular errors, adjust angles and coordinates, determine missing distances, and apply techniques like compass rule, Bowditch adjustment, and equal shifts to analyze the data and solve for unknown values.
Tacheometric surveying is a method of rapidly determining horizontal and vertical positions of points using optical measurements rather than traditional tape or chain measurements. A tacheometer, which is a transit theodolite fitted with a stadia diaphragm, is used to measure the horizontal and vertical angles to a stadia rod or staff held at survey points. Formulas involving the stadia interval, staff intercept readings, and calculated constants are used to determine horizontal distances and elevations from the instrument to points. Measurements can be taken with horizontal lines of sight or inclined lines of sight when the staff is held vertically or normal to the line of sight.
Introduction, electromagnetic spectrum, electromagnetic distance measurement, types of EDM instruments, electronic digital theodolites, total station, digital levels, scanners for topographical survey, global positioning system.
Curves are used to gradually change the direction of transportation routes like roads, railways and pipelines. They connect straight tangents and are usually circular arcs. There are different types of curves classified based on their shape and connection of tangents like simple, compound, reverse etc. Elements like radius, deflection angle, length of curve, tangent length etc are used to design circular curves. Various surveying methods like Rankine's, two theodolite etc are used to establish curves on the ground based on their elements and principles. Compound curves consist of two simple circular curves bending in the same direction and joining at a common point of compound.
i) Curves are used in transportation lines like roads, railways and canals to provide gradual changes in direction and grade instead of abrupt turns. They are classified as horizontal or vertical based on the plane they operate in.
ii) A simple circular curve consists of a single arc of a circle connecting two straight lines or tangents. It is designated by its radius or degree, which indicates the sharpness of the curve.
iii) Curves are set out through linear methods using chains and tapes or angular instrumental methods. This involves locating the tangent points and laying out pegs along the curve at set intervals to join and form the curved alignment.
CHAPTER 5 Earth work quantity and mass haul diagram..pdfMohdMusaHashim1
This document discusses earthwork quantities and mass haul diagrams for roadway design. It begins by defining key terms related to earthwork, such as borrow, waste, free haul, and overhaul. It then describes the various activities involved in earthwork, including clearing, excavation, embankment construction, and slope preparation. Methods for calculating cut and fill volumes using end area and prismoidal formulas are provided. The document outlines the steps to develop a mass haul diagram, including plotting cumulative cut and fill volumes versus station. It describes how mass haul diagrams can be used to determine material distribution, waste and borrow needs, and overhaul quantities. Finally, economic analyses for determining the limit of economical haul are presented.
The document discusses setting out curves for highways. It defines different types of curves, including horizontal and vertical curves. Horizontal curves can be circular, transition, or lemniscate curves. Vertical curves connect different gradients and can be circular arcs or parabolic arcs. Common elements of circular curves are defined, such as tangents, points of intersection, and long chords. Methods for setting out simple circular curves include linear methods using offsets from long chords, bisection of arcs, tangents, or chords produced.
1) Curves are gradual bends provided in transportation infrastructure like roads, railways and canals to allow for a smooth change in direction or grade.
2) There are two main types of curves - horizontal curves which provide a gradual change in direction, and vertical curves which provide a gradual change in grade.
3) Curves are needed to safely guide vehicles and traffic when changing directions or grades, to improve visibility, and to prevent erosion of canal banks from water pressure.
Levelling, also known as heighting, is the process of determining relative height differences between points on the Earth's surface. If the height of one point is known relative to a datum, then the heights of other points can be found relative to the same datum. This is done using a leveling instrument, leveling staff, and following standard procedures such as taking backsight and foresight readings. Care must be taken to eliminate errors from things like atmospheric refraction. Results are typically recorded in a level book or form and can be reduced using methods like height of instrument or rise and fall.
Transition curve and Super-elevation
Transition Curve
Objectives of Transition Curve
Properties Of Transition Curve
Types Of Transition Curve
Length Of Transition Curve
Superelevation
Objective of providing superelevation
Advantages of providing superelevation
Superelevation Formula
Numerical
A compound curve consists of two or more circular arcs that join at common tangent points, turning in the same direction between two main tangents. They are rarely used to provide a gradual transition from straight lines to curves but are mainly found in interchange loops and ramps. Reverse curves have an instantaneous change in direction at the point of reverse curve (PRC) and are seldom used in highway alignment due to this, but can be pleasing for other applications like park roads. Both compound and reverse curves problems can be solved using equations that consider the curve geometry as a five-sided polygon traverse.
Tacheometric surveying is a method of surveying that determines horizontal and vertical distances optically rather than through direct measurement with a tape or chain. It uses an instrument called a tacheometer fitted with a stadia diaphragm to rapidly measure distances. The key principles are that the ratio of perpendicular to base is constant in similar triangles, allowing horizontal distance and elevation to be calculated from observed angles and staff intercept readings. Common tacheometric systems include fixed hair stadia, subtense stadia, and tangential methods. Distance and elevation formulas are derived for horizontal, inclined, and depressed line of sights depending on staff orientation. Tacheometric surveying is well-suited for difficult terrain where direct measurement is challenging
Tacheometric surveying uses a tacheometer to determine horizontal and vertical distances through angular measurements. A tacheometer is a theodolite fitted with stadia hairs and an anallatic lens. The tacheometric formula relates the staff intercept, focal length, stadia interval and additive constant to calculate horizontal distances. Methods include stadia, fixed/movable hair, and non-stadia techniques. Determining the tacheometer constant involves measuring distances and staff intervals at stations to solve equations. Errors arise from incorrect stadia intervals or graduations. Tacheometric surveying provides distances in rough terrain but with less precision than other methods.
The document provides information about stress distribution in soil due to self-weight and surface loads. It discusses Boussinesq's formula for calculating vertical stress in soil due to a concentrated surface load. The formula shows that vertical stress is directly proportional to the load, inversely proportional to depth squared, and depends on the ratio of radius to depth. A table of coefficient values used in the formula for different ratios of radius to depth is also provided.
Horizontal curves are used in highway design to provide a gradual transition between two intersecting roadways. A simple curve is an arc of a circle, with the radius determining the sharpness of the curve. Key elements of a simple curve include the radius, tangent distance, intersection angle, and stationing of the point of curvature and point of tangency. Compound curves consist of two simple curves joined together curving in the same direction. Reversed curves have two simple curves joined and curving in opposite directions, connected at the point of reversed curve.
This document discusses vertical curves used in transportation design. Vertical curves provide a smooth transition between different road or rail grades. They are designed using parabolic equations to maintain a constant rate of change in slope. The key points are:
- Vertical curves connect two different grades using a parabolic shape.
- Their design ensures a constant rate of change in slope for driver comfort.
- The general parabolic equation and methods for computing curve elements like high/low points and elevations at different points are presented.
1. The document presents information from a slideshow on tacheometric surveying. It discusses various methods of tacheometric surveying including fixed hair, movable hair, tangential, and subtense bar methods.
