Ahmadi khan

1. Week No.3
Advance Engineering Surveying
Lecture No.3
B-Tech
By:
Engr. Shams Ul Islam
Lecturer, Civil Engineering . Department
CECOS University Peshawar

2. Computation of Volume
The volume can be calculated by using:
• Trapezoidal Rule
• Prismoidal formula
2Engr.Shams Ul Islam (shams@cecos.edu.pk)

3. Trapezoidal Rule
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝐷
2
× 𝐴1 + 𝐴 𝑛 + 2(𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + … … . 𝐴 𝑛−2 + 𝐴 𝑛−1
𝑊ℎ𝑒𝑟𝑒 𝐴1, 𝐴2, 𝐴3,….etc. are the areas of the sections shown in figure
𝐷 = 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑠
3Engr.Shams Ul Islam (shams@cecos.edu.pk)

4. Prismoidal Formula
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝐷
3
× 𝐴1 + 𝐴 𝑛 + 4(𝐴2 + 𝐴4 + ⋯ + 𝐴 𝑛−1) + 2(𝐴3 + 𝐴5 + ⋯ + 𝐴 𝑛−2
𝑊ℎ𝑒𝑟𝑒 𝐴1, 𝐴2, 𝐴3,….etc. are the areas of the sections shown in figure
𝐷 = 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑠
Note:
The prismoidal formula is applicable when there is an odd number of sections.
If the number of sections is even, the end strip is treated separately and the
area is calculated according to the trapezoidal rule. The volume of the
remaining strips is calculated in the usual manner by the prismoidal formula.
Then both the results are added to obtain the total volume.
4Engr.Shams Ul Islam (shams@cecos.edu.pk)

5. Problem
The following offsets were taken at 15 m intervals from a survey line to an irregular
boundary line.
3.50,4.30, 6.75, 5.25, 7.50, 8.80, 7.90, 6.40, 4.40, 3.25 m
The common width of the survey line is 20 m for all the sections. Calculate the volume
by:
(1)Trapezoidal rule (2) Prismoidal Formula
5Engr.Shams Ul Islam (shams@cecos.edu.pk)

6. Solution
(1) By Trapezoidal Rule
First of all, find the area of all the sections.
𝐴1 =
𝑂1 + 𝑂2
2
× 𝑑 =
3.50 + 4.30
2
× 15 = 58.5𝑚2
𝐴2 =
𝑂2 + 𝑂3
2
× 𝑑 =
4.30 + 6.75
2
× 15 = 82.875𝑚2
Similarly
𝐴3 = 90.00𝑚2
𝐴4 = 95.63𝑚2
𝐴5 = 122.25𝑚2
𝐴6 = 125.25𝑚2
𝐴7 = 107.25𝑚2
𝐴8 = 81.00𝑚2
𝐴9 = 57.38𝑚2
6Engr.Shams Ul Islam (shams@cecos.edu.pk)

7. Now by using Trapezoidal Rule
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝐷
2
× 𝐴1 + 𝐴 𝑛 + 2(𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + … … . 𝐴 𝑛−2 + 𝐴 𝑛−1)
𝑉𝑜𝑙𝑢𝑚𝑒 =
20
2
× 58.5 + 57.38 + 2 × (82.88 + 90 + 95.63 + 122.25 + 125.25 + 107.25 + 81
𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟓𝟐𝟒𝟑. 𝟕𝟓 𝒎 𝟑
Now by using Prismoidal Formula
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝐷
3
× 𝐴1 + 𝐴 𝑛 + 4(𝐴2 + 𝐴4 + ⋯ + 𝐴 𝑛−1) + 2(𝐴3 + 𝐴5 + ⋯ + 𝐴 𝑛−2
𝑉𝑜𝑙𝑢𝑚𝑒 =
20
3
× 58.5 + 57.38 + 4(82.88 + 95.63 + 125.25 + 81) + 2(90 + 122.25 + 107.25
𝑽𝒐𝒍𝒖𝒎𝒆 = 𝟏𝟓𝟐𝟗𝟐. 𝟓 𝒎 𝟑
7Engr.Shams Ul Islam (shams@cecos.edu.pk)

8. Assignment No.1
The following offsets were taken at 25 m intervals from a survey line to
an irregular boundary line.
14.50,12.30, 5, 5.50, 12.50, 10, 9.50, 4.40, 4.45m.
Find the Area by:
(1) Average ordinate Method (2) Simpson’s Rule (3) Trapezoidal Rule
The common width of the survey line is 10 m for all the sections.
Calculate the volume by:
(1)Trapezoidal rule (2) Prismoidal Formula
8Engr.Shams Ul Islam (shams@cecos.edu.pk)