Volumes: Average End Area and Prismoidal¶

What is worked out: the earthwork of a design route between its cross sections: the cutting, the filling and each template layer, interval by interval; their totals (gross) and, with the factor in column F, their net; the cut used in embankment and the borrow or surplus; and the mass ordinate behind the mass haul diagram. This page gives the method exactly as the Volume abstract works it, so you can check any number in it with a calculator.

The areas themselves come from the cross sections: see Cross-section areas.

What goes in¶

• The sections in the range, in chainage order: every section from From to To under Chainage in the Abstract settings panel, both ends included (the whole route when they are not set). Two chainages within half a millimetre are the same chainage.
• Each item's area at each section, as the sections have it now: the cutting area and the filling area (the Earthwork series), and the area of each template layer. Each area is first rounded to 3 decimals, half away from zero, the way the sheet shows it. The quantities are worked from those shown areas, so the sheet is what a checker recomputes.
• The distance between two neighbouring sections: the difference of their chainages, rounded to 0.001 m.

A section whose areas cannot be worked out counts as an area of 0 for every item. The Checks card does not flag that, so look at any section whose areas in the Chainage-wise sheet are all 0.000.

Fewer than 2 sections in the range: there is nothing to work out, and the Checks card says Fewer than 2 sections … nothing to work out.

Average End Area¶

The volume between two sections is the mean of their two areas times the distance between them:

V = D × (A₁ + A₂) ÷ 2

• A₁, A₂: the item's area at the two sections, m²
• D: the distance between them, m

It is done for every interval and every item (cutting, filling, each layer) in the same way. The Chainage-wise sheet shows each step: Mean cutting m² and Mean filling m² are (A₁ + A₂) ÷ 2, and Cutting m³ and Filling m³ are the mean times Distance m. In the workbook from Export to Excel, each of these is a cell formula a checker can follow, and the sheet's heading says Method: Average end area (quantities are formulas).

Prismoidal¶

CSE's Prismoidal method is Simpson's rule, taken over pairs of equal intervals. For three sections at equal distances D, the volume of the pair (length 2D) is:

V = D ÷ 3 × (A₁ + 4 A₂ + A₃)

A₂ is the area at the middle section. CSE does not make up a mid-area or apply a prismoidal correction to end areas: the middle of each pair is a section you surveyed, and its area is used as it is.

Which intervals pair up¶

Going along the range from its first section:

1. Two intervals one after the other make a pair when they are the same length (within half a millimetre) and neither is in a stretch left out.
2. A pair takes both its intervals; the next pair starts after it.
3. An interval that cannot be paired is worked out by average end area.

So with sections every 25 m and an odd number of intervals, the last interval is by average end area. With intervals of 20, 20, 30, 20, 20 m, the two 20 m pairs are by Simpson's rule and the 30 m interval by average end area. Where every interval is a different length, the two methods give the same totals.

How a pair is split between its two intervals¶

The Chainage-wise sheet has a quantity on every row, so the pair's volume is split between its two intervals:

• first interval: D ÷ 12 × (5 A₁ + 8 A₂ − A₃)
• second interval: D ÷ 12 × (−A₁ + 8 A₂ + 5 A₃)

The two add up to D ÷ 3 × (A₁ + 4 A₂ + A₃), Simpson's rule for the pair, so the totals are exact. Where an area jumps sharply, one interval of a pair can come out small or even negative (areas 0, 0 and 3.000 m² 30 m apart give −7.500 and 37.500 m³). The pair's total (30.000 m³) is the right one.

With Prismoidal, Mean cutting m² and Mean filling m² still show (A₁ + A₂) ÷ 2, but the quantity is no longer the mean times the distance. In the Excel workbook the quantities are values, not formulas, and the heading says Method: Prismoidal (values: Simpson's rule is not a cell formula).

When the two methods differ most¶

Over one pair, Prismoidal less Average End Area is:

V(Prismoidal) − V(Average End Area) = −D ÷ 6 × (A₁ − 2 A₂ + A₃)

• When the middle area is the mean of the two outer ones, the area changes steadily along the pair, and the two methods agree exactly.
• When the middle area is less than the mean of the outer two (a dip), Prismoidal gives less. When it is more (a hump), Prismoidal gives more.
• The difference grows with the distance D and with how unevenly the area changes. It is largest where a cutting or a bank starts or ends (an area rising from 0), at a sudden deep cut or high bank at one section, and where sections are far apart.
• Intervals of different lengths, and the interval left over, are worked out the same way by both methods.

On a route with sections close together and ground that changes gradually, the totals differ by little. If they differ by a lot, look at the sections where the areas jump.

Gross, factor and net¶

On the Abstract sheet:

• Gross m³: the item's total over the range, the sum of its interval quantities.
• Factor (column F): the only cell you type in. It starts at 1.10 for Earthwork in embankment (for consolidation) and 1.00 for Earthwork in cutting and for every layer. A factor is a number from 0.00 to 9.99, kept to two decimals (rounded half away from zero).
• Net m³ = Gross × Factor.

