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  • INTRODUCTION OF SURVEYING
    • INTRODUCTION
    • REFERENCE ELLIPSOID
    • BASIC MEASUREMENTS
    • The geoid
    • PROTECTION AND REFERENCING
    • CONTROL NETWORKS
    • The ellipsoid
    • BASIC SETTING-OUT PROCEDURES USING COORDINATES
    • LOCATING POSITION
    • COORDINATE SYSTEMS
    • USE OF GRIDS
    • PLOTTING DETAIL
    • Geodetic coordinates
    • SETTING OUT BUILDINGS
    • Computer-aided design (CAD)
    • Cartesian coordinates
    • Error and uncertainty
    • Plane rectangular coordinates
    • SIGNIFICANT FIGURES
    • Height
    • ERRORS IN MEASUREMENT
    • WEIGHT MATRIX
    • LOCAL SYSTEMS
    • Probability
    • ERROR ANALYSIS
    • Deviation of the vertical
    • INDICES OF PRECISION
    • VARIANCE-COVARIANCE MATRIX OF THE PARAMETERS
    • COMPUTATION ON THE ELLIPSOID
    • COMBINATION OF ERRORS
    • Uncertainty of addition or subtraction
    • Eigenvalues, eigenvectors and error ellipses
    • BLUNDER DETECTION
    • RELIABILITY OF THE OBSERVATIONS
    • PRACTICAL CONSIDERATIONS
    • ESTIMATION IN THREE DIMENSIONS

  • LEVELLING
    • LEVELLING
    • OPTICAL METHODS
    • CURVATURE AND REFRACTION
    • MECHANICAL METHODS
    • EQUIPMENT
    • Weiss quadrilateral
    • INSTRUMENT ADJUSTMENT
    • PARAMETER VECTOR
    • Single wires in two shafts
    • Automatic level
    • DESIGN MATRIX AND OBSERVATIONS VECTOR
    • GYRO-THEODOLITE
    • PRINCIPLE OF LEVELLING
    • Plan network
    • SOURCES OF ERROR
    • Distance equation
    • LEVELLING APPLICATIONS
    • Direction & Angle equation
    • Direct and Indirect contouring
    • Controlling earthworks
    • RECIPROCAL LEVELLING
    • PRECISE LEVELLING
    • Parallel plate micrometer
    • ERROR ELLIPSES
    • Field procedure
    • Booking and computing
    • DIGITAL LEVELLING
    • Factors affecting the measuring procedure
    • TRIGONOMETRICAL LEVELLING

  • CONTOURING
    • TAPES
    • Introduction of Satellite positioning
    • FIELD WORK
    • GPS SEGMENTS
    • Measuring in catenary
    • GPS
    • DISTANCE ADJUSTMENT
    • SATELLITE ORBITS
    • Sag
    • BASIC PRINCIPLE OF POSITION FIXING
    • ERRORS IN TAPING
    • DIFFERENCING DATA
    • Tension,Sag and Slope
    • GPS OBSERVING METHODS
    • ELECTROMAGNETIC DISTANCE MEASUREMENT (EDM)
    • Kinematic positioning
    • ERROR SOURCES
    • Global datums
    • GPS SYSTEM FUTURE
    • DATUM TRANSFORMATIONS
    • GALILEO
    • ORTHOMORPHIC PROJECTION
    • APPLICATIONS
    • ORDNANCE SURVEY NATIONAL GRID
    • (t – T) correction
    • PRACTICAL APPLICATIONS
    • Contouring
    • HEIGHTING WITH GPS

  • Theodolite Surveying
    • PLANE RECTANGULAR COORDINATES
    • PRINCIPLE OF LEAST SQUARES
    • PRINCIPLE OF LEAST SQUARES
    • TRAVERSING
    • LINEARIZATION
    • LEAST SQUARES APPLIED TO SURVEYING
    • Reconnaissance
    • NETWORKS
    • LINEARIZATION
    • Sources of error
    • Traverse computation
    • TRIANGULATION
    • Resection and intersection
    • Resection
    • NETWORKS
    • INSTRUMENT ADJUSTMENT
    • FIELD PROCEDURE
    • Setting up using the optical plumb-bob
    • MEASURING ANGLES
    • Measurement by directions
    • SOURCES OF ERROR