2. Formulas are provided for calculating horizontal distance, vertical distance, and elevation of points using these different tacheometric surveying methods under various sighting conditions such as inclined or depressed lines of sight.
3. The document also discusses tacheometric constants, anallatic lenses, and procedures for conducting field work in a tacheometric survey including selecting instrument stations, taking observations of vertical angles and staff readings, and shifting to subsequent stations.
This document describes the elements and setting out procedure of a compound curve. A compound curve consists of two circular arcs connected by a common tangent. It provides equations to calculate the seven elements (radii, tangents, deflection angles) of the curve given four known values. The setting out procedure involves locating the points of tangency and common tangent using deflection angles, then setting out each arc using Rankine's method starting from the first and second points of tangency.
This document discusses the theory of errors in survey measurements. It begins with an introduction to the types of errors that can occur, including mistakes, systematic errors, and accidental errors. It then provides definitions for key terms like direct observation, indirect observation, conditioned quantity, true value, most probable value, true error, and residual error. The document goes on to explain statistical formulas used to calculate the probable error of single observations, the probable error of the mean, standard deviation, standard error, and precision. It also covers the law of accidental errors and laws of weights. Finally, it discusses the theory of least squares and provides examples of determining probable errors and finding the most probable value of quantities.
This document discusses methods for measuring horizontal distances in chain surveying. It describes direct measurement using tapes, chains, and EDM instruments. Common sources of error in tape measurements include temperature fluctuations, sag, tension applied, and incorrect standardized length. Corrections are made to account for these errors. Perpendicular and oblique offsets from chain lines to features are also measured. Instruments like open cross staffs, adjustable cross staffs, and optical/prism squares are used to lay out right angles when measuring offsets. Proper procedures are outlined for collecting distance and offset measurements in the field.
This document provides information about errors in measurements and their propagation. It defines different types of errors like gross errors, systematic errors, and random errors. It discusses concepts like most probable value, standard deviation, variance, standard error of mean, most probable error, confidence limits, and weight. It distinguishes between precision and accuracy. It also describes the propagation of errors, where the error in a computed quantity is evaluated as a function of the errors in the measured quantities used to compute it.
s2 5 tutorial traversing - compass rule and transit ruleEst
This document contains 7 questions related to surveying techniques. Each question provides field data like angles, distances, and coordinates from closed traverse surveys. The questions ask to calculate angular errors, adjust angles and coordinates, determine missing distances, and apply techniques like compass rule, Bowditch adjustment, and equal shifts to analyze the data and solve for unknown values.
Tacheometric surveying is a method of rapidly determining horizontal and vertical positions of points using optical measurements rather than traditional tape or chain measurements. A tacheometer, which is a transit theodolite fitted with a stadia diaphragm, is used to measure the horizontal and vertical angles to a stadia rod or staff held at survey points. Formulas involving the stadia interval, staff intercept readings, and calculated constants are used to determine horizontal distances and elevations from the instrument to points. Measurements can be taken with horizontal lines of sight or inclined lines of sight when the staff is held vertically or normal to the line of sight.
Introduction, electromagnetic spectrum, electromagnetic distance measurement, types of EDM instruments, electronic digital theodolites, total station, digital levels, scanners for topographical survey, global positioning system.
Curves are used to gradually change the direction of transportation routes like roads, railways and pipelines. They connect straight tangents and are usually circular arcs. There are different types of curves classified based on their shape and connection of tangents like simple, compound, reverse etc. Elements like radius, deflection angle, length of curve, tangent length etc are used to design circular curves. Various surveying methods like Rankine's, two theodolite etc are used to establish curves on the ground based on their elements and principles. Compound curves consist of two simple circular curves bending in the same direction and joining at a common point of compound.
i) Curves are used in transportation lines like roads, railways and canals to provide gradual changes in direction and grade instead of abrupt turns. They are classified as horizontal or vertical based on the plane they operate in.
ii) A simple circular curve consists of a single arc of a circle connecting two straight lines or tangents. It is designated by its radius or degree, which indicates the sharpness of the curve.
iii) Curves are set out through linear methods using chains and tapes or angular instrumental methods. This involves locating the tangent points and laying out pegs along the curve at set intervals to join and form the curved alignment.
Curves are used in transportation routes to provide a gradual change in direction between straight segments. There are several types of curves including simple, compound, and reverse curves. Curves are necessary to avoid obstacles, reduce grading costs, and make alignments more comfortable for transportation. Curves are set out using various linear and angular methods involving measurement of offsets, chords, or angles from tangent lines or radii. Common linear methods include setting offsets perpendicular to chords or tangents drawn from the curve center.
This document discusses simple circular curves, which are curves consisting of a single arc with a constant radius connecting two tangents. It defines key elements of circular curves such as deflection angle, radius of curvature, chord length, and tangent length. Circular curves are used to impose curves between two straight lines in roads and railways. The document also discusses designating curves by their degree or radius, with degree defined as the angle subtended by a 30m chord at the curve's center. Fundamental geometry rules for circular curves are provided.
Location horizontal and vertical curves Theory Bahzad5
Setting out of works
horizontal and vertical curves
Horizontal Alignment
ØAn introduction to horizontal curve &Vertical curve.
ØTypes of curves.
ØElements of horizontal circular curve.
ØGeometric of circular curve
Ø Methods of setting out circular curve
Ø Setting out of horizontal curve on ground
Ø Vertical curve Definition.
ØElements of the vertical curves.
ØAvailable methods for computing the elements of vertical curves
Types of Curves
1- Horizontal Curves
2- Vertical Curves
Horizontal Curves
are circular curves. They connect tangent lines around
obstacles, such as building, swamps, lakes, change
direction in rural areas, and intersections in urban areas.
-Compound Curve.
-Reverse curve.
-Transition or Spiral Curves.
-Horizontal Curve: Simple circular
curve
-Elements of horizontal curves.
-Formulas for simple circular
curves.
-Properties of circular curves.
example:A horizontal curve having R= 500m, ∆=40°, station P.I=
12+00 ,prepare a setting out table to set out the curve
using deflection angle from the tangent and chord length
method, dividing the arc into 50m stations.
:Example H.W
A Horizontal curve is designed with a 600m radius and is
known to have a tangent of 52 m the PI is Station
200+00 determent the Stationing of the PT?
-PROCEDURE SETTING OUT Practical .
Vertical Curves
Elevation and Stations of main points on the Vertical Curve .
Assumptions of vertical curve projection.
Example: A vertical parabola curve 400m long is to be set
between 2% (upgrade) and 1% (down grade), which meet
at chainage of 2000 m, the R.L of point of intersection of
the two gradients being (500.00 m). Calculate the R.L of
the tangent and at every (50m) parabola.