The factor changes the net only: the gross and the Chainage-wise quantities stay as they are.

Cut used in embankment, borrow or surplus¶

Under Balance, from the nets of the earthwork:

• Cut used in embankment = the less of cutting net and embankment net.
• Borrow needed for embankment = embankment net − cutting net, when that is more than 0.
• Surplus cut to dispose of = cutting net − embankment net, when the cutting net is the larger.

This is a balance of the whole range. It does not ask whether a cut is suitable for filling, or how far it would have to be carried. The layers have their own gross and net but are not in the balance. With Earthwork unticked under Series, there is no Balance group and no mass haul.

Mass ordinate and the mass haul diagram¶

The mass ordinate is the cut to date less the fill to date, each times its factor:

M (first section) = 0

M = M (section before) + V(cut) × F(cut) − V(fill) × F(fill)

V(cut) and V(fill) are the interval's quantities by the method chosen, and F(cut), F(fill) are the factors in column F. A positive ordinate means more cut than fill so far. The ordinate at the last section equals cutting net − embankment net, so it is the surplus (positive) or the borrow (negative) of the range. When From is set, the ordinate starts at 0 at the first section in the range.

Under the diagram on the Mass haul sheet:

• Balance points: where the ordinate crosses 0, found on the straight line between the two sections either side. A section whose ordinate is exactly 0, between two of opposite sign, is a balance point too. The 0 at the start is not one, nor a touch of 0 at the end. They are shown to the whole metre.
• Largest ordinate: the largest either way, at its section (the first one, when two are equal), in whole m³.
• Borrow at the end or Surplus at the end: the ordinate at the last section, in whole m³.

CSE draws the mass haul diagram and finds its balance points. It does not work out haul in m³·m, free haul or overhaul.

Stretches left out¶

An interval that lies inside a stretch under Left out (a culvert, a bridge, an existing road) counts nothing, for every item and by either method:

• its quantities are 0 (a dash in the printed statement), and the row is grey with Left out: Culvert (or its own reason) in Remark;
• the mass ordinate runs on flat across it;
• it is not counted in the length (the Distance m total is the length counted);
• with Prismoidal, an interval left out ends the pairing: the intervals either side pair only with intervals on their own side.

A stretch runs from one section to another, so it covers whole intervals. When one interval is more than 1.5 times the usual one (the median interval), the Checks card asks A culvert or bridge? with Leave out… and Keep it in.

Conventions and rounding¶

Because each quantity is kept in full, the shown values of a column can add up to a total that differs from the shown total in the last decimal.

Worked example¶

Four sections 30 m apart, with these areas (the default factors: 1.00 cutting, 1.10 embankment):

Average End Area¶

Prismoidal¶

The three intervals are all 30 m: 0/000–0/060 is a pair, and 0/060–0/090 is left over, so it is worked out by average end area.

The pair, cutting: 30 ÷ 3 × (0.000 + 4 × 0.850 + 1.900) = 10 × 5.300 = 53.000, split as

• 0/000–0/030: 30 ÷ 12 × (5 × 0.000 + 8 × 0.850 − 1.900) = 2.5 × 4.900 = 12.250
• 0/030–0/060: 30 ÷ 12 × (−0.000 + 8 × 0.850 + 5 × 1.900) = 2.5 × 16.300 = 40.750

The pair, filling: 30 ÷ 3 × (1.800 + 4 × 0.950 + 0.300) = 10 × 5.900 = 59.000, split as

• 0/000–0/030: 2.5 × (5 × 1.800 + 8 × 0.950 − 0.300) = 2.5 × 16.300 = 40.750
• 0/030–0/060: 2.5 × (−1.800 + 8 × 0.950 + 5 × 0.300) = 2.5 × 7.300 = 18.250

Check the difference over the pair with the formula above: for the cutting, −30 ÷ 6 × (0.000 − 2 × 0.850 + 1.900) = −5 × 0.200 = −1.000, and 53.000 is 1.000 less than 12.750 + 41.250 = 54.000. The filling is the same: −5 × (1.800 − 1.900 + 0.300) = −1.000.

Mass ordinate¶

The ordinate crosses 0 between 0/060 and 0/090. By Average End Area the balance point is at 60 + 30 × 12.000 ÷ (12.000 + 52.800) = 65.556 m; by Prismoidal at 60 + 30 × 11.900 ÷ (11.900 + 52.900) = 65.509 m. Both are shown as 0/066 under the diagram, with Largest ordinate 53 m³ at 0/090 and Surplus at the end 53 m³.

The abstract¶

The surplus equals the mass ordinate at 0/090, as it should.

Related¶

• Volume abstract: the factors, the range, stretches left out, Excel and the print.
• Cross-section areas: where the areas come from.
• Review the cross sections: the sections behind the numbers.
• Levels between surveyed points: sections at chainages you did not survey.