Formula for calculation of Volume & Examples - Computation of area & volume, Surveying and Levelling | Surveying and Levelling Notes- Agricultural Engg - Agricultural Engineering PDF Download

FORMULA FOR CALCULATION OF VOLUME

D= common distance between the sections
A. trapezoidal rule volume (cutting or filling), V=D/2(A1+An+2(A2+A3+….+An-1))

Volume( cutting or filling), V= D/3{A₁+ A_n +4(A₂+ A₄+ A_n-1)+ 2(A₃+ A₅+….+ An_n-1)} i.e. V=common distance  {area of 1^st section+ area of last section+ 4(sum of areas of even sections) 3 + 2 (sum of areas of odd sections)
Note: the prismoidal formula is applicable whrn 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.

WORKS OUT PROBLEM
Problem 1: an embankment of width 10 m and side slopes 1 ½:1 is required to be made on a ground which is level in a direction transverseto the centre line. The central heights at 40 m intervals are as follows:
0.90,1.25,2.15,2.50,1.85,1.35, and 0.85
Calculate the volume of earth work according to
(i) Trapezoidal formula
(ii) Prismoidal formula
Solution:  the c/s areas are calculated by
∆= (b+sh)*h
∆₁= (10+1.5*0.90)*0.90 = 10.22 m²
∆₂= (10+1.5*1.25)*0.90 = 14.84 m²
∆₃= (10+1.5*1.25)*2.15 = 28.43 m²
∆₄= (10+1.5*2.50)*2.50 = 34.38 m²
∆₅= (10+1.5*1.85)*1.85 = 23.63 m²
∆₆=(10+1.5*1.35)*1.35 = 16.23 m²
∆₇=(10+1.5*0.85)*0.85= 9.58 m²
(a) Volume according to trapezoidal formula
V = 40/2{10.22+ 9.58+2(14.84+28.43+34.38+23.63+16.23)}
= 20{19.80+235.02} = 5096.4 m²
(b) Volume calculated in prismoidal formula:
V = 40/3 {10.22+9.58+4(14.84+34.38+16.23)+2(28.43+23.63)}
= 40/3 (19.80+ 261.80+104.12) = 5142.9 m²
Problem the areas enclosed by the contours in the lake are as follows:

Calculate the volume of water between the contours 270 m and 290 m by:
(i) Trapezoidal formula
(ii) Prismoidal formula
Volume according to trapezoidal formula:
=5/2{2050+31500+2(8400+16300+24600)}
=330,250 m³