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