Cross-section areas¶

What the program works out: at each chainage, for each series other than the NSL, the filling and cutting area between that series and the series it extends to, and the net area. The area statement on the drawing shows the whole working, so a checker can follow every figure.

Most often the series is a design and it extends to the ground (NSL). A layer can extend to another series instead, such as a GSB layer measured against the subgrade (see Add layers). The method is the same either way. On this page, the series is the one being measured and its target is the one it extends to.

The two lines¶

1. The series' line runs from its left toe to its right toe: its own points, with the side slopes run out to the target at each end (see Where the side slopes meet the ground).
2. The target's line is the target's points between the two toes, with the toes added.
3. Where the lines cross. Every point between the toes where the series' line crosses the target's line is added to both lines. Between two crossings, the series is either all above its target (filling) or all below it (cutting).

The method: trapeziums down to a level¶

Each piece of a line, from one point to the next, makes a trapezium down to a level line:

area of a piece = 1/2 (L1 + L2) × (O1 − O2)

where L1 and L2 are the levels of its two points and O1 − O2 is the horizontal width between them.

• The levels are R.L.s, so the trapeziums reach down to R.L. 0. With Calculate areas with respect to the datum ticked (Section Settings), each level is first taken less the drawing's datum. The figures in the working get smaller; the filling, cutting and net area stay the same.
• The statement writes the point farther from the centre line first, and both offsets without their sign. When a piece crosses the centre line, its width is still the full distance between the two offsets.
• A piece with an area of 0 (two points at the same offset) is left out.

The program then walks the pieces from the left toe, one crossing at a time:

• Initial area is the sum of the target's pieces in that stretch: the area under the target.
• Final area is the sum of the series' pieces in that stretch: the area under the series.
• If the final area is the larger, the series is above its target: the stretch is filling. Otherwise it is cutting.

The stretch after the last crossing, up to the right toe, is sorted the same way.

Then:

Gross Fill = Final area − Initial area, over the filling stretches

Gross Cut = Initial area − Final area, over the cutting stretches

Net Area = Gross Fill − Gross Cut

Each block's total (initial filling, final filling, initial cutting, final cutting) is rounded to 3 decimals before the subtraction. The net area is labelled (Filling) when it is more than 0, and (Cutting) otherwise.

The area statement¶

Tick Area statement of in the Areas, m² section of the Quantities panel and pick the series: its statement is drawn with the section. Print Sections prints it too (Print area details only, or Print section with area details). It reads:

• The heading: Area Calculation Between the series & its target.
• Filling, with two blocks side by side: INITIAL AREA (the target's pieces) and FINAL AREA (the series' pieces). Each line is one piece, its working and its result; the block's total is under the results. Then Gross Fill -: and the filling area.
• Cutting, laid out the same way, ending with Gross Cut -: and the cutting area.
• The last line: Net Area: (fill − cut )= net Sqm. and (Filling) or (Cutting).

A part with no area (no filling, say) is left out. Where the two blocks don't fit side by side, the final area goes under the initial area. All figures are shown to 3 decimals.

Worked example¶

A road at one chainage. The ground is level at R.L. 100.000 to the left of the centre line and rises 1 in 4 to the right:

The design is a formation 8 m wide at R.L. 100.500, with side slopes of 2 (1 V : 2 H) both sides:

The lines. The side slopes meet the ground at a left toe (-5.000, 100.000) and a right toe (6.000, 101.500); Where the side slopes meet the ground works them out. The formation crosses the ground where the ground rises to 100.500: 100.000 + 0.25 × O = 100.500, so O = 2.000.

• The design's line: (-5, 100.000), (-4, 100.500), (0, 100.500), (2, 100.500), (4, 100.500), (6, 101.500)
• The ground's line: (-5, 100.000), (0, 100.000), (2, 100.500), (6, 101.500)

From the left toe to the crossing at 2, the design is above the ground: filling. From 2 to the right toe it is below: cutting.

Filling

Gross Fill = 703.250 − 700.500 = 2.750

Cutting

Gross Cut = 404.000 − 403.000 = 1.000

Net Area: (2.750 − 1.000) = 1.750 Sqm. (Filling)

A check by another route. Take the height of the design over the ground at each point: 0 at -5, 0.500 at -4, 0.500 at 0, 0 at 2, −0.500 at 4, 0 at 6. The filling is 1/2 (0 + 0.5) × 1 + 1/2 (0.5 + 0.5) × 4 + 1/2 (0.5 + 0) × 2 = 0.250 + 2.000 + 0.500 = 2.750. The cutting is 1/2 (0 + 0.5) × 2 + 1/2 (0.5 + 0) × 2 = 0.500 + 0.500 = 1.000. The same.

Edge cases¶

• No line to measure. A series with no points at a chainage, or a target with fewer than two points there, has no toes and no area: 0.
• A template layer. A layer from a template (a pavement course, a railway's blanket) counts as filling only. Its area is measured exactly between its line and what it rests on, from toe to toe, not by the trapeziums at 3 decimals; its cutting is 0 and its net area is its filling. If the working to 3 decimals would come to another figure, the statement adds (Measured exactly between the lines; the working to three decimals gives …). A part of the layer below what it rests on is reported as an overlap, not counted. The statement says Below, the name of what the layer rests on, the area, and Sqm, an overlap, not counted (a layer counts as fill only).
• Rounding of the totals. Each block's total is rounded to 3 decimals before the subtraction, as the statement shows it, so the gross area can differ by 0.001 from an exact area worked out in one step.

Related¶

• Where the side slopes meet the ground
• Review the cross sections
• Reach a target area
• Volumes