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The document discusses methods for calculating the volume of irregular shapes using survey data. It describes the trapezoidal rule and prismoidal formula. The trapezoidal rule uses the average of the start and end areas plus the sum of all intermediate areas. The prismoidal formula weights end areas higher and alternating interior areas differently. An example calculation applies both methods to survey offset data to compute the volume. Students are assigned to use various area calculation methods with additional offset data and compute the volume using trapezoidal and prismoidal formulas.

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Week No.3 Advance Engineering Surveying Lecture No.3 B-Tech By: Engr. Shams Ul Islam Lecturer, Civil Engineering . Department CECOS University Peshawar
Computation of Volume The volume can be calculated by using: โ€ข Trapezoidal Rule โ€ข Prismoidal formula 2Engr.Shams Ul Islam (shams@cecos.edu.pk)
Trapezoidal Rule ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 2 ร— ๐ด1 + ๐ด ๐‘› + 2(๐ด2 + ๐ด3 + ๐ด4 + ๐ด5 + โ€ฆ โ€ฆ . ๐ด ๐‘›โˆ’2 + ๐ด ๐‘›โˆ’1 ๐‘Šโ„Ž๐‘’๐‘Ÿ๐‘’ ๐ด1, ๐ด2, ๐ด3,โ€ฆ.etc. are the areas of the sections shown in figure ๐ท = ๐‘กโ„Ž๐‘’ ๐‘๐‘œ๐‘š๐‘š๐‘œ๐‘› ๐‘‘๐‘–๐‘ ๐‘ก๐‘Ž๐‘›๐‘๐‘’ ๐‘๐‘’๐‘ก๐‘ค๐‘’๐‘’๐‘› ๐‘ ๐‘’๐‘๐‘ก๐‘–๐‘œ๐‘›๐‘  3Engr.Shams Ul Islam (shams@cecos.edu.pk)
Prismoidal Formula ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 3 ร— ๐ด1 + ๐ด ๐‘› + 4(๐ด2 + ๐ด4 + โ‹ฏ + ๐ด ๐‘›โˆ’1) + 2(๐ด3 + ๐ด5 + โ‹ฏ + ๐ด ๐‘›โˆ’2 ๐‘Šโ„Ž๐‘’๐‘Ÿ๐‘’ ๐ด1, ๐ด2, ๐ด3,โ€ฆ.etc. are the areas of the sections shown in figure ๐ท = ๐‘กโ„Ž๐‘’ ๐‘๐‘œ๐‘š๐‘š๐‘œ๐‘› ๐‘‘๐‘–๐‘ ๐‘ก๐‘Ž๐‘›๐‘๐‘’ ๐‘๐‘’๐‘ก๐‘ค๐‘’๐‘’๐‘› ๐‘ ๐‘’๐‘๐‘ก๐‘–๐‘œ๐‘›๐‘  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)
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)
Solution (1) By Trapezoidal Rule First of all, find the area of all the sections. ๐ด1 = ๐‘‚1 + ๐‘‚2 2 ร— ๐‘‘ = 3.50 + 4.30 2 ร— 15 = 58.5๐‘š2 ๐ด2 = ๐‘‚2 + ๐‘‚3 2 ร— ๐‘‘ = 4.30 + 6.75 2 ร— 15 = 82.875๐‘š2 Similarly ๐ด3 = 90.00๐‘š2 ๐ด4 = 95.63๐‘š2 ๐ด5 = 122.25๐‘š2 ๐ด6 = 125.25๐‘š2 ๐ด7 = 107.25๐‘š2 ๐ด8 = 81.00๐‘š2 ๐ด9 = 57.38๐‘š2 6Engr.Shams Ul Islam (shams@cecos.edu.pk)
Now by using Trapezoidal Rule ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 2 ร— ๐ด1 + ๐ด ๐‘› + 2(๐ด2 + ๐ด3 + ๐ด4 + ๐ด5 + โ€ฆ โ€ฆ . ๐ด ๐‘›โˆ’2 + ๐ด ๐‘›โˆ’1) ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = 20 2 ร— 58.5 + 57.38 + 2 ร— (82.88 + 90 + 95.63 + 122.25 + 125.25 + 107.25 + 81 ๐‘ฝ๐’๐’๐’–๐’Ž๐’† = ๐Ÿ๐Ÿ“๐Ÿ๐Ÿ’๐Ÿ‘. ๐Ÿ•๐Ÿ“ ๐’Ž ๐Ÿ‘ Now by using Prismoidal Formula ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 3 ร— ๐ด1 + ๐ด ๐‘› + 4(๐ด2 + ๐ด4 + โ‹ฏ + ๐ด ๐‘›โˆ’1) + 2(๐ด3 + ๐ด5 + โ‹ฏ + ๐ด ๐‘›โˆ’2 ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = 20 3 ร— 58.5 + 57.38 + 4(82.88 + 95.63 + 125.25 + 81) + 2(90 + 122.25 + 107.25 ๐‘ฝ๐’๐’๐’–๐’Ž๐’† = ๐Ÿ๐Ÿ“๐Ÿ๐Ÿ—๐Ÿ. ๐Ÿ“ ๐’Ž ๐Ÿ‘ 7Engr.Shams Ul Islam (shams@cecos.edu.pk)
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) Simpsonโ€™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)

