Earthwork Volumes and Mass Haul: Method

Earthwork Volumes and Mass Haul

This page describes the four computations of the tool: the volume of cut and fill between cross sections and the mass diagram that balances them; the elements and staking of a simple circular curve; the elevations of a parabolic vertical curve; and the end area of a roadway template against a surveyed ground line. The methods are those of the public-domain surveying manuals of the U.S. Army (FM 5-233, Construction Surveying) and the U.S. Navy (NAVEDTRA 14337, Engineering Aid 2), and the default inputs of every sub-tool are a printed example of those manuals, so the page reproduces their numbers on first load.

Stations

Distances along the centre line are measured from the origin of the survey in full stations of 100 ft (USCS) and written as the full station plus the remainder: 1,641.33 ft is station 16+41.33, 350 ft is 3+50 (a plus station). The tool takes every station as a plain distance (type 1641.33) and prints the +00 form. In SI the full station is taken as 100 m and the label is written kilometre + metres (1+641.33); the degree of curve is then the central angle of a 100 m arc or chord and the haul unit is the station-m³.

End areas

The cross section at a station is a vertical plane perpendicular to the centre line; the area between the ground line and the design template is the end area, computed separately for cut (ground above grade) and fill (ground below grade), as standard practice requires (FM 5-233 chapter 4, NAVEDTRA 14337 chapter 10). For a template with a flat subgrade of width W at the grade elevation and side slopes of s horizontal to 1 vertical, the catch point on each side is where the side-slope line from the hinge (the edge of the subgrade) first meets the ground line. The tool integrates the difference between the ground line and the template polyline segment by segment, which is exact for piecewise-linear lines and is the same computation as the manual's area by coordinates (the cross-multiplication of successive coordinates, which gives twice the area of a closed polygon). Two closed forms are printed beside it when they apply: the level section A = c (W + s c) for a ground line at the uniform height c above grade, and the three-level section A = (W/4)(h_l + h_r) + (c/2)(d_l + d_r), where h and d are the heights and centre-line distances of the slope stakes and c the centre-line cut; the latter is exact only when the ground runs straight from each slope stake to the centre line, and the difference from the integrated area is the effect of the intermediate ground points. The manual's station 305 is the default: its coordinate method gives 540.35 ft² with each product rounded to a tenth (540.32 ft² exactly) and its three-level formula 507.71 ft²; the slope stakes of that section sit at about 1.2:1 from the hinges although the field notes call the side slope 1.5:1, and the default slopes reproduce the stakes.

Volumes between sections

Average end area. V = L (A₁ + A₂)/2 between two sections L apart, the pay method of every route-survey manual; in USCS the cubic feet are divided by 27 for cubic yards. It treats the solid as a prism whose section varies linearly in area and overstates the volume of a solid whose dimensions vary linearly (a frustum): for a pyramid it gives 1.5 times the true volume, for a frustum of 10 × 10 to 5 × 5 about 7 % too much.

Prismoidal formula. V = L (A₁ + 4 A_m + A₂)/6 with A_m the area of the section midway between the two ends, exact for any solid bounded by planes (a prismoid) and in particular whenever every dimension of the section varies linearly. A_m is not the mean of the end areas: the tool takes a measured middle section where one is entered, and otherwise the similar-sections value A_m = [(√A₁ + √A₂)/2]², the area of the section whose linear dimensions are the means of the end sections, which turns the prismoidal formula into the exact frustum volume L (A₁ + √(A₁A₂) + A₂)/3. The prismoidal correction printed is the average-end-area volume less the prismoidal one.

Shrinkage and swell. A yard of earth in place yields more when loosened and less when compacted into a fill (NAVEDTRA 14337 Table 10-1: 100 yd³ of in-place sand give 111 yd³ loose and 95 yd³ compacted; a compacted fill of loam needs 1.11 yd³ of bank cut per yard of fill, blasted rock 0.77). The fill factor multiplies every fill volume to express it in bank measure before the cumulative column is formed, so cut and fill are compared in the same measure; the manual's table of cumulative yardage applies no factor and the default is 1.00.

Mass diagram

The table of cumulative yardage lists the stations, the end areas, the volumes of cut (positive) and fill (negative) between them and the running algebraic sum; sections are added at every plus station where the cut or the fill runs out so that the turning points of the sum lie on the table. Plotted against station, the cumulative sum is the mass diagram: a rising curve is cut, a falling curve fill, a maximum the grade point from cut to fill and a minimum the grade point from fill to cut. Any horizontal line is a balance line: between two of its crossings the cut and the fill are equal, and the ordinate of a loop above or below it is the volume moved within the loop. Material of a loop above the balance line is hauled forward (in the direction of stationing), below it backward. The line of zero yardage is the balance line used; where the diagram ends off that line the net is a surplus to waste or a deficit to borrow.

