Part 9, Volume.pdf

Koya University
Faculty of Engineering
Civil Engineering Department – 2nd Stage
Surveying (SUR5109)
Part 9:
Volume
Assist. Lecturer: Hawkar Ali Haji

A.Volume of geometric shapes:
Faculty of Engineering – Civil Department Surveying Assistant Lecturer: Hawkar Ali Haji
Methods ofVolume Determination

B.Volume from cross-sections (cut and fill):
Cross sections are useful in determining quantities of cut and fill in construction
design.This can be done by some different methods;
1.Volume by means area (Average end area method):
When the end areas of cut or fill have been computed for adjacent stations, the
volume of cut or fill between those stations can be computed by simply averaging the
end areas and multiplying the average end area by the distance between the end area
stations.
𝑉𝑜𝑙𝑢𝑚𝑒 =
(𝐴1 + 𝐴2)
2
∗ 𝑑

This rule is applicable when the amount of 𝐴1 and 𝐴2 are close to each, because the
accuracy of this formula will decrease as the difference between these two amount
will increase. For instance, when (𝐴1= 0) the volume formula will be as the following;
(𝐴1 + 𝐴2)
2
∗ 𝑑 =
0 + 𝐴2
2
∗ 𝑑 ⇒ 𝑉𝑜𝑙𝑢𝑚𝑒 =
1
2
𝐴 ∗ 𝑑
But, in fact when 𝐴1 𝑜𝑟 𝐴2 = 0 this means the volume shape of the earthwork is pyramid;
and the volume of pyramid is given as below;
𝑉𝑜𝑙𝑢𝑚𝑒𝑝𝑦𝑟𝑎𝑚𝑖𝑑 =
1
3
𝐴 ∗ 𝑑
𝑒𝑟𝑟𝑜𝑟 𝑟𝑎𝑡𝑖𝑜 =
1
2 𝐴 ∗ 𝑑 − (
1
3 𝐴 ∗ 𝑑 )
1
3
𝐴 ∗ 𝑑
∗ 100% = 50%

For (n) number of cross-section areas, the formula will be as below;
(𝐴1+𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + ⋯ + 𝐴𝑛)
𝑛
∗ 𝐿

2. Volume by End Area Method (Trapezoidal rule) or (Simpson`s rule):
𝑉𝑜𝑙𝑢𝑚𝑒𝑡𝑜𝑡𝑎𝑙 = 𝑑/2 ∗ 𝐴1 + 𝐴𝑛 + 2 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + ⋯ + 𝐴𝑛−1

Example

3.Volume by Prismoidal Formula
If the volume of earth between two successive cross sections be considered a
prismoid then a more precise formula (the prismoidal formula) may be used. It is
generally considered that, all thing being equal, use of this formula gives the most
accurate estimate of volume.
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑑/6 (𝐴1 + 4𝐴𝑚 + 𝐴2)
Where 𝐴1 and 𝐴2 are the areas of the two end faces distances 𝑑 apart, 𝐴𝑚 is the
area of the section mid-way between.

𝑉𝑜𝑙𝑢𝑚𝑒 1−3 =
𝑑
6
(𝐴1 + 4𝐴2 + 𝐴3)
𝑉𝑜𝑙𝑢𝑚𝑒 3−5 =
𝑑
6
𝐴3 + 4𝐴4 + 𝐴5
𝑉𝑜𝑙𝑢𝑚𝑒 𝑡𝑜𝑡𝑎𝑙 =
𝑑
6
𝐴1 + 4𝐴2 + 𝐴3 + 𝐴3 + 4𝐴4 + 𝐴5 =
𝑑
6
(𝐴1 + 𝐴5 + 4𝐴2 + 4𝐴4 + 2𝐴3)
𝑉𝑜𝑙𝑢𝑚𝑒 𝑡𝑜𝑡𝑎𝑙 = 𝑑/3 ∗ 𝐴1 + 𝐴𝑛 + 4 𝐴𝑒𝑣𝑒𝑛 + 2 𝐴𝑜𝑑𝑑

C.Volume From Contour Lines:
It is possible to calculate volumes using the horizontal areas contained by contour
lines. Owing to the relatively high cost of accurately contouring large areas, the
method is of limited use, but where accurate contours are available, as for instance, in
reservoir sites, they may be conveniently used.

The contour interval will determine the distance d in the (End Area or Prismoidal)
formula. and for accuracy this should be as small as possible, preferably 2 ft to 5 ft
(1m – 2m).The accuracy of the volume depends basically on the contour vertical
interval. Generally, the closer the contour interval the more accurate is the volume.
𝑉𝑜𝑙𝑢𝑚𝑒 𝑡𝑜𝑡𝑎𝑙 = 𝑑/3 ∗ 𝐴1 + 𝐴𝑛 + 4 𝐴𝑒𝑣𝑒𝑛 + 2 𝐴𝑜𝑑𝑑

D.Volume from Spot Height (Volume of Borrow pits):
-This is a method by means of which the earthworks involved in the construction of
large tanks, basement, borrow pits, etc., and similar works with vertical sides may be
calculated. The computation is simplified if the formation is to be to a fixed level or to
fixed falls, but even basements with several levels present little difficulty.
- Having located the outline of the structure on the ground, the engineer divided up
the area into squares or rectangles, marking the corner points as described in the
method of contouring by spot levels. Level are taken at each of these corner points,
and by subtracting from these the corresponding formation levels, a series of heights
are obtained from which the mean heights of a series of vertical truncated prisms of
earth can be found.
The volume of each prism is given by the plan area multiplied by the mean height of
the prism.

𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐴.
ℎ1 + 2 ℎ2 + 3 ℎ3 + 4 ℎ4
4
𝑤ℎ𝑒𝑟𝑒:
𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒
𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑜𝑛𝑒 𝑠𝑞𝑢𝑎𝑟𝑒
ℎ = 𝑑𝑒𝑝𝑡ℎ 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡𝑠

Example: In the shown figure, the reduced levels of a rectangular plot which is to be excavated to a
uniform reduced level of 120 ft above datum. Assuming the sides to be vertical, calculate the volume
of earth to be excavated different two methods.
Solution: Create a table include all stations with necessary height (ℎ𝑛) that should be excavated
according to our design line in the field, then number of rectangles that involved in each station (𝑛),
finally summate the multiplication of (ℎ𝑛 ∗ 𝑛).
Example

𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑨.
𝒉𝟏 + 𝟐 𝒉𝟐 + 𝟑 𝒉𝟑 + 𝟒 𝒉𝟒
𝟒
= 𝟓𝟎 ∗ 𝟒𝟎 .
𝟐𝟒𝟐. 𝟔𝟏
𝟒
= 𝟏𝟐𝟏𝟑𝟎𝟓 𝒇𝒕𝟑
𝟏𝟐𝟏𝟑𝟎𝟓
𝟐𝟕
= 𝟒𝟒𝟗𝟐. 𝟕𝟕 𝒚𝒅𝟑
Example

Example: Find the volume of cut and fill from the following X-sections by Average
End Area method?
Example

Example: The following was done for the grid of the borrow pit after and before
excavation. If the length of square sides was (20m), and then the depth of cut at each
corner was computed as shown in the figure. Find the total volume of excavation?
Example

Solution:
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 = 20 ∗ 20
Example

Example: The area between the contours curves measured by plan meter as shown
below: Find the volume by:
1. End area method
2. Simpson's one-third rule (Prismoidal rule)
Example

Solution:
1. End area method
2. Simpson's one-third rule (Prismoidal rule)
Example

Part 9, Volume.pdf

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