1. Introduction
Persons engaged in surveying are often called to determine volumes of various types of material.
The most common unit of volume is cubic having edges of unit length. Cubic feet, cubic yard or cubic meter.

2. Method of Volume Measurement
Three principal systems are used : 1- The cross section method 2- The unit area method 3- The contour - area method

3. Method of Volume Measurement
Cross Section Method
The cross section method is employed almost exclusively for computing volumes on linear construction projects such as highways, railroads, and canals.
Cut slopes of 1:1 ( 1 horizontal to 1 vertical ) and fill slope of 1-1/2:1 satisfactory for ordinary loam soil.

4. Method of Volume Measurement

5. Method of Volume Measurement
End –areas
excavation (cut) and embankment (fill).
Volumes will be determine by:
Average area method &
Prismoidal Formula
• Fill from STA to 11+21
• Transition from fill to cut will be STA to STA
• Cut from to

6. Types Of Cross Sections

7. Types Of Cross Sections

8. Types Of Cross Sections
In flat terrain the level section (a) is suitable . The three level section (b) is generally used where ordinary ground level is prevail. Rough topography may require a five level section ( c ).
More practically an irregular section (d). A transition section (e), and slide hill section (f) occur when passing from cut to fill and on slide –hill locations.

9. Method of Volume Measurement
It is necessary to make separate computations for cut and fill end areas when they occur in the same section., since they must always be tabulated independently for pay purposes. Payment is normally made only for excavation except on projects consisting primarily of embankment such as levees, earth dams, some military fortifications, and highways built up by continuous fill in flat areas

10. Types Of Cross Sections
Cut roadway wider than Fill roadway? bc>bf
To provide space for drainage ditches .
• Slope sides in fill usually flatter than in cut?
Because in cut the soil remain in its natural state.
• Its necessary to make separate volume computations for cut & fill ?
separately computation for pay purpose since normally payment made for excavation only except some cases which need primarily fill such as earth dams

11. Types Of Cross Sections
The width of base b , the finished roadway is fixed by project requirements and it is usually wider in cuts than on fills to provide for drainage ditches.
Slide slops in fill are flatter than those in cuts where soil remains in its natural state.

12. Method of Volume Measurement
Average End Area Formula
A1 and A2 are end areas at two stations separated by a horizontal distance L.
The volume between tow stations then equal to :
Ve = (1)

13. Method of Volume Measurement
EX. Compute the volume of excavation between station with an end area of 66 m2 and station with an end area of 48 m2 . Solution Ve=

14. Method of Volume Measurement
Prismoidal Formula
The prismoidal formula applies to volumes of all geometric solids that can be considered prismoids.

15. Method of Volume Measurement
(2)
Where:
VP : is the prismoidal volume in cubic yard
A1 and A2: are areas of successive cross sections taken in the field.
Am : is the area of computed section midway between A1 and A2
L : is the horizontal distance between A1 and A2

16. Method of Volume Measurement
The difference between the volumes obtained by the average–end-area formula and the prismoidal formula is called the prismoidal correction Cp . This applicable for three level section.
(3)
Where :
Cp is the volume of the prismoidal correction in cubic meter
C1 and C2 are center heights in cut or in fill
W1 and W2 are widths of sections from slope intercept to slope intercept

17. Method of Volume Measurement
If the product ( c1 - c2 ) ( w1 – w2 ) is minus , the prismoidal correction (Cp) is added rather than subtracted from the end-area volume.
For sections other than 3 level. Equation (3) may be not accurate enough , therefore equation (2) is recommended.

18. Volume Measurement Example 1
compute the volume using the prismoidal formula and by average end areas for the following three level sections of a roadbed having a base of 24 ft and side slopes of 1.5:1

19. Volume Measurement m2 m2 m2
Applying Premisoidal formula lead to :

20. Volume Measurement Applymg end area formula lead to :
Applying Prismoidal correction lead to :
The prismoidal correction applied to the end area formula
yields a volume of x 103 m3 (22.56 x )

21. Volume Measurement
Usually for expansive rock cut , the prismoidal method can be used. Prismoidal formula gives volume smaller than that volume found by average end-area.

22. Simpson”s Rule or Parabolic Rule
The sum of the first and last ordinates add twice the sum of the third,fifth,seventh(odd ordinates) and four the sum of the 2nd,4th,6th, (even ordinates). Multiply all by x/3. where x the distance between the ordinates.

23. Example 9
Eleven effects were taken from chain line to curved boundary at 10 m intervals and the lengths of the offsets from the left end are in meters, 3.8,5.1,6.5,6.8,5.9,6.2,7.0,6.6,5.8, and 4.2. determine the area between the chain line , the curved boundary and the first and last offsets by parabolic rule

24. Example 8
2) Complete the missing data and compute Ve &Vp. The roadbed is level. The base is 10 m. side slope 2:1
Station L C R
A ( m2)
10+00
C4.2
32.6
C3.0
C3.6
30.8
-----
11+00
170 m
12+00
C7.8
23.7
C5.3
C7.4
23.0
12+50
C6.5
21.8
C6.0
C7.5
23.2
13+00
C5.8
24.8
C6.6
C7.0
23.5
14+00
C2.4
27.2
0.0 20
15+00
94.5

25. Volume Computations
Sources of Error in Determining Volumes Some common errors in determining areas of section and volumes of earthwork are 1- making errors in measuring field cross-sections, e.g. not being perpendicular to the centerline. 2- making errors in measuring end areas. 3- failing to use the prismoidal formula where it is justifies. 4- carrying out areas of cross sections beyond the limit justified by the field data.

26. Volume Computations
Mistakes Some typical mistakes made in earthwork calculations are : 1- Confusing algebraic signs in end area computations using the coordinate method. 2- Using average end-area method for pyramidal or wedge-shaped solids. 4- Mixing cut and fill quantities.