Project Sitework

A. Excavation and embankment

Earthwork: Cut/Fill, Shrinkage, Swell and the Mass Diagram

Average-end-area volumes, the bank/loose/compacted state changes that govern borrow and haul, and how the mass-haul diagram sets balance points and overhaul.

Earthwork is where a civil project's biggest single cost is won or lost, and where the PE rewards engineers who can keep three different volumes of the ·same· dirt straight. A cubic yard of soil shrinks when you compact it into an embankment and swells when you load it loose into a truck, so 'how much fill do I need' and 'how many truckloads is that' are two different numbers — and confusing them is the classic exam trap. This concept builds the chain from cross-section areas to volumes to the mass-haul diagram, the graphical tool that tells you where to balance cut against fill and when hauling becomes overhaul. It follows the NCEES PE Civil Reference Handbook §2.1 Earthwork Construction and Layout and §2.1.4 Earthwork Balancing and Haul Distances.

Volumes from cross sections: average end area

Field earthwork volumes come from cross sections cut at regular stations (commonly every

100\,\text{ft}

). The workhorse is the average-end-area method: average the two end areas and multiply by the distance between them. It is exact for a prism and slightly ·over·-estimates when the section is changing nonlinearly (the more accurate prismoidal formula uses the midsection area). On the exam, use average end area unless the problem hands you a midsection area or says 'prismoidal.' Treat cut and fill as separate volume tallies — never net them inside a single station because cut and fill soils are paid and moved differently.

V = \dfrac{A_1 + A_2}{2}\,L

Three states of soil: bank, loose, compacted

The same particles occupy different volumes in the ground (bank measure, BCY), loose in a truck (loose measure, LCY), and compacted in the fill (compacted measure, CCY). What is conserved across all three is dry weight, not volume, so the master relationship is

\gamma_B V_B = \gamma_L V_L = \gamma_C V_C

. Because loose soil is fluffed up it has the lowest unit weight (

\gamma_L < \gamma_B

), and engineered fill is the densest (

\gamma_C > \gamma_B

). Anchor every conversion to this weight identity and you can never invert a ratio.

\gamma_B V_B = \gamma_L V_L = \gamma_C V_C

Swell, shrinkage, and load factor

Swell measures the bulking from bank to loose; shrinkage measures the densification from bank to compacted. The PE Civil Reference Handbook defines them on a unit-weight basis:

\text{swell}\,(\%) = (\gamma_B/\gamma_L - 1)\times 100

and

\text{shrinkage}\,(\%) = (1 - \gamma_B/\gamma_C)\times 100

. The load factor

LF = \gamma_L/\gamma_B

converts bank volume to loose haul volume (

V_L = V_B/LF

), and the shrinkage factor

\gamma_B/\gamma_C

converts bank to compacted. The handbook explicitly warns that published definitions vary, so on test day default to its forms and, when in doubt, fall back to weight conservation.

\text{swell}\,(\%) = \left(\dfrac{\gamma_B}{\gamma_L}-1\right)100, \qquad \text{shrinkage}\,(\%) = \left(1-\dfrac{\gamma_B}{\gamma_C}\right)100

Borrow, waste, and the import/export decision

Once volumes are in a common measure you can compare project cut to project fill. If compacted-fill demand exceeds available cut (expressed as compacted equivalent), the deficit is made up from a borrow pit; if cut exceeds fill, the surplus is wasted (exported). The trap is comparing apples to oranges: convert everything to one state — usually compacted (in-place) yards for balancing and loose yards for hauling/trucking. A borrow quantity must be grossed up from compacted demand to bank (to size the pit) and again to loose (to count trucks).

The mass-haul diagram

Plot cumulative earthwork volume up-station, with cut positive and fill negative (after applying shrinkage so cut and fill are in the same measure). Rising segments are net cut, falling segments are net fill. Where the curve crosses its baseline the cumulative volume returns to zero — a ·balance point·: cut between two balance points exactly fills the embankment between them. Peaks and valleys (maxima/minima) line up with ·grade points· on the profile where existing and final grade meet. Any horizontal chord of the curve cuts off equal cut and fill volumes; the horizontal distance from the cut's centroid to the fill's centroid is the average haul distance, and dividing the enclosed area by the curve height gives that haul distance directly.

Freehaul, overhaul, and balancing

Contracts include a ·freehaul· distance: moving material up to that distance is part of the base unit price. Material hauled farther incurs ·overhaul·, priced per station-yard (one cubic yard moved one

100\,\text{ft}

station beyond freehaul). The overhaul distance is the centroid-to-centroid haul minus the freehaul distance, and the pay quantity is overhaul volume times overhaul distance. Choosing where to set the horizontal balance lines — and whether to borrow locally rather than overhaul a long way — is the optimization the mass diagram makes visible.

