Areas and VolumesIntroduction

Faculty of Applied Engineering and Urban Planning

Civil Engineering Department

2nd Semester 2008/2009

Surv

eyin

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Areas and Volumes

Planimeters

Digital Planimeter Optical Polar Planimeter

Planimeters

Note : the accuracy of the results obtained from using planimeter in the measurement of areas depends mainly on the original accuracy drawn map, as well as on the experience of the operator when tracing boundary of the figure.

Questions?!

Areas

1. Regular Figures

• Mathematical Formulae

• Method of Coordinates

2. Irregular Figures

• Graphical Method

• Trapezoidal Rule

• Simpson’s One-Third Rule

Mathematical Formulae

sin(C)ba2

1or bh

2

1 2a ba

bah2

1

n

180cotan

4

1 2 2dπ4

1

Mathematical Formulae

21

22 rrπ

baπ bh3

2

2rπ360

1

sin

180

πr

2

1 2

Mathematical Formulae

15 m20 m

Questions?!

Areas by Method of Coordinates

Areas by Method of Coordinates

Areas by Method of Coordinates

Areas by Method of Coordinates

Areas by Method of Coordinates

Area = 0.5(| 136840.01 – (-84890.94) |)= 110865.48 ft2

Questions?!

Areas of Irregular Figures

1. Graphical Method

2. Trapezoidal Rule

3. Simspon’s One-Third Rule

Content

Graphical Method

Trapezoidal Rule

Trapezoidal Rule

Trapezoidal Rule

Example

Simspon’s One-Third Rule

ONLY used with Odd number of offsets(i.e. Even number of Intercepts)

Intercept

Off

set

Simspon’s One-Third Rule

Example

Trapezoidal Rule

Area Calculation

Simspon’s One-Third Rule

• Volumes by:

1. Average-End-Area Method

2. Prismoidal Method

3. Contour Maps

• Volume from Spot Levels

Content

Cut and Fill

Average-End-Area Method

Average-End-Area Method

Prismoidal Method

Calculating Volumes from Contour Map

Calculating Volumes from Contour Map

Calculating Volumes from Contour Map

Questions?!

Volume from Spot Levels

Volume from Spot Levels

Volume from Spot Levels

Mass Haul Diagram

A mass haul diagram is of great value both in planning and construction.

The diagram is plotted after the earthwork quantities have been computed, the ordinates showing aggregate volumes in cubic metres while the horizontal base line, plotted to the same scale as the profile, gives the points at which these volumes obtain.

Mass Haul Diagram

Most materials are found to increase in volume after

excavation ('bulking'), but after being re-compacted by

roller or other means, soils in particular might be found

to occupy less volume than originally, i.e. a 'shrinkage'

has taken place when compacted in the in situ volume.

Definitions

(1) Haul refers to the volume of material multiplied by the distance moved, expressed in ‘station meters'.

(2) Station meter (stn m) is 1 m3 of material moved 100 m, Thus, 20 m3 moved 1500 m is a haul of 20 × 1500/100 = 300 stn m.

(3) Waste is the material excavated from cuts but not used for embankment fills.

(4) Borrow is the material needed for the formation of embankments, secured not from roadway excavation but from elsewhere. It is said to be obtained from a ‘borrow pit’.

(5) Limit of economical haul is the maximum haul distance. When this limit is reached it is more economical to waste and borrow material.

Bulking and shrinkage

Excavation of material causes it to loosen, and thus its excavated volume will be greater than its in situ volume. However, when filled and compacted, it may occupy a less volume than when originally in situ.

For example, light sandy soil is less by about 11% after filling, whilst large rocks may bulk by up to 40%.

To allow for this, a correction factor is generally applied to the cut or fill volumes.

Construction of the MHD

A MHD is a continuous curve, whose vertical ordinates, plotted on the same distance scale as the longitudinal section, represent the algebraic sum of the corrected volumes (cut +, fill −).

Mass Haul Diagram

Mass Haul Diagram