Thank you all
Prepared by:
Asst. Prof. Salar K.Hussein
Mr. Kamal Y.Abdullah
Asst.Lecturer. Dilveen H. Omar
Erbil Polytechnic University
Technical Engineering College
Civil Engineering Department
This document provides an overview of curves and modern surveying techniques. It begins with an introduction to simple circular curves, including their definition, notation, and elements. It describes how to set out a simple circular curve using linear and angular methods. It also discusses transition curves, which provide a gradual transition between straight and curved sections, and types of transition curves like spirals and lemniscates. Finally, it covers vertical curves and the formula to calculate the length of a vertical curve based on the total grade change and rate of change of grade. The document thus provides a comprehensive look at horizontal and vertical curves used in highway and railway design as well as common surveying methods for laying them out.
This document provides information about circular curves used in highways and railways. It discusses the different types of curves including simple, compound, and reverse curves. It defines key elements of circular curves such as radius, deflection angle, tangent length, and mid-ordinate. It presents the relationships between radius and degree of curvature. Finally, it describes various methods for setting out circular curves in the field, including linear methods using offsets and angular methods using a theodolite.
This document provides information on surveying circular curves. It defines different types of curves including simple, compound, and reverse curves. It also defines various curve elements such as point of intersection, point of curve, point of tangency, radius, and deflection angle. Methods for designating a curve by radius or degree of curvature are presented along with relationships between these values. Formulas are given for calculating the length of a curve, tangent length, length of chord, external distance, and mid-ordinate based on the radius and deflection angle. Finally, linear and angular methods for setting out a simple circular curve in the field are described.
Introduction, classification of curves, Elements of a simple circular, designation of curve, methods of setting out a simple circular curve, elements of a compound and reverse curves, transition curve, types of transition curves, combined curve, types of vertical curves
This document provides definitions and explanations of terms related to horizontal curves. It discusses the following:
- Horizontal curves are used to connect two straight lines when there is a change in direction of a road or railway alignment. Circular curves are the most common type of horizontal curve.
- Key terms defined include degree of curve, radius, relationship between radius and degree, superelevation, and centrifugal ratio.
- Different types of horizontal curves are described, including simple circular, compound, reverse, and transition curves.
- Notation used in circular curves is explained, such as tangent points and lengths, deflection angle, and radius.
- Properties of simple circular curves are outlined, including
The document discusses different types of curves used in transportation and infrastructure projects. It describes horizontal curves and vertical curves, which provide gradual changes in direction and grade. Curves are needed on roads, railways, and canals to allow vehicles to transition between paths meeting at an angle, as well as to facilitate gradual changes in elevation and improve visibility and safety. The document outlines key elements of horizontal curves, including the point of intersection, deflection angle, radius, point of curvature, point of tangency, and length of curve.
- Surveying involves making field measurements on or near the Earth's surface to determine relative positions of points or establish points. It includes preliminary surveys to collect data, layout surveys to define proposed construction locations, and construction surveys to provide line and grade during construction.
- Control surveys establish horizontal and vertical reference points and lines that preliminary and construction surveys are referenced to. Horizontal control may be tied to grid monuments, property lines, or baselines while vertical control uses benchmark elevations from leveling surveys.
- Route surveys initially layout highways as a series of tangents joined by circular curves. Compound curves consist of two or more joining circular arcs between main tangents turning in the same direction. Reverse curves connect lines through
CURVE SURVEYING By Denis Jangeed
Type of Curves
Methods for Setting Out of Circular Curve
Broken-back Curve
Elements of Circular Curve
Elements of simple and compound curves, Types of curves, Elements of
circular, reverse, and transition curves. Method of setting out simple,
circular, transition and reverse curves, Types of vertical curves, length of
vertical curves, setting out vertical curves. Tangent corrections.Reverse Curve
Point of tangency Tangent distance Mid ordinate Length of Tangent Length of Chord
Linear Method
1.By Ordinates or Offsets from the Long Chord
2.Perpendicular Offset From Tangent
3.By Offset From Chord Produced (Deflection Distance)
4.Radial Offset From Tangent
5 Successive Bisection Of Chords
Angular Method
Tape & Theodolite/Rankine Method / Tangential / Deflection Angles
Two-theodolite Method
Tachometer Method
Degree of curvature
Curves are usually fitted to tangents by choosing a D (degree of curve) that will place the centerline of the curve on
or slightly on or above the gradeline. Sometimes D is chosen to satisfy a limited tangent distance or a desired curve
length. Each of these situations is discussed below:
Choosing D to fit a gradeline (the most common case).
When joining two tangents where the centerline of the curve is to fall on or slightly above the gradeline,
the desired external is usually used to select D.
This document discusses horizontal curves in surveying. It covers the objectives of learning about horizontal curve layout, types of curves like simple, compound, and reverse curves. It defines degree of curve and how it is calculated based on the arc or cord length. It describes the elements of a circular curve like point of curvature, point of tangency, radius, chord length, and central angle. Methods for laying out a circular curve are discussed, including linear methods using offsets and bisection, and angular methods like Rankine's method and two theodolite method. Key questions about why curves are needed and defining the degree of curve are also answered.
The document discusses circular curves and their use in highway and railway alignment. It defines key terms related to circular curves like deflection angle, chord, radius, and introduces different types of horizontal curves - simple circular curves, compound curves, reverse curves, spiral curves, and lemniscate curves. It also discusses vertical curves like valley and summit curves. The document provides formulas to calculate length of tangent, external distance, middle ordinate, length of chord, length of curve, degree of curve, and minimum radius of curvature for circular curves. It includes examples of problems calculating radius, offset distance, and degree of curve given different curve elements.
This document provides information about curve ranging in surveying engineering. It begins with describing the expected learning outcomes of understanding how to calculate positions for horizontal and vertical curves. It then discusses different types of horizontal curves used in road and railway design, including simple, compound, transition and reverse curves. The key geometry and elements of circular curves like tangent length, external distance, mid-ordinate and curve length are defined. Several example problems are provided to show how to tabulate data and calculate values needed to lay out horizontal and vertical curves using surveying methods and tools. Vertical curves called summit and valley curves are also introduced to smoothly connect changes in roadway gradients.
Overview:
The vertical alignment of a road consists of gradients(straight lines in a vertical plane) and vertical curves. The vertical alignment is usually drawn as a profile, which is a graph with elevation as vertical axis and the horizontal distance along the centre line of the road as the the horizontal axis.
This document discusses layout calculations for railway yards. It provides information on:
1) When layout calculations are needed, such as for yard remodeling, new construction projects, or rectifying defective layouts.
2) Key considerations for layout calculations including turnout parameters, permissible speeds and radii based on IRPWM guidelines.
3) Methods for calculating layouts for different track connections including turnouts taking off from straight or curved tracks, connections to parallel and non-parallel tracks, crossovers, and connections between curved and straight tracks. Diagrams illustrate example calculations.
The document serves as a reference for engineers on best practices and regulatory guidelines for performing layout calculations to design efficient and safe railway yard track configurations.
