Earthwork Volume Computations

Earthwork Volume Computations

Learning Objectives

Understand the definition and purpose of earthworks in engineering surveying.
Learn methods for calculating earthwork volumes including the average end-area and prismoidal methods.
Apply the borrow pit method for extensive grading volume computations.
Interpret the mass haul diagram for planning earth movement.

This lesson covers the methods for calculating cut and fill volumes, which are essential for route grading and large-scale site leveling in civil engineering projects.

In engineering surveying, earthwork involves the computation of volumes of soil or rock that must be excavated (cut) or added (fill) to achieve the design grades of highways, railways, canals, or building foundations.

Volume by Cross-Sections

To determine volumes, cross-sections of the ground and the proposed formation level are taken at regular intervals along a route. The area of each cross-section is calculated, and volumes are determined between successive sections.

1. Average End-Area Method

Average End-Area Method

The most common method used in practice. It assumes the volume between two cross-sections is a prism whose base is the average of the two end areas.

End-Area Formula

Computes the estimated volume of earthwork between two parallel cross-sections using their average area.

V = \frac{A_1 + A_2}{2} \cdot L

Variables

Symbol	Description	Unit
$V$	Volume of earthwork (cut or fill)	$m^3$
$A_1$	Cross-sectional area at the start of the segment	$m^2$
$A_2$	Cross-sectional area at the end of the segment	$m^2$
$L$	Distance between the two cross-sections	m

Interactive Simulation

Alter the cross-sectional areas and the length between them to compute the estimated earthwork volume using the average end-area method.

End-Area Volume Simulator

Estimate earthwork volume between two cross-sections.

Area 1 (

A_1

) (m²)50.0

Area 2 (

A_2

) (m²)100.0

Length (

L

) (m)20.0

Results

V = \frac{A_1 + A_2}{2} \times L

V = \frac{50 + 100}{2} \times 20 = 1500.0 \text{ m}^3

Overestimation Warning

This method tends to slightly over-estimate volumes, particularly when the end areas differ significantly.

2. Prismoidal Formula

Prismoidal Method Overview

A more precise method. It is used when a higher degree of accuracy is required, such as in solid rock excavation or when end areas are vastly different. It requires a middle section area ( $A_m$ ).

Prismoidal Formula

Computes a highly accurate earthwork volume using a mid-section area.

V = \frac{L}{6} \cdot (A_1 + 4A_m + A_2)

Variables

Symbol	Description	Unit
$V$	Volume of earthwork	$m^3$
$L$	Distance between the end sections	m
$A_1$	First end area	$m^2$
$A_2$	Second end area	$m^2$
$A_m$	Area of the cross-section midway between A_1 and A_2 (computed from average linear dimensions, not average areas)	$m^2$

C_p

Instead of using the Prismoidal formula directly, it is often easier to compute the End-Area volume and apply a correction.

Prismoidal Correction

V_{prismoidal} = V_{end\_area} - C_p

Interactive Volume Calculator

Interactive Simulation

Explore the volume calculations dynamically using the earthwork volume simulator below.

Earthwork Volume Calculator

Area 1 (

A_1

) in

m^2

120

Area 2 (

A_2

) in

m^2

180

Middle Area (

A_m

) in

m^2

145

Distance (

L

) in

m

End-Area Method

7500.00 m³

Formula: $V = L \\\\cdot \\\\frac{A_1 + A_2}{2}$

Prismoidal Formula

7333.33 m³

Formula: $V = L \\\\cdot \\\\frac{A_1 + 4A_m + A_2}{6}$

Borrow Pit Method

Used for computing volumes of extensive grading operations, like leveling a building site. The area is divided into a grid of squares or rectangles. Elevations are taken at the corners before and after excavation.

Borrow Pit Formula

Computes total excavation volume for a large area divided into grid units.

V = \frac{A}{4} \cdot (\Sigma h_1 + 2\Sigma h_2 + 3\Sigma h_3 + 4\Sigma h_4)

Variables

Symbol	Description	Unit
$V$	Total volume	$m^3$
$A$	Area of one grid unit (square or rectangle)	$m^2$
$h_1$	Corner heights used by exactly 1 grid rectangle	m
$h_2$	Corner heights used by exactly 2 grid rectangles	m
$h_3$	Corner heights used by exactly 3 grid rectangles	m
$h_4$	Corner heights used by exactly 4 grid rectangles	m

Mass Haul Diagram (MHD)

Mass Haul Diagram Concept

A graphical representation of the cumulative volume of earthwork along a project's centerline. It is a fundamental tool for planning the transportation of excavated material (cut) to areas requiring fill.

Properties of Mass Haul Diagram

Ascending Curve: Indicates a section of net cut (volume increases).
Descending Curve: Indicates a section of net fill (volume decreases).
Peak: Change from cut to fill.
Trough (Lowest Point): Change from fill to cut.
Horizontal Line (Balance Line): A line drawn horizontally intersects the curve at points where the total cut volume exactly equals the total fill volume (balancing).

Key Hauling Concepts

Freehaul Distance ( $FHD$ ): A specified distance over which the contractor is paid a flat rate for excavation, regardless of how far the material is moved within this limit.
Overhaul ( $OH$ ): The transportation of material beyond the freehaul distance. Contractors are paid extra for overhaul (usually calculated in station-meters or station-yards).
Borrow: Soil imported from outside the project limits when there is not enough cut to satisfy the fill requirements.
Waste: Excess excavated soil that must be disposed of off-site when there is more cut than fill.

Key Takeaways

Earthwork Volumes: Calculate cut and fill for route grading.
End-Area Method: Standard method. Averages the areas and multiplies by length. Overestimates slightly.
Prismoidal Formula: Highly accurate. Requires a true mid-section area.
Borrow Pit Method: Uses a grid for large, wide excavation areas.
Mass Haul Diagram: Plans earth movement, identifies cut/fill balance points, and helps calculate overhaul.