Haul. The haul of a loop (volume × distance) is the area between the mass curve and the balance line. The free-haul distance (500 ft in NAVEDTRA 14337 Figure 10-10) is the distance included in the excavation price; drawn in a loop as the horizontal line of that length cut off by the curve, the volume above it (peak less the line's ordinate) moves less than the free-haul distance, and the band between the line and the balance line is the overhauled material. By the planimeter method the overhaul in station-yards is [band area − (band volume)(free-haul distance)]/100; the centroid method gives the same number as the band volume times the distance between the centroids of its cut and its fill masses less the free-haul distance, and the tool prints both. The tool solves the exact piecewise-linear ordinate (bisection for the level at which the loop is exactly the free-haul distance wide, exact trapezoidal integrals for the areas), so the two methods agree to round-off. With a borrow cost per cubic yard and an overhaul rate per station-yard the limit of economic haul is LEH = D_fh + 100 c_b/r: the line of that length drawn in the loop marks the volume (below it) cheaper to waste at the cut and replace by borrow at the fill than to haul, and the overhaul is then counted between the two lines only. The manual's Figure 10-10 does without the LEH and wastes the cut 0+00 to 1+00 and borrows the fill 6+00 to 8+00 of its first two loops by inspection, which is what a limit of economic haul close to the free-haul distance gives.

Simple horizontal curve

Two tangents meeting at the PI with the intersection angle I (equal to the central angle) are joined by a circular arc from the PC to the PT. The sharpness is the degree of curve D: by the arc definition (highways) the central angle subtended by a 100-ft arc, R = 5,729.58/D; by the chord definition (railroads, and the military manuals) the angle subtended by a 100-ft chord, R = 50/sin(D/2) (5,729.65 ft at 1°; FM 5-233 page 3-4). Elements (FM 5-233 page 3-6): T = R tan(I/2), L = 100 I/D (the arc length under the arc definition, the length along the chords under the chord definition; the true arc R I is printed too), E = R (1/cos(I/2) − 1) = T tan(I/4), M = R (1 − cos(I/2)), LC = 2 R sin(I/2), PC = PI − T, PT = PC + L. The curve is staked by deflection angles from the PC: for a chord C the deflection is d = (D/2)(C/100) degrees = 0.3 C D minutes, exact for the arc definition and the customary approximation for the chord definition; the chords are 100, 50, 25 or 10 ft for D up to 3°, 8°, 16° and above, with subchords at the ends, and the deflections accumulate to I/2 at the PT, the check on the arithmetic. The true chord of each deflection is 2 R sin d. The default is the FM 5-233 example (PI 18+00, I 45°, D 15°, chord definition: R 383.07, T 158.67, L 300.00, E 31.56, M 29.16, LC 293.19 ft, PC 16+41.33, PT 19+41.33); the NAVEDTRA 14337 arc-definition example (I 75°, D 15°, half stations) is in the test script.

Vertical curve

Grade lines g₁ and g₂ (percent, rising positive in the direction of stationing) meeting at the PVI are joined by a parabola, because its grade changes at a constant rate and its offsets from the tangent vary as the square of the distance. For an equal-tangent curve of length L the PVC and PVT lie L/2 before and after the PVI; the middle offset at the PVI is e = L (g₂ − g₁)/8 (L in stations, grades in percent, e in feet: FM 5-233 writes V_m = L/8 (G₂ − G₁) and checks it as ½[(elev PVC + elev PVT)/2 − elev PVI]); the elevation of any station is its tangent elevation plus (x/l)² e, with x measured from the PVC on the back tangent and from the PVT on the forward tangent, e positive for a sag and negative for a summit; the rate of change of grade is r = (g₂ − g₁)/L percent per station; the second differences of the elevations at equal intervals are constant, the check used by both manuals. The turning point (low point of a sag, high point of a summit) lies x = g l/A from the end of the tangent of lesser slope, where A = g₂ − g₁, when that falls within the curve. For an unsymmetrical curve with tangents l₁ and l₂ the middle offset is e = l₁ l₂ (g₂ − g₁)/(2 (l₁ + l₂)) and each half is its own parabola with offsets (x/l₁)² e and (x/l₂)² e; the two share the grade at the PVI. The default is the FM 5-233 Figure 3-20 sag (−10 % to +5 %, 600 ft, PVI 90+00 at 50.00 ft: e 11.25, low point 91+00 at 60.00); the NAVEDTRA 14337 crest and unsymmetrical examples are in the test script.

Units and conventions

End areas and elevations go through the site's unit dictionary (ft² / m², ft / m); stations, grades, slopes, the degree of curve, the soil factor and money are plain numbers and are not converted on a unit-system switch (re-enter them). Volumes are reported in cubic yards (USCS) or cubic metres (SI), haul in station-yards or station-m³. The side slopes are horizontal : 1 vertical. The curves turn left on the drawing; the field sense is set by the alignment.

Limits

Not covered: the balancing of several loops against a shifted balance line to minimise the total cost, side-borrow and side-cast within a station, haul on grades and equipment cycle times, clearing and stripping, borrow-pit grids (FM 5-233 chapter 4 section III); compound, reverse and spiral curves, superelevation and sight distance; vertical curves through a fixed point and the minimum lengths for comfort or sight distance; crowned or ditched templates. The sources are filed with the pages read in the site's References/Earthwork/ folder; the test script checks every printed value of the examples and the closed forms against an independent numpy / scipy script.