AREAS and VOLUMES

doc. Ing. Hana Staňková, Ph.D

EARTHWORKS

Estimation of areas and volumes is basic to most engineering schemes such asroute alignment, reservoirs, tunnels, etc

Earthwork volumes must be estimated to: 1- To enable route alignment to be located 2- To enable contract estimates of time and cost to be made for proposed work 3- To form the basis of payment for work carried out

Digital ground models (DGM)), in which the ground surface is definedmathematically in terms of x, y and z coordinates, are now stored in thecomputer memory

The data banks may be updated with new survey information at any time

We need knowledge of the fundamentals of areas and volumes, not only toproduce the software necessary, but to understand the input data required

DEM

Earth's surface is a mathematically irrecoverable area, it needs to begeneralized (simplified). DEM has the task of describing this surface in digitalform and allowing for further operations above the result.

There are points in space and other data (eg Edge definition, etc.). The surfaceis mostly smooth (common smooth surfaces, in simplification), but also sharp(breaks, cuts, edges, artificial terrain shapes).

Types of DEM: Digital Terrain Model (Digital Representation of Earth Surface Relief in

Computer Memory, composed of data and interpolation algorithm, which allows,among other things, to derive the height of intermediate points) (Terminologickýslovník ČÚZK).

Digital Surface Model (A special case of a digital relief model, usuallyconstructed using automated means (eg image correlation in photogrammetry),to display the terrain surface and the top surfaces of all objects on it (roofs, treecrowns, etc.) (TS ČÚZK)) .

DEM types by surface type: Polyhedral terrain model (irregular triangular network, TIN model). Terrain raster model (square mesh, collapsed quadrilaterals). Terrain plate model (more complex, higher order approximation).

PLOTTED AND COMPUTED AREAS The computation of areas may be based on data scaled from plans or

drawings, or direct from the 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, Area = [ s( s– a)( s– b)( s– c)]1/2 where s= (a+b+c)/2

2) Where the area is irregular, a sheet of gridded tracing material may besuperimposed over it and the number of squares counted to estimate theirregular boundaries

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 withextra areas ‘given’

4) If the area is a polygon with straight sides it may be reduced to a triangle ofequal area

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

5)The most common method of measuring areas from plans is to use an instrumentcalled a planimeter.

This comprises two arms, which are free to move relative to each otherthrough the hinged point but fixed to the plan by a weighted needle. There is agraduated measuring wheel and P the tracing point. As P is moved around theperimeter of the area, the measuring wheel partly rotates and partly slides over theplan with the varying movement of the tracing point

The measuring wheel is graduated circumferentially into 10 divisions, each ofwhich is further sub-divided by 10 into one-hundredths of a revolution, whilst avernier enables readings to one thousandths of a revolution. The wheel isconnected to a dial which records the numbered revolutions up to 10

COMPUTED AREAS Using appropriate field data it may be possible to define the area by its

rectangular coordinates There are three different methods to calculate the area Firstly: The area enclosed by the traverse ABCDA in the image can be found

by taking the area of the rectangle a’cDd and subtracting the surroundingtriangles, etc., as follows:

The following rule may be used when the total coordinates only are given

Multiply the algebraic sum of the northing of each station and the one followingby the algebraic difference of the easting of each station and the one following

The value of 22000 m2 is more correct considering the number of significantfigures involved in the computations

This latter rule is the one most commonly used and is easily remembered if written as follow

A= 0.5[ N A( E B– E D) + N B( E C – E A) + N C ( E D– E B) + N D( E A–E C )]

= 0.5[0 + 71(163) + 148(263 – 71) + –25(0 – 163)]

= 0.5[11 573 + 28 416 + 4075] = 22 032 m2

Surveyors are often required to compute volumes of earthwork

either in cut or in fill when planning a highway system. There are

basically 3 methods for this:

Cross section method Unit area or borrow pit method Contour area method

VOLUMES

1. Cross Section Method

A cross section is a profile of the

ground at right angles to the longitudinal

line, serving mainly to allow the

calculation of the volumes of

earthworks. This method employed for

computation the volume of earthwork to

be handled in highway, railway and

reservoir construction projects. Cross

sections must be taken at regular

intervals, then the volumes of cut or fill

are obtained.

CROSS-SECTION Finding the areas of cross-sections is the first step in obtaining the volume of

earthwork to be handled in route alignment projects

The centre-line of the route, defined in terms of rectangular coordinates at 10to30 m intervals

Ground levels are obtained along the centre-line and also at right-angles to theline. The levels at right-angles to the centre-line depict the ground profile

The shape of the cross-section is defined in terms of vertical heights (levels) athorizontal distances each side of the centre-line; thus no matter howcomplexthe shape, these parameters can be treated as rectangular coordinates andthe area computed

The whole computational procedure, including the road design andoptimization, would then be carried out on the computer to produce volumes

Where there are no computer facilities the cross-sections may beapproximated to the ground profile to afford easy computation. The particularcross-section adopted would be dependent upon the general shape of theground

Whilst equations are available for computing the areas and side widths they tend to be over-complicated and the following method using ‘rate of approach’ is recommended

i.e. (1/5 + 1/2) –1 x= 10 x/7 = y1

Similarly, to find distance y2in triangle ADC,subtract the two grades, invert them and multiply by x:

e.g.(1/5 – 1/2) –1 x= 10 x/3 = y2

The rule, therefore is:

1. When the two grades are running in opposing directions (as in ABC ), add (signs opposite + –)

2. When the two grades are running in the same direction (as in ADC ), subtract (signs same)

Volume Through Transition

In the case of railroad or

highway construction there is

both cut and fill. However, it is

transitional. For example, there

may be a number of full fill-

sections and a number of full

cut-sections. In between there

may be a section of partial fill

and cut. This is shown in the

figure.

Volume from Spot Levels

This method is useful in thedetermination of volumes of largeopen excavations for tanks,basements, borrow pits, and forground levelling operations such asplaying fields and building sites.Having located the outline of the sites,the area into squares, rectangles ortriangles. Marking the corner pointsand then determine the reduced level.By subtracting from the observedlevels the corresponding formationlevels, a series of heights can befound.

Volume by Simpson's Cubature Formula

Volume can also be computed by a non-linear profile formula

known as Simpson's cubature formula which is applicable only

when the grid has an even number of intervals in each direction.

METHOD OF COMPUTATION OF VOLUMES• The importance of volume assessment has already been outlined

Many volumes encountered incivil engineering appear, at first glance, to be rather complex in shape

PRISM• The two ends of the prism are equal and parallel, the resulting sides thus

being parallelograms Vol = AL

WEDGE• Volume of wedge = L/6(sum of parallel edges × vertical height of base)

=L/6[(a+b+c) × h]• When a = b = c V = AL/2

Pyramid

• Volume of pyramid = AL/3

• V = L/6( A1+ 4 Am + A2)

• where A1 and A2 are the end areas and Am is the area of the section situated mid-way between the end areas

Equations can all be expressed as the common equation:

PRISM

• In this case A1= Am= A2

• V = L/6( A+ 4 A+ A) =( Lx6A) / 6 = AL

WEDGE

• In this case Am is the mean of A1and A2, but A2= 0. Thus Am = A/2

PYRAMID

• In this case A = Am/4 and A2= 0