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This document discusses various methods for transporting and placing concrete, including using mortar pans, wheelbarrows, cranes, trucks, conveyor belts, chutes, skips and hoists, transit mixers, pumps and pipelines. It also discusses considerations for pumpable concrete and proper placement of concrete in forms, under water, or using conveyor belts, slipforms, or pumps. Common recommendations for underwater placement include using tremies, pumps, or bottom dump buckets with low water velocity and temperature, low w/c ratio, and high cement content.
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The document discusses workability of fresh concrete. It defines workability as the ability of concrete to fill forms properly under vibration without reducing quality. Workability depends on water content, mix proportions, aggregate size and shape, and use of admixtures. The ideal workability allows for proper compaction and finishing with minimal bleeding or segregation. Common tests to measure workability include slump test, compacting factor test, flow test, and Vee Bee consistometer test. Factors like water-cement ratio, aggregate size and shape, grading, and admixtures affect workability.
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2. Curves
2
• Curves are usually employed in lines of
communication in order that the change in
direction at the intersection of the straight lines shall
be gradual.
• The lines connected by the curves are tangent to it
and are called Tangents or Straights.
• The curves are generally circular arcs but parabolic
arcs are often used in some countries for this
purpose.
• Most types of transportation routes, such as
highways, railroads, and pipelines, are connected
by curves in both horizontal and vertical planes.
3. Curves
• The purpose of the curves is to deflect a vehicle
travelling along one of the straights safely and
comfortably through a deflection angle θ to enable
it to continue its journey along the other straight.
θ
3
10. Curves
Classification of Curves
1) Simple Curves: Consist of single Arc Connecting two straights.
2)Compound curves: Consist of 2 arcs of different radii, bending
in the same direction and lying on the same sides of their
common tangents, their centers being on the same side of the
curve.
3)Reverse curves: Consist of 2 arcs of equal or unequal radii,
bending in opposite direction with common tangent at their
junction (meeting Point), their center lying on the opposite sides
of the curve.
10
12. Curves
A C
R R
O
E 90o
90o
T1 T2
∅
𝟐
90o - ∅/2
Nomenclature of Simple Curves
B’
B
∅
I
F
∅
𝟐
∅
∅
𝟐
12
13. Curves
Nomenclature of Simple Curves
1)Tangents or Straights: The straight
lines AB and BC which are
connected by the curves are called
the tangents or straights to curves.
2)Point of Intersection: (PI.) The
Point B at which the 2 tangents AB
and BC intersect or Vertex (V).
3)Back Tangent: The tangent lineAB
iscalled 1st tangent or Rear
tangents or Backtangent.
4)Forward Tangent: The tangents
line BC iscalled 2nd tangent or
Forward tangent.
13
14. Curves
Nomenclature of Simple Curves
5)Tangents Points: The points T1 and
T2 at which the curves touches the
straights.
5)Point of Curve (P
.C): The
beginning of the curve T1 is called
the point of curve or tangent curve
(T
.C).
6)b) Point of tangency (C.T): The
end of curve T2 iscalled point of
tangency or curve tangent (C.T).
6) Angle of Intersection: (I) The angle
ABC between the tangent lines AB
and BC. Denoted by I.
14
15. Curves
Nomenclature of Simple Curves
7)Angle of Deflection (∅): Then
angle B`BC by which the forward
(head tangent deflect from the
Rear tangent.
8)Tangent Length: (BT1 and BT2)
The distance from point of
intersection B to the tangent points
T1 and T2. These depend upon the
radii of curves.
9)Long Cord: The line T1T2 joining
the two tangents point T1 and T2 is
called long chord. Denoted by l.
15
16. Curves
Nomenclature of Simple Curves
10)Length of Curve: the arc T1FT2 is
called length of curve. Denoted by
L.
11)Apex or Summit of Curve: The mid
point F of the arc T1FT2 iscalled
Apex of curve and lies on the
bisection of angle of intersection. Itis
the junction oflines radii.
12)External Distance (BF): The
distance BF from the point of
intersection to the apex of the curve
iscalled Apex distance or External
distance.
16
17. 1 2 2 1
∟ BT T = ∟ BT T =
• Ifthe curve deflect to the right of the
direction of the progress of survey it is
called Right-hand curve and id to the
left , it is called Left-handcurve.
• The ∆BT1T2 is an isosceles triangle and
therefore the angle
∅
2
Curves
Nomenclature of Simple Curves
13)Central Angle: The angle T1OT2
subtended at the center of the curve
by the arc T1FT2 iscalled central
angle and isequal to the deflection
angle.
14)Mid ordinate (EF): Itisa ordinate
from the mid point of the long chord
to the mid point of the curve i.e
distance EF. Also called Versed sine
of the curve.
17
18. 2
1 1
In ∆T OB, tan (∅
)= BT / OT 1
1= 1 2
BT OT tan(∅
)
1 = 2= 2
BT BT R tan(∅
)
c) Length of Chord(l):
2
1 1
In ∆T OE, sin( ∅
) = T E /OT 1
1 1 2
T E = OT sin(∅
)
1 2
T E = R sin(∅
)
Curves
Elements of Simple Curves
a) ∟T1BT2 + ∟B`BT2=180o
I + ∅= 180o
∟T1OT2 = ∅= 180o – I
b) Tangent lengths: (BT1 , BT2)
18
19. 1 2
l = 2 T E = 2 R sin(∅
)
d) Length of Curve (L):
L = length of arcT1FT2
L = R ∅(rad) = 𝜋R∅
1
8
0
Or
L /2 𝜋R = ∅/360
L =2 𝜋R ∅/ 360 = 𝜋R∅
180
e) Apex distance or Externaldistance:
BF = BO – OF
2
1 1
In ∆OT B, cos(∅
)= OT / BO
Curves
Elements of Simple Curves
19
r
S =r x 𝜃
𝜃
∅
L
360o
2𝜋
R
R R
20. 1 2 2
BO = OT / cos(∅
)= R / cos(∅
)
2
BO = Rsec(∅
)
BF = R sec(∅/2)– R
2
BF = R ( sec(∅
)–1)
BF = R (
1
cos(∅
)
– 1)
2
f) Mid ordinate or Versed sine of curve:
EF= OF – OE
In ∆T1OE, cos(∅/2)= OE / OT1
OE = OT1 cos(∅/2)= R cos(∅/2)EF =
R – R cos(∅/2)
EF = R (1 –cos(∅
))
Curves
Elements of Simple Curves
20
2
21. Curves
Designation Of Curves
• InU.K a curve isdefined by Radius which it expressed in
terms of feet or chains(Gunter chain) e.g 12 chain curve, 24
chain curve.
• When expressed in feet the radius is taken as multiple of 100
e.g 200, 300, 400...
• InUSA, Canada, India and Pakistan a curve is designated by
a degree e.g 2 degree curve , 6 degree curve.
Degree Of Curves
Degree of curve is defined in 2 ways
1) Arc Definition
2) Chord Definition
21
22. 1) Arc Definition:
“ The degree of a curve isthe
central angle subtended by 100 feet
of arc”.