\text{Overhaul} = V_{oh}\times(d_{\text{CG-CG}} - d_{\text{freehaul}})

Exam strategy

First decide which soil state the question wants and convert everything to it before you do anything else — fill demand in compacted yards, trucking in loose yards, pit size in bank yards. Write the weight-conservation identity

\gamma_B V_B=\gamma_L V_L=\gamma_C V_C

in the margin so a forgotten swell/shrinkage direction can't bite you. For average end area, keep cut and fill columns separate and watch the units: areas in

\text{ft}^2

times length in

\text{ft}

gives

\text{ft}^3

, then divide by 27 for cubic yards. On mass-diagram questions, remember balance points are baseline crossings (zero cumulative volume) and grade points are the peaks/valleys — and that average haul distance = enclosed area ÷ ordinate height.

Key equations

Average end-area volume

V = \dfrac{A_1+A_2}{2}\,L

Volume between two cross sections; $A_1,A_2$ in $\text{ft}^2$ , $L$ in $\text{ft}$ , $V$ in $\text{ft}^3$ (÷27 for cy). Slightly overestimates for nonlinear sections.

Prismoidal volume

V = \dfrac{L}{6}\left(A_1 + 4A_m + A_2\right)

More accurate volume using midsection area $A_m$ ; use when the problem supplies $A_m$ or asks for prismoidal.

Weight conservation across states

\gamma_B V_B = \gamma_L V_L = \gamma_C V_C

Master identity. $B$ = bank (in-situ), $L$ = loose (hauled), $C$ = compacted (fill). Dry weight is conserved, volume is not.

Swell (percent)

\text{swell}\,(\%) = \left(\dfrac{\gamma_B}{\gamma_L}-1\right)\times 100

Bulking from bank to loose. $\gamma_L<\gamma_B$ so swell is positive; loose volume $V_L = V_B(\gamma_B/\gamma_L)$ .

Shrinkage (percent)

\text{shrinkage}\,(\%) = \left(1-\dfrac{\gamma_B}{\gamma_C}\right)\times 100

Densification from bank to compacted. $\gamma_C>\gamma_B$ ; compacted volume $V_C = V_B(\gamma_B/\gamma_C)$ .

Load factor

LF = \dfrac{\gamma_L}{\gamma_B}, \qquad V_L = \dfrac{V_B}{LF}

Converts bank to loose for trucking; $LF<1$ . Some texts define load factor as the reciprocal — confirm against weight conservation.

Relative compaction

RC = \dfrac{\gamma_{d,\text{field}}}{\gamma_{d,\max}}\times 100

Field acceptance ratio of field dry density to laboratory maximum (Proctor); specs commonly require $\ge 95\%$ .

Water to add for moisture

G = \dfrac{\gamma_{d}\,V\,(w_{des}-w_{borrow})}{8.33}

Gallons of water; $\gamma_d$ in $\text{lb/ft}^3$ , $V$ in $\text{ft}^3$ , $w$ as decimal moisture content, $8.33\,\text{lb/gal}$ .

Borrow-pit grid square volume

V = \dfrac{A_{grid}}{4}\,(a+b+c+d)

Cut volume of one full grid square from corner cut depths $a,b,c,d$ ; $A_{grid}$ is the plan area of the square.

Triangular spoil bank volume

V = \dfrac{H^2 L}{\tan R}

Loose spoil pile of length $L$ , height $H$ , angle of repose $R$ ; base width $B = 2H/\tan R$ .

Overhaul quantity

\text{Overhaul} = V_{oh}\,(d_{\text{CG-CG}}-d_{fh})

Pay quantity in station-yards: overhaul volume times the centroid-to-centroid haul beyond the freehaul distance $d_{fh}$ .

Average haul distance

d_{avg} = \dfrac{\text{area under mass curve}}{\text{ordinate height}}

Graphical haul distance for a balanced segment of the mass-haul diagram.

Worked examples

Average-end-area cut volume

Problem. Two adjacent cross sections $100\,\text{ft}$ apart show cut end areas of $120\,\text{ft}^2$ and $180\,\text{ft}^2$ . Find the cut volume in cubic yards.

Solution. Average the end areas and multiply by the spacing:

V = \dfrac{A_1+A_2}{2}L = \dfrac{120+180}{2}(100) = 150\times100 = 15{,}000\,\text{ft}^3.

Convert to cubic yards:

15{,}000/27 = 555.6\,\text{cy}

. **Answer:

556\,\text{cy}

(bank measure).** Sanity check: a uniform

150\,\text{ft}^2

section over

100\,\text{ft}

is

15{,}000\,\text{ft}^3

, and

27\,\text{ft}^3=1\,\text{cy}

, so a few hundred cy is the right order of magnitude.

