Earthwork volumes, swell and shrinkage on the PE Civil WRE exam

Earthwork is dry-land arithmetic inside a water exam. Borrow-pit take-offs belong to Project Planning, and bank/loose/compacted conversions and mass diagrams run through Project Sitework, the largest area on the exam. The math is light, so the points come down to calculating in the right soil state.

What you actually need to own

• The borrow-pit grid. One grid square's volume is its area times the average of its four corner cut depths. Over a whole grid, weight each corner by how many squares share it: V = (A/4)(Σh₁ + 2Σh₂ + 3Σh₃ + 4Σh₄), where hₙ is a depth at a corner shared by n squares. With ft² and ft, divide by 27 for cubic yards. The Handbook's Earthwork Volumes section also gives wedge and cone shapes for partial squares at the edge.
• Three soil states. Bank (in place), loose (hauled) and compacted (placed in the fill):
  • loose = bank × (1 + swell);
  • compacted = bank × (1 − shrinkage);
  • load factor = loose unit weight ÷ bank unit weight = bank volume ÷ loose volume, so load factor = 1 / (1 + swell).
• The Handbook's definitions. Published definitions of the swell and shrinkage factors vary. The Handbook's Excavation and Embankment section defines them through unit-weight ratios: swell factor = bank unit weight ÷ loose unit weight (greater than 1, equal to 1 + swell), and shrinkage factor = bank unit weight ÷ compacted unit weight (less than 1, equal to 1 − shrinkage). It notes that exam questions follow these forms unless a problem says otherwise, so use the Handbook's versions.
• Cross-section volumes. Average end area: V = L(A₁ + A₂)/2. Prismoidal: V = L(A₁ + 4Aₘ + A₂)/6, where Aₘ is the area of the actual midsection, not the average of the two ends. With 100-ft stations and end areas in ft², divide by 27.
• The mass diagram. It plots cumulative volume up-station, with cut positive and fill negative. A rising line is cut and a falling line is fill. Peaks and valleys sit at grade points. Where the curve crosses the baseline, cut and fill balance. Area divided by height gives the average haul. Overhaul distance is the distance between the centers of mass, minus the freehaul distance.
• Truckloads. Trucks carry loose volume: loads = bank × (1 + swell) ÷ truck capacity, rounded up. Bank 1,000 yd³ at 25% swell is 1,250 loose yd³, or 90 loads in 14-yd³ trucks.

Where people lose points

• Corner weights. An interior corner shared by four squares counts four times. An edge corner counts twice. A corner at a notch in an irregular pit can count three times.
• Load factor read as swell. A load factor of 0.75 means 33% swell (1/0.75 − 1), not 25%.
• The wrong state for the question. The pit is measured in bank, trucks haul loose, and the embankment is specified compacted. To find the borrow needed for a fill, divide the compacted volume by (1 − shrinkage) to get bank. Then apply swell only if the question is about hauling.
• ft³ vs. yd³. A factor of 27 lost inside a grid or end-area sum.
• Mixing states on a mass diagram. Before accumulating, convert fill to the same measure as the cut, usually bank through shrinkage. Otherwise the balance points shift.
• Rounding trucks down. A fraction of a load is still a trip.

How to study it

Draw the bank → loose → compacted chain once, with swell on one arrow and shrinkage on the other. Then label every number you compute with its state. Work a small borrow-pit grid by hand, marking each corner's multiplier, until the weighting is automatic. Tie haul volumes to the cost and scheduling side of Project Planning for unit-cost problems, and see lateral earth pressure and trench safety for the excavation side of the same site.

Part of the Project Planning area of the PE Civil WRE exam. → Start practicing free — the whole practice bank, free with an account.