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=A1+A22⋅L V = \frac{A_1 + A_2}{2} \cdot L

Variables

SymbolDescriptionUnit
VVVolume of earthwork (cut or fill)m3m^3
A1A_1Cross-sectional area at the start of the segmentm2m^2
A2A_2Cross-sectional area at the end of the segmentm2m^2
LLDistance between the two cross-sectionsm

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 (A1A_1) (m²)50.0
Area 2 (A2A_2) (m²)100.0
Length (LL) (m)20.0

Results

V=A1+A22×LV = \frac{A_1 + A_2}{2} \times L
V=50+1002×20=1500.0 m3V = \frac{50 + 100}{2} \times 20 = 1500.0 \text{ m}^3
A1A2L = 20

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 (AmA_m).

Prismoidal Formula

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

V=L6⋅(A1+4Am+A2) V = \frac{L}{6} \cdot (A_1 + 4A_m + A_2)

Variables

SymbolDescriptionUnit
VVVolume of earthworkm3m^3
LLDistance between the end sectionsm
A1A_1First end aream2m^2
A2A_2Second end aream2m^2
AmA_mArea of the cross-section midway between A_1 and A_2 (computed from average linear dimensions, not average areas)m2m^2

Prismoidal Correction (CpC_p)

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

Prismoidal Correction

Vprismoidal=Vend_area−Cp 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

120
180
145
50

End-Area Method

7500.00 m³

Formula: V=LcdotfracA1+A22V = L \\\\cdot \\\\frac{A_1 + A_2}{2}

Prismoidal Formula

7333.33 m³

Formula: V=LcdotfracA1+4Am+A26V = 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=A4⋅(Σh1+2Σh2+3Σh3+4Σh4) V = \frac{A}{4} \cdot (\Sigma h_1 + 2\Sigma h_2 + 3\Sigma h_3 + 4\Sigma h_4)

Variables

SymbolDescriptionUnit
VVTotal volumem3m^3
AAArea of one grid unit (square or rectangle)m2m^2
h1h_1Corner heights used by exactly 1 grid rectanglem
h2h_2Corner heights used by exactly 2 grid rectanglesm
h3h_3Corner heights used by exactly 3 grid rectanglesm
h4h_4Corner heights used by exactly 4 grid rectanglesm

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 (FHDFHD): 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 (OHOH): 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.
Earthwork Volume and Mass-Haul Workflow

Choose a corridor or grid method, compute volumes, then interpret cut, fill, borrow, waste, and haul conditions where the data support them.

For corridor work, use the average end-area method for routine intervals and the prismoidal method when the required accuracy or area change warrants it; prismoidal volume requires a measured or computed middle area. Keep cut and fill volumes separate. Build a station-based mass-haul diagram with cut positive and fill negative. Equal ordinates identify balanced cut-to-fill segments; the final ordinate relative to the initial ordinate shows net cut or fill. The mass-haul branch applies to route sections, while the grid branch sums site excavation cells. Freehaul is contract-defined, so defer overhaul classification until its distance is known.

Earthwork Volume and Mass-Haul WorkflowChoose a corridor or grid method, compute volumes, then interpret cut, fill, borrow, waste, and haul conditions where the data support them.. For corridor work, use the average end-area method for routine intervals and the prismoidal method when the required accuracy or area change warrants it; prismoidal volume requires a measured or computed middle area. Keep cut and fill volumes separate. Build a station-based mass-haul diagram with cut positive and fill negative. Equal ordinates identify balanced cut-to-fill segments; the final ordinate relative to the initial ordinate shows net cut or fill. The mass-haul branch applies to route sections, while the grid branch sums site excavation cells. Freehaul is contract-defined, so defer overhaul classification until its distance is known.. Earthwork survey data → Corridor sections or site grid?; Corridor sections or site grid? — Corridor → Calculate section areas by station; Corridor sections or site grid? — Grid → Compute grid-cell excavation volumes; Compute grid-cell excavation volumes → Sum grid cells; report excavation; Sum grid cells; report excavation → Report earthwork quantities and disposition; Calculate section areas by station → Higher accuracy or marked area change?; Higher accuracy or marked area change? — Yes → Middle-section area available?; Higher accuracy or marked area change? — No → Average end-area volume; Middle-section area available? — Yes → Prismoidal volume; Middle-section area available? — No → Survey midpoint; compute middle area; Survey midpoint; compute middle area → Prismoidal volume; Average end-area volume → Record interval cut / fill volumes; Prismoidal volume → Record interval cut / fill volumes; Record interval cut / fill volumes → Accumulate cut (+) and fill (−); Accumulate cut (+) and fill (−) → Plot cumulative volume vs station; Plot cumulative volume vs station → Read rises, falls and balance line; Read rises, falls and balance line → Pair equal ordinates for balanced hauls; Pair equal ordinates for balanced hauls → Final ordinate equals initial?; Final ordinate equals initial? — Yes → Project totals balance; Final ordinate equals initial? — No → Final ordinate higher?; Final ordinate higher? — Yes → Plan waste for surplus cut; Final ordinate higher? — No → Plan borrow for fill deficit; Project totals balance → Contract freehaul distance known?; Plan waste for surplus cut → Contract freehaul distance known?; Plan borrow for fill deficit → Contract freehaul distance known?; Contract freehaul distance known? — Yes → Compare balanced haul with freehaul; Contract freehaul distance known? — No → Hold: confirm contract freehaul; Compare balanced haul with freehaul → Haul exceeds freehaul distance?; Haul exceeds freehaul distance? — Yes → Classify excess haul as overhaul; Haul exceeds freehaul distance? — No → Haul within freehaul distance; Classify excess haul as overhaul → Report earthwork quantities and disposition; Haul within freehaul distance → Report earthwork quantities and disposition

