Earthwork Volume Computations

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

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

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

End-Area Variables

  • VV: Volume of earthwork (cut or fill) in cubic meters (m3m^3).
  • A1,A2A_1, A_2: Cross-sectional areas at the ends of the segment (m2m^2).
  • LL: Distance between the two cross-sections (mm).
Note: This method tends to slightly over-estimate volumes, particularly when the end areas differ significantly.

2. Prismoidal Formula

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

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

Prismoidal Variables

  • AmA_m: Area of the cross-section exactly midway between A1A_1 and A2A_2.
  • It is not the average of the areas, but the area computed from the average of the corresponding linear dimensions of the end sections.

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

Earthwork Volume Calculator

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

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)

Grid Variables

  • AA: Area of one grid unit (square or rectangle).
  • h1h_1: Sum of heights (cut/fill) used by 1 rectangle.
  • h2h_2: Sum of heights used by 2 rectangles.
  • h3h_3: Sum of heights used by 3 rectangles.
  • h4h_4: Sum of heights used by 4 rectangles.

Mass Haul Diagram (MHD)

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.
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.
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