Let R =Radius of Curve
D =Degree ofCurve
Then =
𝐷 100
360 2𝜋R
R =
5729.58
𝐷
(feet)
Itisused in highways.
Curves
Degree Of Curves
R R
O
100 feet
D
D
100 feet
360o
2𝜋
R
R R
22
23. From ∆OPM
sin(
2
) =
𝑂𝑀
=
𝐷 𝑀𝑃 50
𝑅
5
0
sin(𝐷
)
R= (feet)
2
Itis used in Railway.
Example: D =1o
1) Arc Def:
2) Chord Def:
R = 5729.58 / D
= 5729.58 feet
R = 50 / sin (D/2)
Curves
R R
O
Degree Of Curves
2) Chord Definition:
“ The degree of curve is the central
angle subtended by 100 feet of chord”.
100 feet
D
M N
50 ‘ P 50 ‘
D/2
23
= 5729.65 feet
24. Simple Curves
Method of Curve Ranging
• There are a number of different methods by which a
centerline can be set out, all of which can be summarized in
two categories:
• Traditional methods: which involve working along the
centerline itself using the straights, intersection points and
tangent points forreference.
• The equipment used for these methods include, tapes and
theodolites or totalstations.
• Coordinate methods: which use control networks as
reference. These networks take the form of control points
located on site some distance away from the centerline.
• For this method, theodolites, totals stations or GPS receivers
can be used.
24
25. Along the arc it is practically not possible that is why measured
along thechord.
Simple Curves
Method of Curve Ranging
The methods for setting out curves may be divided into 2
classes according to the instrument employed .
1) Linear or Chain & Tape Method
2) Angular or InstrumentalMethod
Peg Interval:
Usual Practice--- Fix pegs at equal interval on the curve
20 m to 30 m ( 100 feet or one chain)
66 feet ( Gunter’s Chain)
Strictly speaking this interval must be measured as the Arc
intercept b/w them, however it is necessarily measure along
the chord. The curve consist of a series of chords rather than
arcs.
25
26. Peg Interval:
For difference in arc and chord to be negligible
𝑅
Length of chord >2
0
of curve
R = Radius of curve
Length of unit chord =30 m for flate curve ( 100 ft)
(peg interval) 20 m for sharp curve (50ft)
10 m for very sharp curves (25 ft or less)
Simple Curves
Method of Curve Ranging
26
27. 1 2 2
BT = BT =R tan( ∅
)
4) Locate T1 and T2 points by measuring
the tangent lengths backward and
forward along tangent linesAB and BC.
Method of Curve Ranging
Location of Tangent points:
To locate T1 and T2
1)Fixed direction of tangents, produce
them so as to meet at point B.
2)Set up theodolite at point B and
measure T1BT2 (I).
Then deflection angle ∅ = 180o –I
3)Calculatetangents lengths by
Simple Curves
27
28. Procedure:
• After locating the positions of the tangent points T1
and T2 ,their chainages may be determined.
• The chainage of T1 is obtained by subtracting the
tangent length from the known chainage of the
intersection point B. And the chainage of T2 is found
by adding the length of curve to the chainage of T1.
• Then the pegs are fixed at equal intervals on the
curve.
• The interval between pegs is usually 30m or one chain
length.
• The distances along the centre line of the curve are
continuously measured from the point of beginning of
the line up to the end .i.e the pegs along the centre
line of the work should be at equal interval from the
beginning of the line up to the end.
Simple Curves
Method of Curve Ranging
28
29. Procedure:
• There should be no break in the regularity of their spacing
in passing from a tangent to a curve or from a curve to
the tangent.
• For this reason ,the first peg on the curve is fixed at such a
distance from the first tangent point (T1) that its chainage
becomes the whole number of chains i.e the whole
number of peg interval.
• The length of the first sub chord isthus less than the peg
interval and it is called a sub-chord.
• Similarly there will be a sub-chord at the end of the curve.
• Thusa curve usually consists of two sub-chords and a no.
of fullchords.
Simple Curves
Method of Curve Ranging
29
30. Method of Curve Ranging
Simple Curves
3
0
1 2 2 1
∟ BT T = ∟ BT T =
Important relationships for
Circular Curves for Setting Out
• The ∆BT1T2 is an isosceles triangleand
therefore the angle
∅
2
The following definition can be given:
• The tangential angle 𝛼at T1 to any point
X on the curve T1T2 is equal to half the
angle subtended at the centre of curvature
O by the chord from T1 to that point.
• 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.
• I.e. T1OT2 = 2(α + β + 𝛾)and itfollows
that BT1T2= (α + β + 𝛾).
31. Problem 01: Two tangents intersect at chainage of 6 +26.57. it is
proposed to insert a circular curve of radius 1000ft. The deflection
angle being16o38’. Calculate
a) chainageof tangents points
b) Lengths of long chord , Mid ordinate and External distance.
Solution:
1 2 2
Tangent length = BT = BT = R tan(∅
)
BT1 = BT2 = 1000 xtan(16o38`/2)
= 146.18 ft
Length of curve = L = 𝜋R ∅
180o
L = 𝜋x 1000 x 16o38` = 290.31ft
180o
Chainage of point of intersection
minus tangent length
chainage of T1
plus L
Chainage of T2
= 6 + 26.56
= -1 + 46.18
= 4 + 80.39
= + 2 + 90.31
= 7 + 70.70
Simple Curves
31
32. Problem 01:
Solution:
Length of chord = l = 2 Rsin(∅
)
2
l = 2 x 1000 x sin(36o38`/2) = 289.29ft
Mid ordinate = EF = R (1 – cos(∅
)
2
EF= 1000 x ( 1 – cos(36o38`/2)) = 10.52 ft
2
Ex. distance = BF = R ( sec ∅
- 1)
2
BF = 1000 x ((1/cos(∅
)– 1) = 10.63ft
Simple Curves
32
33. Problem 02: Two tangents intersect at chainage of 14 +87.33,
with a deflection angle of 11o21`35``.Degree of curve is 6o.
Calculate chainage of beginning and end of the curve.
Solution:
D = 6o
R = 5729.58 / D ft = 954.93 ft
1 2 2
Tangent length = BT = BT = R tan(∅
)
BT1 = BT2 = 954.93 xtan(11o21`35``/2)
BT1 = BT2 = 94.98 ft
Length of curve = L = 𝜋R ∅
180o
L = 𝜋x 954.93 x 11o21`35`` = 189.33 ft
180o
Chainage of intersection point B
minus tangent length BT1
Chainage of T1
plus L
Chainage T2
= 14 + 87.33
= - 0 + 94.96
= 13 + 92.35
= + 1 + 89.33
= 15 + 81.68
Simple Curves
33
34. Method of Curve Ranging
1) Linear or Chain & Tape Method
• These methods use the chain surveying tools only.
