Earthwork and Volume Computations (FE Civil Surveying)

NCEES FE Civil spec: Surveying, C. Earthwork and volume computations. FE Reference Handbook: Civil Engineering chapter, Surveying section (earthwork formulas, mass haul diagram).

What the FE Civil Spec Covers

Section C covers the volume of soil to be cut or filled along a road or site, computed from cross sections taken at stations. Expect the average end area method, the prismoidal formula, simple pyramid volumes, conversions between cubic feet and cubic yards, and swell and shrinkage factors that link excavated volume to hauled and compacted volume. Mass haul diagram reading also appears.

Key Concepts

Stations: 1 station = 100 ft in US practice, so station 12+00 to 13+00 is 100 ft. Metric stationing is often 1 km per station (0+020 is 20 m).
End areas: the cut or fill area of each cross section, found by coordinates or by counting trapezoids.
Swell: loose volume = bank volume times (1 + swell). Shrinkage: compacted volume = bank volume times (1 minus shrinkage).
Mass haul: cumulative cut (positive) minus fill (negative) plotted against station. Rising segments are cut, falling segments are fill.

Core Formulas

Average end area, end areas A1 and A2 a distance L apart:

V = \frac{L ( A _{1} + A _{2} )}{2}

Prismoidal formula, with A_m the area of the section halfway between:

V = \frac{L ( A _{1} + 4 A _{m} + A _{2} )}{6}

Pyramid or cone (for example where a cut runs out to zero):

V = \frac{h A _{base}}{3}

Unit conversion:

1 yd^{3} = 27 ft^{3}

Where It Is in the FE Reference Handbook

The Civil Engineering chapter, Surveying section, lists the average end area, prismoidal and pyramid formulas next to a sample profile and a mass haul diagram. Swell and shrinkage factors are not tabulated, so problems give them. Solid volume formulas are also in the Mathematics chapter.

Worked Example

Cut end areas are 220 ft² at station 12+00 and 340 ft² at station 13+00. The area of a section measured at 12+50 is 275 ft². (a) Find the cut volume by average end area and by the prismoidal formula, in cubic yards. (b) The soil swells 20%. How many 12 yd³ truckloads are needed to haul the average end area volume?

Step 1, average end area. L = 1300 minus 1200 = 100 ft.

V = \frac{100 ( 220 + 340 )}{2} = 28, 000 ft^{3} = \frac{28 , 000}{27} = 1037 yd^{3}

Step 2, prismoidal. Use the measured mid-section, not the average of the end areas:

V = \frac{100 ( 220 + 4 \times 275 + 340 )}{6} = 27, 667 ft^{3} = 1025 yd^{3}

Step 3, loose volume and trucks.

V_{loose} = 1037 \times 1.20 = 1244 yd^{3} \frac{1244}{12} = 103.7 \Rightarrow 104 loads

The average end area result (1037 yd³) is about 1.2% higher than the prismoidal result (1025 yd³), which is typical. Round truckloads up, since a partial load is still a trip.

Exam Strategy

Read whether the question wants bank, loose or compacted volume, and whether it wants cubic feet, cubic yards or cubic meters. Many wrong answer choices are the right volume in the wrong condition or unit.

When a cut section transitions to a fill section between stations, the cut tapers to zero somewhere in between. Treat that end as a pyramid (V = hA/3) or apply the average end area formula with one area equal to zero, depending on what the question specifies; the two give different answers, so follow the stated method.

On a mass haul diagram, the vertical distance between a peak or valley and the balance line is the volume moved, and the horizontal distance between balance points is the maximum haul distance. Free haul is the distance contractors move material without extra payment; anything beyond it is overhaul, usually priced per station-yard.

Common Mistakes

Averaging A1 and A2 and calling it A_m. In the prismoidal formula A_m must be the actual middle section.
Dividing cubic feet by 9 (that is square feet to square yards). Cubic yards need 27.
Applying swell to compacted fill or shrinkage to loose haul volume. Start from bank volume.
Reading the station difference wrong: 12+00 to 13+50 is 150 ft.
Adding cut and fill areas in the same section instead of computing them separately.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

Two earthwork cross-sectional end areas are 40 m² and 60 m², spaced 100 m apart. What is the volume by the average-end-area method? See the worked solution or practice the Surveying deck
1,000 m³ of bank soil swells by 25% when excavated. Trucks carry 10 m³ loose each. The number of truckloads needed is: See the worked solution or practice the Construction & Scheduling deck
A rectangular trench is 100 m long, 1.5 m wide, and 2 m deep, with vertical sides. What is its in-place excavation volume? See the worked solution or practice the Construction & Scheduling deck

Related Calculators and Examples

Frequently Asked Questions

Is the average end area method exact?

No. It is exact only when the two end areas are equal or the solid is a prism. For a pyramid-like shape it overestimates the volume. The prismoidal formula is more accurate, but it needs the true mid-section area.

What is the difference between bank, loose and compacted volume?

Bank volume is soil in place before excavation. Loose volume is larger because excavated soil swells. Compacted volume is usually smaller than bank volume because compaction densifies the soil (shrinkage). Always state which condition a volume refers to.

What does a balance point on a mass haul diagram mean?

The mass haul ordinate is the running total of cut minus fill. Where the curve crosses a horizontal balance line, the cut and fill between those stations balance, so material can be moved within that stretch without borrow or waste.

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