V = \dfrac{120+180}{2}(100) = 15{,}000\,\text{ft}^3 = 556\,\text{cy}

Borrow, haul, and truckloads

Problem. An embankment needs $10{,}000\,\text{cy}$ of compacted fill. Borrow soil has unit weights $\gamma_B = 105\,\text{pcf}$ (bank), $\gamma_L = 90\,\text{pcf}$ (loose), $\gamma_C = 122\,\text{pcf}$ (compacted). Find the bank volume to excavate, the loose volume to haul, and the number of $12\text{-cy}$ truckloads.

Solution. Use weight conservation

\gamma_B V_B = \gamma_L V_L = \gamma_C V_C

. Bank volume:

V_B = V_C\dfrac{\gamma_C}{\gamma_B} = 10{,}000\dfrac{122}{105} = 11{,}619\,\text{cy (bank)}

. Loose volume:

V_L = V_C\dfrac{\gamma_C}{\gamma_L} = 10{,}000\dfrac{122}{90} = 13{,}556\,\text{cy (loose)}

. Truckloads:

13{,}556/12 = 1{,}130

loads. **Answers:

11{,}600\,\text{BCY}

,

13{,}600\,\text{LCY}

,

1{,}130

loads.** Check: shrinkage

=(1-105/122)\times100 = 13.9\%

and swell

=(105/90-1)\times100 = 16.7\%

; loose > bank > compacted in volume, as it must be.

V_B = 10{,}000\dfrac{122}{105}=11{,}619\,\text{cy},\quad V_L = 10{,}000\dfrac{122}{90}=13{,}556\,\text{cy}

Overhaul payment

Problem. A balanced reach of a mass-haul diagram moves $6{,}000\,\text{cy}$ of overhaul material. The centroid of the cut mass is $1{,}400\,\text{ft}$ from the centroid of the fill mass, and the contract freehaul distance is $500\,\text{ft}$ . At an overhaul price of $0.40 per station-yard, find the overhaul cost.

Solution. Overhaul distance beyond freehaul:

d = 1{,}400 - 500 = 900\,\text{ft} = 9\,\text{stations}

. Overhaul quantity:

6{,}000\,\text{cy}\times 9\,\text{sta} = 54{,}000\,\text{station-yards}

. Cost:

54{,}000\times\$0.40 = \$21{,}600

. **Answer: $21{,}600.** Check: only the haul beyond the

500\,\text{ft}

freehaul is paid as overhaul; the first

500\,\text{ft}

is already in the base price, so using the full

1{,}400\,\text{ft}

would over-bill by

6{,}000\times5\times0.40=\$12{,}000

.

\text{Overhaul} = 6{,}000\,(9) = 54{,}000\,\text{sta-yd}

Common pitfalls

•Mixing soil states: comparing bank cut directly to compacted fill, or counting truckloads off compacted yards. Convert everything to one state first, and gross compacted fill UP to bank/loose for borrow and trucking.
•Inverting swell or shrinkage: loose volume is LARGER than bank ( $\gamma_L<\gamma_B$ ), compacted is SMALLER ( $\gamma_C>\gamma_B$ ). When unsure, fall back to $\gamma_B V_B=\gamma_L V_L=\gamma_C V_C$ .
•Forgetting the ÷27: areas in $\text{ft}^2$ times length in $\text{ft}$ give $\text{ft}^3$ ; divide by 27 to report cubic yards.
•Confusing balance points with grade points: balance points are where cumulative volume returns to zero (baseline crossings); grade points are the peaks/valleys where existing and final grade meet.
•Billing the full centroid-to-centroid distance as overhaul. Overhaul distance is the haul MINUS the freehaul distance; the freehaul portion is in the base price.
•Netting cut and fill within one station of average-end-area work. Keep cut and fill as separate tallies — they are different pay items and move differently.
•Using the prismoidal formula when only end areas are given. Default to average end area unless a midsection area is supplied or 'prismoidal' is stated.

References

NCEES PE Civil Reference Handbook — §2.1 Earthwork Construction and Layout
NCEES PE Civil Reference Handbook — §2.1.2 Earthwork Volumes
NCEES PE Civil Reference Handbook — §2.1.4 Earthwork Balancing and Haul Distances
Caterpillar Performance Handbook — swell/shrinkage and load-factor conventions for haul estimating

B. Construction site layout and control

Site Layout, Staking and Construction Control

Horizontal and vertical control, differential leveling with HI/BS/FS bookkeeping, and translating design grades into offset, grade, and slope stakes in the field.