• These methods are used for the short curves which doesn’t
require high degree of accuracy.
• These methods are used for the clear situations on the road
intersections.
a) By offset or ordinate from Long chord
b) By successive bisections ofArcs
c) By offset from theTangents
d) By offset from the Chords produced
Simple Curves
34
35. 1 1
OT = R , T E =l
2
OE = OD – DE = R – Oo
R2 = (l/2)2 + (R – Oo)2
Method of Curve Ranging
1) Linear or Chain & Tape Method
a) By offset or Ordinate from long chord
ED = Oo = offset at mid point ofT1T2
PQ = Ox = offset at distance x from E , so that EP = x
OT1 = OT2 = OD = R = Radius of thecurve
Exact formula for offset at any point on the
chord line may be derived as:
By Pathagoras theorem
∆OT1E, OT1
2 = T1E2 +OE2
Simple Curves
35
36. In ∆OQQ1,
2 2
OQ2= QQ1 + OQ1
OQ1 = OE + EQ1 = OE + Ox
OQ1 = ( R – Oo) + Ox
R2 = x2 + {(R – Oo) + Ox ) }2
Ox = √(R2–x2) – (R - Oo)
Ox = √(R2–x2) – (R – (R- √(R2–(l/2)2))
x
l
2
2 2 2 2
O = √(R –x ) – (√(R – ( ) )) -- 1 exact formula
Method of Curve Ranging
1) Linear or Chain & Tape Method
a) By offset or Ordinate from long chord
DE = Oo = R – √ (R2 – ( l )2) -------A
2
Ineqn A two quantities areusually or
must known.
Simple Curves
36
37. formula or x 2
𝑅
O = 𝑥(
𝐿−
𝑥
)
-------- 2 (Approximate formula)
In eqn 1 the distance x is measured from the mid point of the
long chord where as eqn 2 it is measured from the 1st tangent
point T1.
• This method is used for setting out short curves e.g curves for
street kerbs.
.
Method of Curve Ranging
1) Linear or Chain & Tape Method
a) By offset or Ordinate from long chord
When the radius of the arc is larger as compare to the length of
the chord, the offset may be calculated approximately by
Simple Curves
37
38. Method of Curve Ranging
1) Linear or Chain & Tape Method
a) By offset or Ordinate from long chord
Working Method:
To set out thecurve
• Divided the long chord into even
number of equal parts.
• Set out offsets as calculated from the
equation at each of the points of
division. Thusobtaining the required
points on thecurve.
• Since the curve ssymmetrical along ED,
the offset for the right half of the curve
will be same as those for the left half.
Simple Curves
38
39. Problem 03: calculatethe ordinate at 7.5 m interval for a circular
curve given thatl =60 m and R =180 m, by offset or ordinate
from long chord.
Solution:
Ordinate at middle of the long chord = verse sine = Oo
o
l 6
0
2 2
O = R – √(R – ( ) ) = 180 – √(180 – ( )
2 2 2 2
Oo = 2.52m
Various coo2.52 rdinates may be calculated by formula
x
2 2
O = √(R –x ) – (√
l
2
2 2
(R – ( ) ))
x = distance measured from mid point of long chord.
7.5
2 2
O = √(180 –7.5 ) – (√
6
0
2
2 2
(180 – ( ) )) = 2.34 m
15 2
O = √(1802 –152) – (√(1802– (60
)2))= 1.89m
22.5
2 2
O = √(180 –22.5 ) – (√
6
0
2
2 2
(180 – ( ) )) = 1.14 m
30 2
O = √(1802 –302) – (√(1802– (60
)2))= 0m
X (m) Ox (m)
0 2.52
7.5 2.34
15 1.89
22.5 1.14
30 0
Simple Curves
39
40. ED = y’ = R ( 1 – cos(∅
)
𝟐
• Join T1D and DT2 and bisectthem
at F and G respectively.
• Set out offset HF(y’’) and GK(y’’)
each eqn be
𝟒
FH = GK = y’’= R ( 1 – cos (∅
)
Obtain point H and K on a curve.
By repeating the same process,
obtainas many pints as required
on the curve.
Method of Curve Ranging
1) Linear or Chain & Tape Method
b) By Successive Bisection ofArcs
• Let T1 and T2 be the tangents
points. Join T1 and T2 and bisect it
at E.
• Setout offsets ED(y’), determined
point D on the curve equal to
Simple Curves
40
41. Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from theTangents
In this method the offsets are setout either radially or perpendicular to
the tangents BA ad BC according to as the center O of the curve is
accessible orinaccessible.
a) By Radial Offsets: (O is Accessible)
Working Method:
• Measure a distance x from T1 on back
tangent or from T2 on the forwardtangent.
• Measure a distance Ox along radial line
A1O.
• The resulting point E1 lies on thecurve.
Simple Curves
41
42. a) By Radial
Offsets:
EE1 = Ox , offsets at distance x from T1along
tangentAB.
In ∆OT1E
2 2 2
OT1 + T1E = OE
OT1 = R, T1E = x, OE = R + Ox
R2 + x2 = (R + Ox)2
R + Ox = √(R2 +Ox
2)
Ox =√(R2+ X2) – R ---------A (ExactFormula)
O
R E1
A B
R
Ox
T1 E
x
Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from theTangents
In this method the offsets are setout either radially or perpendicular to
the tangents BA ad BC according to as the center O of the curve is
accessible orinaccessible.
Simple Curves
42
43. x
Expanding √(R2 + x ) , O = R ( 1 +
x2
+
x4
2R2 8R4 + …..) - R
Taking any two terms
x2
R x2
eqn (A) Ox = R ( 1 + 2R2 ) –R
Ox = 2 R2
x 2 R
O = x2
-------B ( Approximateformula)
Used for short curve
O
R E1
A B
T1
R
Ox
E
x
Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from the Tangents
a) By Radial Offsets:
Simple Curves
43
44. Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from theTangents
a) By Radial Offsets:: Example
Simple Curves
44
45. Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from theTangents
In this method the offsets are setout either radially or perpendicular to
the tangents BA ad BC according to as the center O of the curve is
accessible orinaccessible.
b) By Offsets Perpendicular to Tangents
(O is Inaccessible)
Working Method:
• Measure a distance x from T1 on back
tangent or from T2 on the forwardtangent.
• Erect a perpendicular of length Ox.
• The resulting point E1 lies on thecurve.
Simple Curves
45
46. tangentAB
∆OE1E2,
2 2 2
OE1 = OE2 + E1E2
OE = R, E E = x, OE = R – O
1 1 2 2 x
R2 = (R – Ox) 2 + x2
(R – Ox)2 = R2 – x2
Ox = R – √(R2– x2) ------- A (ExactFormula)
Expanding √(R2– x2) and neglecting higher
power
x
O =
x2
2 R ------- Approximate Formula
O
R
E1
A B
T1
R
Ox
E
x
E2
Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from theTangents
b) By Offsets Perpendicular to Tangents
EE1 = Ox = T offset at a distance of x measured along
Simple Curves
46
47. Method of Curve Ranging
1) Linear or Chain & Tape Method
3) By Offsets from theTangents
b) By Offsets Perpendicular to Tangents : Example
Simple Curves
47
48. 1 2R
T E = R2 𝛼
, 𝛼= T1E --------- 1
Method of Curve Ranging
1) Linear or Chain & Tape Method
4) By Offsets from ChordProduced
T1E = T1E1 = bo --- 1st chord of length “b1”
EF, FG, etc = successive chords of length b2 and b3,
each equal to length of unit chord.
