Walking a Contour with an A-Frame Level: How Far the Line Drifts After a Hundred Stations
Year: 2026
Type: article
Abstract: An A-frame level — two legs joined at an apex, a crossbar, and a plumb line — marks contour lines across hillsides one frame-length at a time, and it is a standard tool for laying out terraces and swales where no manufactured level is available. Each station, however, transfers the level datum through a fresh act of judgement, so errors accumulate along the walk. This paper computes that accumulation from stated assumptions: a plumb reading standard deviation of 2 mm over an effective plumb length of 0.9 m, a leg placement standard deviation of 2 mm per station, and a calibration bias from a crossbar mark offset by 0.5 mm. For a frame of 2.0 m leg spacing, each station carries a random error of 4.874 mm and a bias of 1.111 mm. After 100 stations (200 m of hillside) the random drift has a standard deviation of 48.74 mm while the bias drift reaches 111.11 mm, giving a design bound of 208.59 mm against a terrace tolerance of 50 mm; on a cross-slope gradient of 0.10 that elevation drift is a horizontal departure of 2.086 m from the true contour. The bias, not the random error, is the controlling term, and over a fixed hillside width the bias drift is exactly independent of frame size, because a wider frame takes proportionally larger bias steps but proportionally fewer of them. The random term alone is minimised near a leg spacing of 1.0 m. Holding a 50 mm tolerance requires checking the frame against a known level — or an equivalent reversal procedure — at intervals the computation fixes: every 13 stations (26 m) for a 2.0 m frame used as found, or every 26 stations if the frame is reversed at every station to cancel the bias. Stated plainly, an unchecked 200 m contour cannot be held to 50 mm by any frame size under these assumptions: the best unchecked bound over 200 m is 195.67 mm, and even with the bias cancelled by reversal the random term alone gives 97.47 mm. All results are computed, not measured. SIMULATED, NOT MEASURED.
Author: Aldred, William Alexander
Language: English
Subfield: Civil and Structural Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225760948
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