Mass haul diagrams on the PE Civil Transportation exam

A mass diagram shows where the earth goes. On the Transportation exam you either build one from end areas and find the balance point and the haul, or read one that's drawn for you: where cut turns to fill, how far earth travels, how much is overhauled, and what is wasted or borrowed. The Handbook's Construction chapter covers it under Earthwork Balancing and Haul Distances, which defines the mass diagram, grade points, balance points, freehaul and overhaul.

What you actually need to own

• The ordinate. The diagram plots cumulative volume moving up-station, with cut positive and fill negative. It rises through cut and falls through fill. A maximum is where cut turns to fill and a minimum is where fill turns to cut. Both sit at grade points, where the final grade meets the existing ground.
• One soil state. Cut is measured bank and fill compacted, so convert before you accumulate. To plot in compacted cubic yards, multiply each cut volume by (1 − shrinkage) and leave the fill as it is. Plotting in bank yards instead (fill ÷ (1 − shrinkage)) works too. Use whichever state the problem asks for, and use one state for every ordinate. The conversions themselves are on earthwork volumes, swell and shrinkage.
• Volumes between stations. Average end area, V = L(A₁ + A₂)/2, ÷ 27 for cubic yards. Where the section passes from cut to fill, split the interval at the grade point, where the end area is zero.
• Balance points and horizontal lines. Where the curve crosses its baseline, cumulative cut equals cumulative fill. Any horizontal line cutting the curve at two points joins a cut and a fill that balance, and its length is how far that earth travels.
• Haul. The height of the diagram is a volume. The area under it is volume × distance, and area ÷ height is the average haul distance. In station-yards (1 yd³ moved 100 ft), haul = area in yd³·ft ÷ 100.
• Freehaul and overhaul. Freehaul is the distance covered by the contract's base price. Overhaul is material moved farther than that. Fit a horizontal line exactly one freehaul long inside the loop: the earth between that line and the baseline travels beyond freehaul. The Handbook defines overhaul distance as the distance between the centers of gravity of that remaining excavation and embankment, minus the freehaul distance. Overhaul quantity = overhaul volume × overhaul distance.
• Waste or borrow. Overhauling a cubic yard costs more for every extra station. Once that cost passes the cost of borrowing a cubic yard, it's cheaper to waste the cut and borrow the fill. The Handbook doesn't print this limit, so a problem that needs it states it, for example as freehaul plus (borrow cost per yd³ ÷ overhaul cost per station-yard) stations. Fit a horizontal line of that length inside the loop: the cut between that line and the baseline is wasted, and the fill it would have supplied is borrowed.

Where people lose points

• Shrinkage on the wrong side. Applying (1 − shrinkage) to the fill instead of the cut raises the ordinates and moves the balance point. In one problem it shifts from STA 26+65 to 27+89 and the haul goes from about 33,400 to 46,800 station-yards.
• A triangle for a flat-topped diagram. Where the road runs at grade, the ordinate holds level. Half the base times the height drops that whole rectangle. Sum the trapezoids.
• Hauling past the balance point. On an unbalanced job, the fill beyond the last balance point is borrow. The distance between the centers of gravity of all the tabulated cut and all the fill overstates the haul.
• Reading a haul limit as a distance from a balance point. Freehaul and the economic limit are lengths of horizontal lines fitted inside a loop, with both ends on the curve. Read the stations where the curve actually reaches that height on each side.
• Freehaul alone as the waste limit. The height of the freehaul line marks earth that's overhauled, not wasted. Waste starts at the economic limit.
• Maximum read as "deepest cut." The deepest cut is where the curve rises most steeply. At a maximum the cut has just run out.

How to study it

Take a short profile with cut, an at-grade stretch and fill. Compute the interval volumes, convert the cut to compacted, plot the ordinates, interpolate the balance point and total the haul from trapezoids. Then add a freehaul and a borrow price and find the waste. The profile that sets the cut and fill comes from vertical curve design, and what goes on top is AASHTO-93 flexible pavement design.

Part of the Geotechnical and Pavement area of the PE Civil Transportation exam. → Start practicing free — the whole practice bank, free with an account.