BT1E = 𝛼= angle b/w tangents T1B and 1st chord
T1E
E1E = O1 = offset from tangentBT1
E2F = O2 = offset from preceding chord T1E
produced.
Arc T1E = chordT1E
T1E = OT1 2𝛼
Simple Curves
48
49. 1 1
2
O = T E /2R =
b1
2
2
𝑅
----- 2
∟E2EF1 = ∟DET1 (verticallyopposite)
∟DET1 = ∟DT1E since DT1 = DE
::: ∟E2EF1 = ∟DT1E =∟E1T1E
The ∆sE1T1E and E2EF2 being nearly
isosceles may be considered similar
Method of Curve Ranging
1) Linear or Chain & Tape Method
4) By Offsets from ChordProduced
Similarly chord EE1 = arc E1E
1st offset O1 = E1E = T1E x 𝛼
O1 = T1E T1E /2R
Simple Curves
49
50. 𝐸
𝐸
2 𝑇
1
𝐸
i.e
𝐸
2
𝐹
1
= 𝐸
1
𝐸 𝐸
2
𝐹
1
= 𝑂1
2 1
𝐸𝐹 =
𝑏𝑂
2 1 2
𝑏 𝑏1
𝑏
2 𝑏
1
2
= x =
𝑏 𝑏
2 1
𝑏
1 𝑏
1 2
𝑅 2
𝑅
F1F being the offset from the tangent atE
is equal to :
F1F =
EF2
2
𝑅
= 2
b 2
2
𝑅
Method of Curve Ranging
1) Linear or Chain & Tape Method
4) By Offsets from ChordProduced
The ∆sE1T1E and E2EF2 beingnearly
isosceles may be considered similar
Simple Curves
50
51. O2 = E2F = 2 1
+
𝑏𝑏 𝑏 2
2
2
𝑅 2
𝑅
O2 = E2F = 2 1 2
𝑏
(
𝑏+𝑏)
2
𝑅
Similarly 3rd offset
3
O = 3 2 3
b ( b + b )
2
𝑅 2 3 4
, since b =b =b …
O3 = 2
b 2
𝑅
, so O3=O4=O5 except for last offset.
n
O = n n
b ( bn −1………...+b )
2
𝑅
Method of Curve Ranging
1) Linear or Chain & Tape Method
4) By Offsets from ChordProduced
Now 2nd offset O2
O2 = E2F = E2F1 + F1F
Simple Curves
51
52. Method of Curve Ranging
2) Angular or Instrumental Methods
1) Rankine’s Method of
Tangential Angles
2) Two Theodolite Method
1) Rankine’s Method of
Tangential Angles
• Inthis method the curve is
set out tangential angle
often called deflection
angles with a theodolite,
chain ortape.
Simple Curves
52
53. Method of Curve Ranging
2) Angular or Instrumental Methods
1) Rankine’s Method of Tangential Angles
Working method:
1. Fix the theodolite device to be at point
T1and directed at point B.
2. Measure the deflection angles 𝛿1and
the chords C1.
3. Connect the ends of the chords to
draw the curve.
Deflection Angles:
The angles between the tangent and the
ends of the chords from point T1.
Simple Curves
53
54. Method of Curve Ranging
2) Angular or Instrumental Methods
1) Rankine’s Method of
Tangential Angles
• AB = Rear tangent to curve
• D,E,F = Successive point on the
curve
• δ1, δ2, δ3….. The tangentialangles
which each of successive chord
…T1D,DE,EF…… makes with the
respective tangents at T1, D, E.
• ∆1 , ∆2, ∆3…… Totaldeflection
angles
• C1, C2, C3…….. Length of the
chord. T1D, DE, EF.
Simple Curves
54
55. C1= T1D = 1
2 π R δ
180
And
δ1
= C1 ∗ 180
2 π R
degree
δ1 2 π R
= C1 ∗ 180 ∗ 60min
1
δ =
1718.9 ∗C1
𝑅
2
δ =
1718.9 ∗C2
𝑅
n
δ = 1718.9 ∗Cn
𝑅
Method of Curve Ranging
2) Angular or Instrumental Methods
1) Rankine’s Method of Tangential Angles
Arc T1D = chord T1D = C1
So, T1D = R x 2 δ1
Simple Curves
55
56. But B T1 E = B T1 D + D T1 E
∆2 = δ1 + δ2 = ∆1 +δ2
∆3 = δ1 +δ2 + δ3 = ∆2 +δ3
∆4 = δ1 +δ2 + δ3 + δ4 = ∆3 +δ4
∆n = δ1 +δ2 +…………. + δn = ∆n-1 +δn
Check:
1 2 2
Total deflection angle BT T = φ , φ = Deflection angle of the curve
This Method give more accurate result and is used in railway &other
important curve.
Method of Curve Ranging
2) Angular or Instrumental Methods
1) Rankine’s Method of Tangential Angles
Total deflection angle for the
1st chord --T1D = BT1D :. ∆1= δ1
2nd Chord --DE = B T1 E
Simple Curves
56
57. Problem 04: Two tangents intersect at chainage2140 m . ∅=18o24`.
Calculate all the data necessary for setting out the curve, with R =
600 m and Peg interval being 20 m by:
1) By deflection angle ∅
2) offsets from chords.
Solution:
BT1 = BT2 = R tan
∅
2
= 600 tan
o
18 24`
2
BT1 = BT2 = 97.18 m
1
8
0
𝑜
Length of curve = L = 𝜋 R ∅
=
𝜋600 1
8
𝑜
2
4̀
1
8
0
𝑜
L = 192.68m
Chainage of point of intersection
Minus Tangent length
Chainage of T1
Plus L
Chainage of T2
= 2140 m
= - 97.18 m
= 2042.82 m
= + 192.68 m
= 2235.50 m
Simple Curves
57
60. Problem 04:
Solution:
2) By Offsets from Chords
n
O = n
b ( bn −1………...+bn)
2
𝑅
1
O = n
2
=
b 1
7
.
1
82
2
𝑅 2𝑥6
0
0
= 0.25 m
2
O = 2 1 2
b (b + b )
2
𝑅
=
2
0 1
7
.
1
8
+
2
0
2𝑥6
0
0
= 0.62 m
3
O = 3 2 3
b (b + b ) b2
2
2
𝑅 𝑅
= = 0.67 m
O3 = O4 = O5 = O6 = O7 = O8 = O9
O10
= b10 (b9 + b10) = 1
5
.
