CALCULATION OF VOLUME OF EARTHWORKS IN ROAD CONSTRUCTION
Defination: Earthworks are engineering works created through the moving and/or
processing of massive quantities of soil or unformed rock. Earthwork is done to
reconfigure the topography of a site to achieve the design levels. Earthwork involves
cutting and filling to achieve the required topography.
Cutting: Cutting is the process of excavating earth material from a work location or
borrow pits to achieve the desired topography.
Filling: The filling is the process of moving the excavated material or additional earth
material to a work location to achieve the desired topography.
Cut to spoil: disposing unwanted excess/unsuitable material to spoil banks that meet
environmental qualifications. Cut to spoil = total cut - fill
Applications of Earthwork: Typically, earthwork is done in the following projects:




Road works
Railways
Irrigation projects such as canals and dams
Other common earthwork applications are land grading to reconfigure the
topography of a site, or to stabilize slopes
Data Preparation in Earthwork Projects
The first step involved in doing an earthwork calculation is surveying the site. By
surveying, the site elevations of the existing earth at various points of the work site are
determined. Further, all the calculations are done depending upon these values.
Preparing for Earthwork Calculation: From the point data available, a 3D Surface can be
created with software and then the volume can be found out directly from such
surfaces.
So, we need to do the calculation in steps and first, we need to arrange the available
survey data either in the grid level format or in the contour map depending upon the
requirement.
1. Finding the grid Levels [For section method, division by square method and
average method]
1
Or
1. Generating contour map [For Contour method]
Finding the grid Levels: For the first three methods, namely Section Method, Average
method and Division by Square method, input elevations are converted into grid levels.
It may be highly difficult to take the measurement exactly at the grid levels and hence
levels are captured at various points of the work site as per the project requirement, and
then those values are interpolated at the required points.
To find the grid levels from the available data, interpolation has to be adopted and there
are different methods to find out the interpolation.
1. Weighted average: Measure the distance from two nearest elevations and then
find the weighted average to interpolate the value at the grid point.
2. Triangulation: Triangulation is the process of connecting points to form a very
large triangular network as per the predefined algorithm (Delaunay triangulation
theory being the most popular). Once the triangulation is done, then each
triangle is considered as one surface and elevation at the grid point is calculated
considering the point that, that particular grid point is on the 3-D surface where
its x and y falls.
In the following example Grid point “A” is falling in the triangle marked by “P1”, “P2”,
and “P3”.
2
Grid levels thus found out become the basis of further calculation.
Methods for deriving Volume Calculation
There are 4 different Volume calculation methods that are in Practice for Earthwork
Calculation.
1.
2.
3.
4.
Section Method (Area Can be computed with Different methods)
Average method
Division of Square method
Contour Method
Each of these methods is explained in detail in the following posts.
Area calculation in Section can be done with 3 different methods i.e., Trapezoidal Rule,
Nett area calculation, Simpsons 1/3rd Rule.
Volume calculation can be done using 3 different methods Average end area Method,
Prismoidal Formula, Simpsons 1/3rd Rule.
Area Calculation - Cross Section (CS) Method
For calculating the volume, always two sets of surface data are required.


Initial Level: Original Ground Level (OGL) (survey data)
Final Level: Proposed/Designed Level (estimate) or Formation Level (survey data)
Let us consider the following example where the survey data is converted into grid
levels as shown below and the final level is 20 meters.
3
For each grid line, cross-sections should be drawn, and the area of cutting and area of
filling should be determined for each section. These areas become the basis of volume
calculation using the section method.
Finding the Intersection Point: When a section is drawn using two sets of grid levels, and
if the section lines are intersecting, then we have to find the intersection point for future
calculations.
4
In the above example, we need to find intersection at point B i.e., distance between A
and B. AB= Horizontal distance between A and C * [Level difference at point A / (Level
difference at Point A + Level difference at Point B)] AB = 5 * [1 / (1+2)] = 1.67
Area Calculation: Once the sections are drawn and intersection points are determined,
then we can find the area of each section using any of the following methods:



Trapezoidal rule
Nett Area method
Simpsons 1/3rd rule
Area Calculation - Trapezoidal Rule
In the trapezoidal method, each segment of the section is divided into various
trapezoids and triangles.
Trapezoidal Area A = 1/2 X a X (b1+b2)
5
Triangle area A = a * b/2
Example 1:
6
Intersection Point
In the above example, intersection point is between 351 and 354
Filling Height=0.1 @ distance 351
Cutting depth=0.2 @ distance 354
Length from 351 to 354 =3
Distance from 351 to intersection point = 3*[0.1/ (0.1+0.2)] =1 i.e., intersection point is
at 352
Cutting Area – Sum of Area of Segment 1, 2, and 3
Sl. No.
Easting
Initial
Final
Level
Level
Difference
Calculation
1
345
20.70
20
0.70
2
348
20.50
20
3
351
20.10
20
4
352
20.00
20
0.50 Segment 1: Area of Trapezoid =
½ * (b1 + b2) * a = ½ * (0.70 +
0.50) * 3
0.1 Segment 2: Area of Trapezoid =
½ * (b1 + b2) * a = ½ * (0.50 +
0.10) * 3
0 Segment 3: Area of Triangle = ½
* b * h = ½ * 0.1 * 1
Total
7
Area (Sq.
meters)
1.80
0.90
0.05
2.75
Filling Area – Sum of Area of Segment 4, 5, 6, and 7
Sl. No.
Easting
Initial
Final
Level
Level
Difference
Calculation
Area
(Sq. meters)
1
352
20.00
20
0.00
2
354
19.80
20
3
357
19.40
20
4
360
19.10
20
5
363
19.00
20
0.20 Segment 4: Area of Triangle = ½
* b * h = ½ * 0.2 * 2
0.60 Segment 5: Area of Trapezoid =
½ * (b1 + b2) * a = ½ * (0.20 +
0.60) * 3
0.90 Segment 6: Area of Trapezoid =
½ * (b1 + b2) * a = ½ * (0.60 +
0.90) * 3
1.00 Segment 7: Area of Trapezoid =
½ * (b1 + b2) * a = ½ * (0.90 +
1.00) * 3
Total
0.20
1.20
2.25
2.85
6.50
Area Calculation - Nett Area Method
Nett Area is the Cutting/Filling area that is calculated by finding out the difference
between the total area from zero base line defined by the Initial Level and the total area
from zero base line defined by the final level. While calculating the area using this
method, the area is once again calculated by dividing the total area into smaller
trapezoids.
Example 2:
8
9
Cutting Calculation
Sl.
No.
Easting
Initial
Level
Consider
Level
Calculation
1
345
20.70
20.70
2
348
20.50
20.50
Segment 1: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.70 + 20.50) * 3
3
351
20.10
20.10
Segment 2: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.50 + 20.10) * 3
4
352
20.00
20.00
Segment 3: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.10 + 20.00) * 1
5
354
19.80
20.00
Segment 4: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.00 + 20.00) * 2
6
357
19.40
20.00
Segment 5: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.00 + 20.00) * 3
7
360
19.10
20.00
Segment 6: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.00 + 20.00) * 3
8
363
19.00
20.00
Segment 7: Area of Trapezoid = ½ *
(b1 + b2) * a = ½ * (20.00 + 20.00) * 3
10
Area(Sq. meters
Area of final level = 6 trapezoids = ½*(20.00+20.00)*3 = 6 * 60 = 360 Cutting Area =
362.75 – 360 = 2.75
Filling Calculation
Sl.
No.
Easting
Initial
Level
Consider
Level
Calculation
1
345
20.70
20.00
2
348
20.50
20.00
Segment 1: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (20.00 + 20.00) * 3
3
351
20.10
20.00
Segment 2: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (20.00 + 20.00) * 3
4
352
20.00
20.00
Segment 3: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (20.00 + 20.00) * 1
5
354
19.80
19.80
Segment 4: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (20.00 + 19.80) * 2
6
357
19.40
19.40
Segment 5: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (19.80 + 19.40) * 3
7
360
19.10
19.10
Segment 6: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (19.40 + 19.10) * 3
8
363
19.00
19.00
Segment 7: Area of Trapezoid = ½ * (b1 + b2)
* a = ½ * (19.10 + 19.00) * 3
Area of final level = 6 trapezoids = ½*(20.00+20.00)*3 = 6 * 60 = 360 Filling Area =
360.00 – 353.50 = 6.50
11
Area(Sq. m
Area Calculation - Simpsons One Third Rule
Simpson’s 1/3rd rule is one of the most popular methods of finding the area for a given
set of points by the method of numerical integration. The basic idea is to divide the Xaxis into equally spaced divisions as shown and to complete the top of these strips of an
area in such a way that we can calculate the area by adding up these strips
Simpson's rule is based on a parabolic model of the function to be integrated (that is
instead of connecting 2 adjacent points merely by a straight line, a parabola is chosen
such that the curve formed by joining these points is extremely smooth and thus helps
in calculating the area).
Where the sum of odd and even terms do not include the first and the last terms
Important points to be considered while applying Simpson’s Rule are:
1. The number of intervals must be an even number.
2. Minimum of 3 points are required
3. Intervals are expected to be equal
12
Example 3:
Cutting Area
Sl. No. Easting
3
Initial
Final
Cutting
Calculation - Area (Sq. meters)
Level Level
Depth
345
20.70
20
0.70 Simpsons 1/3rd Rule= h/3(First Value + last
value + 4 * (Sum of odd values) + 2 *
348
20.50
20
0.50 (Sum of even values) H = 21 / 3 = 3 = 3/3
351
20.10
20
0.1 (0.70 + 0 + 4 * 0.5 + 2 * 0.1)
4
354
19.80
20
0
5
357
19.40
20
0
6
360
19.10
20
0
7
363
19.00
20
0
1
2
Total = 2.9
Filling Area
Sl. No. Easting
4
Initial
Final
Cutting
Calculation - Area (Sq. meters)
Level Level
Depth
345
20.70
20
0 Simpsons 1/3rd Rule= h/3(First Value +
last value + 4 * (Sum of odd values) + 2
348
20.50
20
0 * (Sum of even values)
351
20.10
20
0 H = 21 / 3 = 3
= 3/3 (0 + 1.00 + 4 * 1.1 + 2 * 0.4)
354
19.80
20
0.20
5
357
19.40
20
0.60
6
360
19.10
20
0.90
7
363
19.00
20
1.00
1
2
3
Total = 6.6
13
Summary table
Sl.No
Section at
Trapezoidal Method
Cutting
2.75
Filling
Net Area
Cutting
6.50
Filling
6.50
Cutting
2.9
Filling
1
729
2
732
15.75
15.75
15.9
3
735
78.90
78.90
79
4
738
126.90
126.90
125.8
5
741
72.00
72.00
72.4
6
744
29.55
29.55
30.1
14
2.75
Simpsons Rule
6.6