Earthwork survey data → Corridor sections or site grid?; Corridor sections or site grid? — Corridor → Calculate section areas by station; Corridor sections or site grid? — Grid → Compute grid-cell excavation volumes; Compute grid-cell excavation volumes → Sum grid cells; report excavation; Sum grid cells; report excavation → Report earthwork quantities and disposition; Calculate section areas by station → Higher accuracy or marked area change?; Higher accuracy or marked area change? — Yes → Middle-section area available?; Higher accuracy or marked area change? — No → Average end-area volume; Middle-section area available? — Yes → Prismoidal volume; Middle-section area available? — No → Survey midpoint; compute middle area; Survey midpoint; compute middle area → Prismoidal volume; Average end-area volume → Record interval cut / fill volumes; Prismoidal volume → Record interval cut / fill volumes; Record interval cut / fill volumes → Accumulate cut (+) and fill (−); Accumulate cut (+) and fill (−) → Plot cumulative volume vs station; Plot cumulative volume vs station → Read rises, falls and balance line; Read rises, falls and balance line → Pair equal ordinates for balanced hauls; Pair equal ordinates for balanced hauls → Final ordinate equals initial?; Final ordinate equals initial? — Yes → Project totals balance; Final ordinate equals initial? — No → Final ordinate higher?; Final ordinate higher? — Yes → Plan waste for surplus cut; Final ordinate higher? — No → Plan borrow for fill deficit; Project totals balance → Contract freehaul distance known?; Plan waste for surplus cut → Contract freehaul distance known?; Plan borrow for fill deficit → Contract freehaul distance known?; Contract freehaul distance known? — Yes → Compare balanced haul with freehaul; Contract freehaul distance known? — No → Hold: confirm contract freehaul; Compare balanced haul with freehaul → Haul exceeds freehaul distance?; Haul exceeds freehaul distance? — Yes → Classify excess haul as overhaul; Haul exceeds freehaul distance? — No → Haul within freehaul distance; Classify excess haul as overhaul → Report earthwork quantities and disposition; Haul within freehaul distance → Report earthwork quantities and disposition

  • Earthwork survey data: terminator. Start with longitudinal cross-sections or a site grid and the required volume accuracy.
  • Corridor sections or site grid?: decision. Use station-based cross-sections for route work and weighted grid cells for a borrow pit or site excavation.
  • Compute grid-cell excavation volumes: process. Apply the borrow-pit grid method using each corner depth weighted by the number of adjacent grid cells (one through four).
  • Sum grid cells; report excavation: process. Sum cell volumes consistently in cubic units. This site-grid branch reports excavation volume and does not create a station-based mass-haul diagram.
  • Calculate section areas by station: process. Determine the cut or fill area at each cross-section and the distance between adjacent stations.
  • Higher accuracy or marked area change?: decision. Use the project accuracy requirement and the difference between end areas to decide whether the prismoidal method is warranted.
  • Middle-section area available?: decision. The prismoidal formula requires an actual or computed middle cross-section area, not an assumed average of end areas.
  • Survey midpoint; compute middle area: process. Obtain the middle cross-section and calculate its area before using the prismoidal formula.
  • Average end-area volume: process. Use V = (A₁ + A₂)L/2 for the interval.
  • Prismoidal volume: process. Use V = L(A₁ + 4Aₘ + A₂)/6 with the measured or computed middle area Aₘ.
  • Record interval cut / fill volumes: process. Keep each interval's station limits, method, units, and cut or fill quantity.
  • Accumulate cut (+) and fill (−): process. Add cut and subtract fill volumes successively to obtain the cumulative mass ordinate at each station.
  • Plot cumulative volume vs station: process. Construct the mass-haul diagram with station on the horizontal axis and cumulative volume on the vertical axis.
  • Read rises, falls and balance line: process. A rising curve indicates net cut; a falling curve indicates net fill. Peaks and troughs mark cut/fill changes.
  • Pair equal ordinates for balanced hauls: process. A horizontal balance line intersects at equal cumulative volumes; the enclosed route interval balances cut against fill.
  • Final ordinate equals initial?: decision. Compare endpoint ordinates to determine whether total project cut and fill balance.
  • Project totals balance: process. Equal initial and final ordinates indicate no net project borrow or waste.
  • Final ordinate higher?: decision. With cut positive and fill negative, a higher final ordinate means net cut surplus.
  • Plan waste for surplus cut: process. Excess excavated material beyond fill demand must be disposed of off-site.
  • Plan borrow for fill deficit: process. Import material when project fill demand exceeds available cut.
  • Contract freehaul distance known?: decision. Use the freehaul distance specified for the project contract; do not assume a standard value.
  • Compare balanced haul with freehaul: process. Determine the haul distance for the balanced route segment and compare it with the specified freehaul distance.
  • Haul exceeds freehaul distance?: decision. Only haul beyond the specified freehaul distance is classified as overhaul.
  • Classify excess haul as overhaul: process. Record the contract-defined overhaul quantity or station-distance measure.
  • Haul within freehaul distance: process. Classify the balanced haul as within freehaul.
  • Hold: confirm contract freehaul: terminator. The lesson gives no numerical freehaul distance; obtain it from project contract documents before classifying overhaul.
  • Report earthwork quantities and disposition: terminator. Document the volume method, quantities, and units. Corridor work also records balanced sections, net borrow or waste, and contract-defined freehaul/overhaul; a site-grid branch reports excavation volume.
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.