  • Simple Curves
    • CIRCULAR CURVES
    • Plotted areas
    • RESPONSIBILITY ON SITE
    • PHOTOGRAMMETRY
    • SETTING OUT CURVES
    • PARTITION OF LAND
    • COMPOUND AND REVERSE CURVES
    • CROSS-SECTIONS
    • SHORT AND/OR SMALL-RADIUS CURVES
    • VOLUMES
    • TRANSITION CURVES
    • Effect of curvature on volumes
    • Centrifugal ratio
    • MASS-HAUL DIAGRAMS
    • CONTROLLING VERTICALITY
    • The equation of motion
    • Coefficient of friction
    • CONTROLLING GRADING EXCAVATION
    • Sources of error
    • SETTING-OUT DATA
    • ROUTE LOCATION
    • LINE AND LEVEL
    • Highway transition curve tables (metric)
    • THE OSCULATING CIRCLE
    • Transitions joining arcs of different radii (compound curves)
    • Coordinates on the transition spiral
    • VERTICAL CURVES
    • Vertical curve design
    • Sight distances
    • Permissible approximations in vertical curve computation

Branch : Civil Engineering
Subject : Surveying-I
Unit : Simple Curves

Plotted areas


Introduction:

Estimation of areas and volumes is basic to most engineering schemes such as route alignment, reservoirs, tunnels, etc. The excavation and hauling of material on such schemes is the most significant and costly aspect of the work, on which profit or loss may depend. Areas may be required in connection with the purchase or sale of land, with the subdivision of land or with the grading of land. Earthwork volumes must be estimated to enable route alignment to be located at such lines and levels that cut and fill are balanced as far as practicable; and to enable contract estimates of time and cost to be made for proposed work; and to form the basis of payment for work carried out. The tedium of earthwork computation has now been removed by the use of computers. Digital ground models (DGM), in which the ground surface is defined mathematically in terms of x, y and z coordinates, are stored in the computer memory.

 

 

This data bank may now be used with several alternative design schemes to produce the optimum route in both the horizontal and vertical planes. In addition to all the setting-out data, cross-sections are produced, earthwork volumes supplied and mass-haul diagrams drawn. Quantities may be readily produced for tender calculations and project planning. The data banks may be updated with new survey information at any time and further facilitate the planning and management not only of the existing project but of future ones. To understand how software does each stage of the earthwork computations, one requires a knowledge of the fundamentals of areas and volumes, not only to produce the software necessary, but to understand the input data required and to be able to interpret and utilize the resultant output properly.

AREAS:
The computation of areas may be based on data scaled from plans or drawings, or data gained directly from survey field data.

Plotted areas:


(1) It may be possible to sub-divide the plotted area into a series of triangles, measures the sides a, b, c, and compute the areas using:


Area = [s(s − a)(s − b)(s − c)]1/2


where s = (a b c)/2


The accuracy achieved will be dependent upon the scale error of the plan and the accuracy to which the sides are measured.


(2) Where the area is irregular, a sheet of gridded tracing material may be superimposed over it and the number of squares counted. Knowing the scale of the plan and the size of the squares, an estimate of the area can be obtained. Portions of squares cut by the irregular boundaries can be estimated.

Areas of give and take

3) Alternatively, irregular boundaries may be reduced to straight lines using give-and-take lines, in which the areas ‘taken’ from the total area balance out with extra areas ‘given’.


(4) If the area is a polygon with straight sides it may be reduced to a triangle of equal area. Consider the polygon ABCDE .


Take AE as the base and extend it as shown, Join CE and from D draw a line parallel to CE on to the base at F. Similarly, join CA and draw a line parallel from B on to the base at G. Triangle GCF has the same area as the polygon ABCDE.


(5) The most common mechanical method of measuring areas from paper plans is to use an instrument called a planimeter . This comprises two arms, JF and JP, which are free to move relative to each other through the hinged point at J but fixed to the plan by a weighted needle at F  is the graduated measuring wheel and P the tracing point.

 

 

As P is moved around the perimeter of the area, the measuring wheel partly rotates and partly slides over the plan with the varying movement of the tracing point . The measuring wheel is graduated around the circumference into 10 divisions, each of which is further sub-divided by 10 into one-hundredths of a revolution, whilst a vernier enables readings to one thousandths of a revolution. The wheel is connected to a dial that records the numbered revolutions up to 10. On a fixed-arm planimeter one revolution of the wheel may represent 100 mm2 on a 1:1 basis; thus, knowing the number of revolutions and the scale of the plan, the area is easily computed. In the case of a sliding-arm planimeter the sliding arm JP may be set to the scale of the plan, thereby facilitating more direct measurement of the area.

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