Finding the areas of cross-sections is the first step in obtaining the volume of earthwork to be handled in route alignment projects (road or railway), or reservoir construction, for example.
In order to illustrate more clearly what the above statement means, let us consider a road construction project. In the first instance an accurate plan is produced on which to design the proposed route. The centreline of the route, defined in terms of rectangular coordinates at 10- to 30-m intervals, is then set out in the field. Ground levels are obtained along the centre-line and also at right angles to the line .
The levels at right angles to the centre-line depict the ground profile, as shown in Figure , and if the design template, depicting the formation level, road width, camber, side slopes, etc., is added, then a cross-section is produced whose area can be obtained by planimeter or computation. The shape of the cross-section is defined in terms of vertical heights (levels) at horizontal distances each side of the centreline; thus no matter how complex the shape, these parameters can be treated as rectangular coordinates and the area computed using the rules.
The areas may now be used in various rules (see later) to produce an estimate of the volumes. Levels along, and normal to, the centre-line may be obtained by standard levelling procedures, with a total station, or by aerial photogrammetry. The whole computational procedure, including the road design and optimization, would then be carried out on the computer to produce volumes of cut and fill, accumulated volumes, areas and volumes of top-soil strip,
Cross-sectional area of a cutting
side widths, etc. Where plotting facilities are available the program would no doubt include routines to plot the cross-sections for visual inspection. Cross-sections may be approximated to the ground profile to afford easy computation. The particular cross-section adopted would be dependent upon the general shape of the ground.
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 .
Given: height x and grades AB and CB in triangle ABC.
Required: to find distance y1.
Add the two grades, using their absolute values, invert them and multiply by x.
I.e. (1/2 1/5)−1x = 10x/7 = y1 or (0.5 − 0.2)−1x = 10x/7 = y1
(a) Cutting, (b) embankment, (c) cutting and (d) hillside section
Rate of approach
Similarly, to find distance y2 in triangle ADC, subtract the two grades, invert them and multiply by x.
E.g. (1/2 − 1/5)−1x = 10x/3 = y2 or (0.5 − 0.2)−1x = 10x/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).
N.B. Height x must be vertical relative to the grades