5
0 2
0+
1
5
.
5
0
= 0.46 m
2
𝑅 2𝑥6
0
0
Simple Curves
60
61. ∆1 is b/w
T1E = ∆2
tangent T1B & T1D => BT1D = T1T2D = ∆1
= T1 T2 E
The total tangential angle or deflection angle
∆1, ∆2 ∆3 … ,As calculate in the 1st method.
Method of Curve Ranging
2) Angular or Instrumental Methods
2) Two Theodolite Method
• This method is used when ground is not favorable foraccurate
chaining i.e rough ground , very steep slope or if the curve onewater
• It is based on the fact that angle between tangent & chord is equalto
the angle which that chord subtends in the oppositesegments.
Simple Curves
61
62. Obstacles in Setting Out Simple Curve
• The following obstacles occurring in common practice will be
considered.
1) When the point of intersection of Tangent lines is inaccessible.
2)When the whole curve cannot be set out from the Tangent
point, Vision being obstructed.
3) When the obstacle to chaining occurs.
Simple Curves
62
63. Obstacles in Setting Out Simple Curve
1) When the point of intersection of Tangent lines is inaccessible
• When intersection point falls in lake, river , wood or any other
construction work
1) To determine the value of ∅
2) To locate the points T1 &T2
Calculate θ1
& θ2
by instrument
(theodolite).
∟ BMN = α = 180 – θ1
∟ BNM = β = 180 – θ2
Deflection angle = ∅= α +β
∅= 360o – (sum of measuredangles)
∅= 360o - (θ1+ θ2)
Simple Curves
63
64. =
𝐵
𝑀 𝑀𝑁
sin β sin (180o − (α+β))
BM = 𝑀𝑁 sin β
sin (180o − (α+β))
BN =
𝑀𝑁 sin α
sin (180o − (α+β))
1 2 2
BT & BT = R * tan (
∅
)
MT1 =BT1 - BM
NT2 = BT2 - BN
Obstacles in Setting Out Simple Curve
1) When the point of intersection of Tangent lines is inaccessible
Calculate the BM & BN from ∆BMN
By sine rule
Simple Curves
64
65. Obstacles in Setting Out Simple Curve
2) When the whole curve cannot be set out from the Tangent
point, Vision being obstructed
• As a rule the whole curve is to be set out from T1 however
obstructions intervening the line of sight i.e Building, cluster of tree,
Plantation etc. In such a case the instrument required to be set up
at one or more point along the curve.
Simple Curves
65
66. Obstacles in Setting Out Simple Curve
3) When the obstacle to chaining occurs
Simple Curves
66
67. Simple Curves
67
Assignment
• Obstacles in Setting Out Simple Curve
(Detail procedure)
Page 130 PartI
I
• Example 1 (approximate method)
• Example 2
• Example 3
Page 135 PartI
I
68. Curves
Compound Curves
• A compound curve consist of 2 arcs of different radii
bending in the same direction and lying on the same side
of their common tangent. Then the center being on the
same side of the curve.
RS = Smaller radius
RL = Larger radius
TS = smaller tangent length = BT1
TL = larger tangent length = BT2
𝛼= deflection angle b/w common
tangent and rear tangent
𝛽= angle of deflection b/w common
tangent and forward tangent
N = point of compound curvature
KM = common tangent
68
69. Curves
Compound Curves
Elements of Compound Curve
∅=𝛼+𝛽
1 S 2
KT = KN =R tan(𝛼
)
2 L 2
MN = MT = R tan (𝛽
)
From ∆BKN, by sine rule
=
𝐵
𝐾 𝑀𝐾
sin 𝛽 sin 𝐼
=
𝐵
𝐾 𝑀𝐾
sin 𝛽 sin(180o – (𝛼+𝛽))
BK = 𝑀𝐾 sin 𝛽
sin(180o – (𝛼+𝛽))
BM = 𝑀𝐾 sin𝜶
sin(180o – (𝛼+𝛽))
69
180o - (𝛼+𝛽)=I
180o – ∅=I
70. Curves
Compound Curves
Elements of Compound Curve
TS = BT1 = BK + KT1
TS = BT1 =
𝑀𝐾 sin 𝛽
sin(180o – (𝛼+𝛽)) S 2
+ R tan (𝛼
)
TL = BT2 =
𝑀𝐾 sin 𝜶
sin(180o – (𝛼+𝛽)) L 2
+ R tan (𝛽
)
70
Of the seven quantities RS, RL, TS, TL, ∅,𝛼,𝛽
four must be known.
71. Curves
1 L
𝛼
2
o
KT = KN = R tan ( ) = 600 tan(40 /2)
KT1 = KN = 218.38m
2 S
𝛽
2
o
MN = MT = R tan( ) = 400 tan(35 /2)
MN = MT2 = 126.12 m
KM = MT2 + MN = 218.38 + 126.12
KM = 344.50 m
Compound Curves
Problem: Two tangents AB & BC are intersected by a line KM. the
angles AKM and KMC are 140o & 145o respectively. The radius of
1st arc is600m and of 2nd arc is400m. Find the chainage of
tangent points and the point of compound curvature given that
the chainage of intersection point is 3415 m.
Solution:
𝛼= 180o – 140o = 40o
𝛽= 180o – 145o =35o
∅=𝛼 + 𝛽= 75o
I = 180o – 75o =105o
71
72. Curves
Solution:
Fin ∆BKM, by sin rule
=
𝐵
𝐾 𝑀𝐾
sin 𝛽 sin(𝐼)
sin(𝐼
) s
i
n1
0
5
𝑜
BK = 𝑀𝐾 sin 𝛽
= 3
4
4
.
5
0𝑥𝑠
𝑖
𝑛
3
5
𝑜
= 204.57 m
sin(𝐼
) s
i
n1
0
5
𝑜
BM = 𝑀𝐾 sin 𝛼
= 3
4
4
.
5
0𝑥𝑠
𝑖
𝑛
4
0
𝑜
= 229.25m
L 1
8
0
𝑜
1
8
0
𝑜
TL = KT1 + BK = 218.38 + 204.57 = 422.95 m
TS = MT2 + BM = 126.12 + 229.25 = 355.37 m
L = 𝜋𝑅
𝐿𝛼
= 𝜋𝑥6
0
0𝑥
40
= 418.88 m
S
L = 𝑆
1
8
0
𝑜
𝜋 𝑅 𝛽 𝜋𝑥4
0
0𝑥3
0
1
8
0
𝑜
= = 244.35 m
Compound Curves
72
73. Curves
Solution:
Chainage of intersection point = 3415 m
Minus TL = - 422.95 m
Chainage of T2 = 2992.05 m
Plus L = + 418.88 m
Chainage of compound curvature (N) = 3410.93 m
Plus LS = + 244.35 m
Chainage of T2 = 3655.25 m
Compound Curves
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