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surveying and levelling

  • 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: โ€ข Trapezoidal Rule โ€ข Prismoidal formula 2Engr.Shams Ul Islam (shams@cecos.edu.pk)
  • 3. Trapezoidal Rule ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 2 ร— ๐ด1 + ๐ด ๐‘› + 2(๐ด2 + ๐ด3 + ๐ด4 + ๐ด5 + โ€ฆ โ€ฆ . ๐ด ๐‘›โˆ’2 + ๐ด ๐‘›โˆ’1 ๐‘Šโ„Ž๐‘’๐‘Ÿ๐‘’ ๐ด1, ๐ด2, ๐ด3,โ€ฆ.etc. are the areas of the sections shown in figure ๐ท = ๐‘กโ„Ž๐‘’ ๐‘๐‘œ๐‘š๐‘š๐‘œ๐‘› ๐‘‘๐‘–๐‘ ๐‘ก๐‘Ž๐‘›๐‘๐‘’ ๐‘๐‘’๐‘ก๐‘ค๐‘’๐‘’๐‘› ๐‘ ๐‘’๐‘๐‘ก๐‘–๐‘œ๐‘›๐‘  3Engr.Shams Ul Islam (shams@cecos.edu.pk)
  • 4. Prismoidal Formula ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 3 ร— ๐ด1 + ๐ด ๐‘› + 4(๐ด2 + ๐ด4 + โ‹ฏ + ๐ด ๐‘›โˆ’1) + 2(๐ด3 + ๐ด5 + โ‹ฏ + ๐ด ๐‘›โˆ’2 ๐‘Šโ„Ž๐‘’๐‘Ÿ๐‘’ ๐ด1, ๐ด2, ๐ด3,โ€ฆ.etc. are the areas of the sections shown in figure ๐ท = ๐‘กโ„Ž๐‘’ ๐‘๐‘œ๐‘š๐‘š๐‘œ๐‘› ๐‘‘๐‘–๐‘ ๐‘ก๐‘Ž๐‘›๐‘๐‘’ ๐‘๐‘’๐‘ก๐‘ค๐‘’๐‘’๐‘› ๐‘ ๐‘’๐‘๐‘ก๐‘–๐‘œ๐‘›๐‘  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. ๐ด1 = ๐‘‚1 + ๐‘‚2 2 ร— ๐‘‘ = 3.50 + 4.30 2 ร— 15 = 58.5๐‘š2 ๐ด2 = ๐‘‚2 + ๐‘‚3 2 ร— ๐‘‘ = 4.30 + 6.75 2 ร— 15 = 82.875๐‘š2 Similarly ๐ด3 = 90.00๐‘š2 ๐ด4 = 95.63๐‘š2 ๐ด5 = 122.25๐‘š2 ๐ด6 = 125.25๐‘š2 ๐ด7 = 107.25๐‘š2 ๐ด8 = 81.00๐‘š2 ๐ด9 = 57.38๐‘š2 6Engr.Shams Ul Islam (shams@cecos.edu.pk)
  • 7. Now by using Trapezoidal Rule ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 2 ร— ๐ด1 + ๐ด ๐‘› + 2(๐ด2 + ๐ด3 + ๐ด4 + ๐ด5 + โ€ฆ โ€ฆ . ๐ด ๐‘›โˆ’2 + ๐ด ๐‘›โˆ’1) ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = 20 2 ร— 58.5 + 57.38 + 2 ร— (82.88 + 90 + 95.63 + 122.25 + 125.25 + 107.25 + 81 ๐‘ฝ๐’๐’๐’–๐’Ž๐’† = ๐Ÿ๐Ÿ“๐Ÿ๐Ÿ’๐Ÿ‘. ๐Ÿ•๐Ÿ“ ๐’Ž ๐Ÿ‘ Now by using Prismoidal Formula ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = ๐ท 3 ร— ๐ด1 + ๐ด ๐‘› + 4(๐ด2 + ๐ด4 + โ‹ฏ + ๐ด ๐‘›โˆ’1) + 2(๐ด3 + ๐ด5 + โ‹ฏ + ๐ด ๐‘›โˆ’2 ๐‘‰๐‘œ๐‘™๐‘ข๐‘š๐‘’ = 20 3 ร— 58.5 + 57.38 + 4(82.88 + 95.63 + 125.25 + 81) + 2(90 + 122.25 + 107.25 ๐‘ฝ๐’๐’๐’–๐’Ž๐’† = ๐Ÿ๐Ÿ“๐Ÿ๐Ÿ—๐Ÿ. ๐Ÿ“ ๐’Ž ๐Ÿ‘ 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) Simpsonโ€™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)
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