Earthwork quantities

Earthwork is often the largest single quantity on civil and site development projects, and one of the biggest sources of cost overruns and claims. The quantities drive the estimate, the equipment selection, the schedule and, on unit-price contracts, the payments. Yet soil does not have a fixed volume: the same material occupies one volume in the ground, a larger volume in a truck and a smaller volume after compaction. This lesson explains the states of soil, swell and shrinkage, and the standard methods of computing earthwork volumes: average end areas, the prismoidal formula, the grid method, simple shapes for trenches and footings, and surface-to-surface comparison of digital terrain models.

Bank, loose and compacted volume

Earthwork quantities are always stated in one of three conditions, and a quantity without its condition is meaningless.

• Bank volume (bank cubic metres, BCM): the material in its natural state in the ground, before excavation. Excavation is normally measured and paid in bank volume.
• Loose volume (loose cubic metres, LCM): the material after excavation, in a bucket, truck or stockpile. It is larger because the soil particles are disturbed and contain more voids. Hauling is planned in loose volume.
• Compacted volume (compacted cubic metres, CCM): the material after placement and compaction in a fill. It is usually smaller than the bank volume for soils, because compaction removes more voids than nature left, but larger for blasted rock.

The relationships are expressed as percentages of the bank volume. Swell is the increase from bank to loose; shrinkage is the decrease from bank to compacted. The load factor is the ratio of bank to loose volume, used to convert truck or bucket loads back to bank volume.

The reverse question is equally common: how much bank material is needed to build a fill? Divide the compacted volume by (1 − shrinkage). A 10,000 compacted unit embankment of material with 15 per cent shrinkage needs 10,000 ÷ 0.85 = 11,765 bank units.

Specifications state how quantities are measured for payment. Most Canadian provincial and municipal specifications pay excavation in place, measured by cross-sections or surfaces before and after, and pay imported fill either by compacted volume in place or by weight from scale tickets. Know which applies before you price a job.

Cross-section areas

For linear works such as roads, channels and long embankments, volumes are calculated from cross-sections at regular stations. The area of cut and the area of fill are measured on each cross-section. Software computes the areas from coordinates (the same shoelace method used for traverses), but simple sections can be done by hand by dividing them into triangles and trapezoids.

The average end area method

The average end area method assumes that the volume between two cross-sections is their average area multiplied by the distance between them. It is simple, it is the standard method in most specifications, and it is accurate enough when the sections are close together and similar in shape.

Cut and fill volumes are calculated separately. Where a section changes from cut to fill, the sections between include a transition: the volume tapers to zero, and many estimators treat each part as a pyramid, using one third of the area times the length, rather than an average.

In the table, the cut volume between 0+040 and 0+060 is computed as an average end area even though cut runs out; a more careful estimator would compute the transition separately. The total cut is 680 m³ and the total fill 570 m³, both before applying swell and shrinkage, which the next lesson uses to balance cut against fill.

The prismoidal formula

The average end area method overestimates volume when the shape of the sections changes between stations, for example when one end is wide and shallow and the other narrow and deep, because the true solid is a prismoid rather than a prism. The prismoidal formula uses the area at the midpoint, Am, which must be measured or computed from the mid-section, not averaged.

For most projects the difference is small and the average end area method is specified. The prismoidal formula is used for expensive materials such as rock excavation, and where sections change shape sharply.

The grid, or borrow pit, method

For building pads, parking lots, borrow pits and other areas without a single alignment, volumes are computed on a grid. The site is divided into squares, typically 5 to 20 metres, and the cut or fill depth is found at each grid corner by comparing existing and design elevations. The volume of each square is its area times the average of its four corner depths. Because interior corners are shared by four squares, edge corners by two and outer corners by one, the calculation can be simplified by weighting each depth by the number of squares that share it.

Here A is the area of one grid square and h₁, h₂, h₃ and h₄ are depths at corners shared by one, two, three and four squares respectively. The h₃ term appears only on irregular grids with inside corners.

A finer grid gives a more accurate result on uneven ground. Where the grid crosses from cut to fill, squares containing both must be split, or the cut and fill computed separately from their own depths, setting the other to zero.

Footings, basements and topsoil

Structural excavation for basements and footings is calculated from the excavation geometry, not just the footprint of the structure. Allowances are added for working space around footings and walls (often 0.6 to 1.0 metre), and for side slopes or shoring as required by the excavation safety regulations. A basement excavation with sloped sides is a frustum, and its volume can be approximated by the prismoidal formula using the bottom area, the top area and the mid-depth area.

Topsoil stripping is usually a separate item, computed from the stripped area times the stripping depth. Topsoil is stockpiled for reuse, and stockpiles need space: a stockpile of loose topsoil occupies more volume than the bank quantity and needs room for slopes and access.

Digital terrain models and surface comparison

On most projects today, earthwork quantities are computed by software comparing two digital terrain models (DTMs), surfaces built from triangles connecting surveyed points (a triangulated irregular network, or TIN). The existing ground surface comes from a topographic survey, drone survey or lidar; the design surface comes from the grading design. The software computes the volume between them, cut where the existing is above the design and fill where it is below. The same method measures progress (existing against current surface) and final quantities (original against as-built).

Surface comparison is fast and accurate, but its results are only as good as the surfaces. Common problems include an existing surface that does not include breaklines along ditches, curbs and the tops and toes of slopes, so the triangles cut across them; a surface that is the top of vegetation or snow rather than the ground; design surfaces that represent finished grade when the earthwork stops at subgrade, so the pavement structure and topsoil thickness must be subtracted; and surfaces in different coordinate systems or vertical datums. Always review the surfaces visually and spot-check them against field shots.

• Ask for the earthwork report to state which surfaces were compared, their dates and sources.
• Confirm whether the design surface is finished grade or subgrade.
• Check that quantities are stated as bank, and apply swell and shrinkage separately.
• Compare the software total with a rough hand estimate, such as average depth times area.