The purpose of the land grading calculations is to determine the slope of the land and consequently the final levels after the land grading operations and also to calculate the amount of earthwork involved. The final grade given to the land surface is dependent upon the ultimate purpose for which the land is to be used.
In irrigated agriculture, particularly when surface irrigation methods are to be adopted the general land slopes recommended are given in Table 13.1. Surface irrigation methods can be conveniently designed if the land slopes are within these ranges for the different types of soils.
There are four basic methods available for land grading calculations.
These are:
1. Plane method,
2. Profile method,
3. Plan inspection method, and
4. Contour adjustment method.
1. Plane Method:
In this method, the surface elevations are calculated such that the resulting land surface has a uniform downward slopes and a uniform cross slope. The centroid of the area is found and a plane is passed through this point at an elevation equal to the average elevation of the field.
Because of this condition, irrespective of the slope of the plane, the volume of excavation equals the volume of fill. We need more excavation than fill. The plane is, therefore, lowered sufficiently in order to provide for a proper balance.
In deciding the equation of the plane, two approaches are followed. The first one is to decide the slope to which the land is to be graded and make the subsequent calculations. In the other approach the slope of the land is obtained based upon least square method such that for the given conditions the amount of excavation is a minimum.
Calculations using both these approaches are illustrated in the following example:
Example 1:
Calculate the ground elevations if the area shown in the following figure is to be made perfectly horizontal and for providing slopes in both directions. Assume that the levels are in meters and grid size is 30 m x 30 m.
Solution:
The minimum level in the whole area is 3.8. If the area is cut at this level, the amount of cut at each station will be as follows –
Now the elevations of all other grid points on line ‘O’ can be calculated by adding or subtracting 0.09 (=(0.3 x 30)/100) for each grid point to the right or left of this point respectively.
The elevations of line N can then be obtained by adding 0.15 to the proposed elevation on line ‘O’. The cuts and fills and proposed elevations can be calculated.
Any plane passing through the centroid will give equal amounts of cut and fill. In order to have cut more than the fill, the centroid is lowered from the original level and the grid elevations are calculated. This is a trial and error procedure.
In the above example, the centroid is lowered by 0.03 and the elevations are calculated as follows –
Method of Least Square:
This statistical procedure can be applied to find out the best plane of fit for a given set of readings. The slopes obtained with this procedure will give the least possible cut and fill and may or may not give the desired slopes from irrigation requirements.
The general equation of a plane surface is –
For areas other than rectangular, the centroid is calculated, by dividing the area into rectangles and triangles. The distance to the centroid of the field from any reference line is found by summing up the products obtained by multiplying the area of each figure times the distance from the line of reference to its centroid divided by the area of the entire field.
When the area is divided into grids, the centroid can also be calculated by using the grid numbers on each side. The plane of the best fit is found calculating Sx and Sy by solving the equations 13.7 and 13.8 simultaneously. It is convenient to consider a rectangular area within the boundaries and extending the slopes of the plane to the remaining areas.
Land Levelling Index:
Land grading could be evaluated using levelling index.
The levelling index (L.I.) is defined as –
Maximum uniformity will be achieved at an L.I. value of zero and any increase in L.I. value means a deterioration in levelling quality.
Computer programs for land grading calculations can easily be developed for use on personal computers.
Example 2:
For the field indicated in Fig. 13.2 calculate the land slopes using the least squares method.
Solution:
Referring to Fig. 13.2
Positive values of Sx and Sy indicate that the plane of best fit has a lower elevation towards the origin.
The value of ‘a’ the elevation at the origin is given by –
We can use this equation for finding the elevation at each of the grid points. The calculated cuts/fills are given in Fig. 13.4.
2. Profile Method:
In this method, using the grid point elevation, ground profiles are plotted along the grid lines. Final land profiles are chosen by trial and error such that the cuts and fills balance. The method is suited to levelling design for very flat lands or land with undulating topography on which it is desired to develop a surface.
The profiles are commonly plotted in one direction. The datum lines are kept at proposed profiles. Calculations of cuts and fills from these profiles indicate a satisfactory cut- fill ratio. When the slopes are to be given in both direction two way profiles are useful.
3. Plan Inspection Method:
This method is again a trial and error method. From a study of the initial levels of the land a suitable down slope and cross slope are assumed. The cuts and fills are calculated. The assumed slopes are altered until a satisfactory balance of the cut and fill volumes are obtained.
4. Contour Adjustment Method:
In this method, using the levels, a contour map of the area is drawn. The ground surface expected after grading is shown on the same map with the help of new contour lines (Fig. 13.6). The new contour lines are so drawn such that a uniform desired slope is obtained. The cuts and fills are estimated at the grid points by interpolating between contour lines and by taking the difference in elevation between the original and the new surface.
Calculation of Earthwork:
After the cut and fill at each grid point are calculated, using this information the total earthwork required in the entire field needs to be calculated.
While there are several methods for earthwork calculations, the commonly used methods are –
(1) Summation method, and
(2) Four point method.
(1) Summation Method:
This method assumes that a given cut or fill at a grid point represents an area midway to the next grid point. The sum of all the cuts and fills are determined and multiplied by the area which a grid point represents in order to obtain the volume of earthwork.
Thus in Example 1-
This method is not accurate and is used only to obtain quick estimates of the earthwork involved.
(2) Four-Point Method:
This method suggested by the US SCS (1959) is most commonly used and is sufficiently accurate for land grading calculations. The volume of cut Vc is given as –
For computing the volume of fill Vf, (ΣC)2 in the numerator of the above equation is replaced by (ΣF)2. If the earth work calculations are for a part of the grid, then the full grid volume is reduced in proportion to the reduced area. As for example – if Vc is 1000 cu.m on a 30 m x 30 m grid, the volume for a 15 m x 30 m area would be 500 cu.m.
With the above formula the total volume of cut or fill in a single grid square remains for the same each set of cut and fill depths regardless of whether the total depth occurs at 1,2, 3, or 4 corners. Thus the corresponding volumes for each combination of cut and fill that might occur in a grid can be tabulated for ready reference.
For Example 1, the earthwork calculations using the four-point method (Eq. 13.11) are as follows. Cuts and fills are as in Fig. 13.3.
Earthwork Balance:
In land grading operations it is necessary to provide sufficient excavation to construct the designed fill. A certain volume of earth loosened by excavation increases in volume referred to as ‘swell’.
The same earth when placed in fill and compacted decreases in volume. The decrease from the original volume is referred to as shrinkage. The reasons for the shrinkage are the high organic content and low bulk density of the top soils which are usually excavated and also the compaction due to earth moving equipment used in land grading operations.
Because of the shrinkage, the cut fill ratio should be greater than 1. The ratio is 1.3 to 1.6 for most of the soils but may be as high as 2.0 for soils with high shrinkage values. Cut fill ratios recommended for various soils are given in Table 13.2.
