Measurements of Area & Volume - 1 - Free MCQ Practice Test with solutions,

Methods for measurement of earthwork:
1. Mid Section method  2. Trapezoidal Method  3. Prismoidal Method  4. Simpson's 3/8th rule
Mid Section Method:
In this method the quantity of earthwork is computed with the help of size of mid section.

Mid-Section Area Method

• The mean depth is the average depth of two consecutive sections.
• The area of mid-sections is calculated by using mean depth.
• The volume of the earthwork is calculated by multiplying the mid-section area by the distance between the two sections.

The formula used for finding the amount of earthwork using the mid-sectional method is given by:
Q = A x L
Where,
Q = Volume of earthwork
A = Area of mid-section = BD_m + SD_m²
B = width of the road
D_m = Depth of mid-section = 
S = Slope
L = Length of road
Hence, Q = (BD_m + SD_m²)L

Simpson's Rule:
In this rule, the boundaries between the ends of ordinates are assumed to form an arc of a parabola. Hence, Simpson's rule is sometimes called the parabolic rule.
The rule may be stated as follows,
To the sum of the first and the last ordinate, four times the sum of even ordinates and twice the sum of the remaining odd ordinates are added. This total sum is multiplied by the common distance. One-third of this product is the required area.
The area is given by Simpson's rule:

where O₁, O₂, O₃, .........O_n is the offset
Limitation: 
This rule is applicable only when the number of divisions is even, i.e., the number of ordinates is odd.
Calculation

Simpson's Rule :- If this rule is to be applied, the number of ordinates must be odd. But here the number of ordinate is even (eight)
so, Simpson's rule is applied from 0₁ to 0₇ and the area between 0₇ and 0₈ is found out by trapezoidal rule. 

A₁ = 10 × (5 + 40 + 16.9)
A₁ = 619 m² 
A₂ = 30/2(5.00+5.80)
A₂ = 162 m²
∴ Total area = A₁ + A₂
= 619 + 162 = 781 m²

Simpson's method assumes that the short lengths of the boundaries between the ordinates are parabolic arcs. This method is more useful when the boundary line departs considerably from the straight line.
Earthwork computation is involved in the excavation of channels, digging of trenches for laying underground pipelines, formation of bunds, and earthen embankments, digging of farm ponds, land leveling, and smoothening. In most of the computation, the cross-sectional areas at different interval along the length of the channels and embankments are first calculated and the volume of the prismoids are obtained between successive cross-section either by trapezoidal or prismoidal formula.
Calculation of area is carried out by any one of the following methods:
(a) Mid-ordinate method
(b) Average ordinate method
(c) Trapezoidal rule
(d) Simpson’s rule
Trapezoidal Rule

• Boundaries of the ordinates are presumed to be straight. Therefore the area between the lines is considered to be a trapezoid.

This rule can be used for any number of ordinates, and hence it has no drawbacks or limitations.
Simpson’s Rule:

• An arc is assumed to be present between the boundaries of the ordinates. Therefore it is also known as the parabolic rule. 

Additional Information

• The only limitation to this rule is that it can only be used when the number of ordinates is odd.
• Simpson’s rule gives a more accurate result.
• After the calculation of the cross-sectional areas, volume is also needed to be calculated to determine the capacity

The volume of earthwork by trapezoidal method  = V₁

The volume of earthwork by prismoidal formula = V₂

Prismoidal correction

• The volume by the prismoidal formula is more accurate than any other method
• But the trapezoidal method is more often used for calculating the volume of earthwork in the field.
• The difference between the volume computed by the trapezoidal formula and the prismoidal formula is known as a prismoidal correction.
• Since the trapezoidal formula always overestimates the volume, the prismoidal correction is always subtractive in nature is usually more than calculated by the prismoidal formula, therefore the prismoidal correction is generally subtractive.
• Volume by prismoidal formula = volume by the trapezoidal formula - prismoidal correction

Prismoidal correction (C_P)

Where, D = Distance between the sections, S (Horizontal) : 1 (Vertical) = Side slope, d and d₁ are the depth of earthwork at the centerline

Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'O_n' should be odd. 
The area is given by Simpson's rule:

Important Points

• In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.

Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd. 
The area is given by Simpson's rule:

where O₁, O₂, O₃, .........O_n is the offset
Important Points

• In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.

• Trapezoidal Formula:
  Volume (v) of earthwork between a number of sections having areas A₁, A₂,…, A_n spaced at a constant distance d.

Also can be written as 

Calculation:
Given, A₁ = 50 × 40 = 2000 m², As the side slope is given 2:1 i.e. H:V
So for a depth of 1 m, there is a change of 2 m in a Horizontal Direction.
So at 4 m vertical depth
Bottom Dimension is ( 50 - 16 ) = 34 m & (40 - 16) = 24 m
∴ The bottom area is 34 m × 24 m = 816 m²
Mean area (A_m) = (2000 + 816) / 2 = 1408 m²

According to simple Trapezoidal rule for volume,

Gradient: 
A gradient is the rate of rise or falls along the length of the road with respect to horizontal. It is expressed as ‘1' vertical unit to 'N' horizontal units.

Area of trapezoidal:
According to the trapezoid area formula, the area of a trapezoid is equal to half the product of the height and the sum of the two bases.
Area = ½ x (Sum of parallel sides) x (perpendicular distance between the parallel sides).
Calculation:

Road embankment = 10 m
Average height = 5 m
Difference in elevation(h) = 280 - 220 = 60 m
Average gradient = 1/40

Average cross-sectional area(A) = (1/2) ×(10+30)×5
A = 100 m²
Volume of earthwork = A × L
Volume of earthwork = 100 × 2400
∴ Volume of earthwork = 2,40,000 m³

Trapezoidal Formula:
Volume (v) of earthwork between a number of sections having areas A₁, A₂, …, A_n spaced at a constant distance d.

Simpson’s Formula:
Volume (v) of the earthwork between a number of sections having area A_1, A₂, …, A_n spaced at constant distance d apart is

Calculation:
The areas enclosed by the contours in a lake are as follows:

Given contour interval (d) = 275-270 = 5 m
So using trapezoidal formula:
 8000 m³