Earthwork Area Calculation – Section – Simpsons 1/3rd 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 X-axis into equally spaced divisions as shown and to complete the top of these strips of 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 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:
- Number of intervals must be an even number.
- Minimum of 3 points are required
- Intervals are expected to be equal
Example 3:
Cutting Area
| Sl. No. | Easting | Initial Level | Final Level | Cutting Depth | Calculation – Area (Sq. meters) |
| 1 |
345 |
20.70 |
20 |
0.70 |
Simpsons 1/3rd Rule= h/3(First Value + last value + 4 * (Sum of odd values) + 2 * (Sum of even values)
H = 21 / 3 = 3 = 3/3 (0.70 + 0 + 4 * 0.5 + 2 * 0.1) |
| 2 |
348 |
20.50 |
20 |
0.50 |
|
| 3 |
351 |
20.10 |
20 |
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 |
|
| Total = 2.9 |
Filling Area
| Sl. No. | Easting | Initial Level | Final Level | Cutting Depth | Calculation – Area (Sq. meters) |
| 1 |
345 |
20.70 |
20 |
0 |
Simpsons 1/3rd Rule= h/3(First Value + last value + 4 * (Sum of odd values) + 2 * (Sum of even values)
H = 21 / 3 = 3 = 3/3 (0 + 1.00 + 4 * 1.1 + 2 * 0.4) |
| 2 |
348 |
20.50 |
20 |
0 |
|
| 3 |
351 |
20.10 |
20 |
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 |
|
| Total = 6.6 |
Summary table
|
Sl. |
Section |
Trapezoidal Method |
Net Area |
Simpsons Rule |
|||
|
No. |
at |
Cutting |
Filling |
Cutting |
Filling |
Cutting |
Filling |
|
1 |
729 |
2.75 |
6.50 |
2.75 |
6.50 |
2.9 |
6.6 |
|
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 |
|||