Chain Line Across a Sloping Field with an Obstacle
A two-person party is chaining a straight base line AB on a uniform slope using a 30 m steel tape that is standard at 20 °C and 50 N tension. The measured slope distance between A and B is 186.420 m, and the average angle of inclination read with a clinometer is 6°20'. Midway, a 4 m wide irrigation ditch interrupts the line; the banks are level with the ground surface and the ditch runs perpendicular to AB. Tape temperature during measurement is 34 °C, and the tape has a known coefficient of thermal expansion of 11.2 × 10⁻⁶ per °C. The party also reports that the tape was dragged over stony ground and shows a 3 mm kink near the 12 m mark. The surveyor must deliver a corrected horizontal distance and decide how to handle the ditch.
1. Compute the corrected horizontal distance from A to B, and state which corrections you applied and which you deliberately left out.
Read the worked answer
Step 1 - Slope correction. For a uniform slope, horizontal distance H = L·cos θ, where L = 186.420 m and θ = 6°20' = 6.3333°.
cos 6.3333° ≈ 0.99390, so H ≈ 186.420 × 0.99390 ≈ 185.283 m.
Equivalently, the slope correction is L(1 − cos θ) ≈ 186.420 × 0.00610 ≈ 1.137 m, subtracted from the slope length.
Step 2 - Temperature correction. Ct = L·α·(Tm − T0) = 186.420 × 11.2 × 10⁻⁶ × (34 − 20) = 186.420 × 11.2 × 10⁻⁶ × 14 ≈ +0.0292 m.
The tape was warmer than standard, so it was longer than nominal and the measured length is too short; the correction is positive.
Step 3 - Corrections deliberately omitted.
- Tension: the tape was used at its standard 50 N, so no tension correction applies.
- Sag: the tape was laid on the ground (not suspended), so sag correction is zero.
- Kink: a 3 mm kink cannot be corrected by formula; it makes the tape read short by an unknown small amount and the tape should be condemned or the kink section avoided. Flag it as a systematic error that biases the result low.
Step 4 - Combined result.
H_corrected ≈ 185.283 + 0.029 ≈ 185.312 m.
Report the value as 185.31 m, with a note that the kink introduces an unquantified negative bias of order a few millimetres and that the tape should be checked against a standard.
2. Explain how the party should chain across the 4 m ditch without breaking the line, and what error each method introduces.
Read the worked answer
The ditch is narrow (4 m) with level banks, so the standard short-obstacle methods apply. The key requirement is that the line AB is not deflected and that the offset measurements are taken perpendicular to AB.
Method 1 - Perpendicular offset (most accurate here).
- Set pegs at C and D on the line where the ditch banks begin, so CD spans the ditch.
- At C, erect a perpendicular (3-4-5 triangle or an optical square) and measure a convenient offset, e.g. 3 m to C'.
- At D, erect a perpendicular on the same side and set D' at the same offset 3 m.
- Chain C'D' along the level ground beside the ditch. Because CC' = DD' and both are perpendicular to AB, C'D' = CD exactly (a rectangle).
Error: only the perpendicular-setting error and the offset-measurement error; typically a few millimetres if a 3-4-5 triangle is used over 3 m.
Method 2 - Similar triangles. Set a point E off the line such that CE and DE can both be chained, then use the geometry of the triangle to compute CD. This is more computation and more sensitive to angular error than Method 1 for a short span, so it is not preferred for a 4 m ditch.
Method 3 - Random line / ranging. Not justified here; ranging a random line is for obstacles that block sight as well as chain, such as a building or a wood. A 4 m ditch can be straddled.
Judgement: Use Method 1. Record the offset values in the field book so the computation is auditable, and keep the offset short (3 m) to limit the effect of any angular error in setting the perpendicular.
3. The kink is discovered only after the survey. What should the surveyor do about the already-reduced distance, and how should the field book record it?
Read the worked answer
A 3 mm kink is a physical defect, not a correctable systematic error. The tape's effective length between graduations is altered unpredictably, and the kink may straighten or deepen with handling, so the error is not constant and cannot be modelled.
Actions:
- Do not silently adjust the number. The reduced distance 185.31 m should stand as computed, but it must be annotated as provisional pending a tape check.
- Compare the tape with a standard. Lay the tape on a flat concrete surface or a calibration baseline at 50 N tension and 20 °C and compare graduation by graduation, especially around the 12 m mark. If the kink causes a measurable shortening, apply a tape-length (absolute) correction to the whole line.
- Re-measure the affected span. If the kink lies within the measured 186.420 m, re-chain the section from 0 to 30 m containing the kink with a sound tape. The rest of the line may be accepted if the tape check shows no other defect.
- Assess fitness for purpose. For a base line feeding a traverse, a few millimetres is tolerable; for a cadastral boundary or a setting-out control line, it is not, and re-measurement is mandatory.
Field book record: enter the date, tape identification, temperature, tension, slope angle, the raw 186.420 m, each correction with its sign, the reduced value, and a clear marginal note: "3 mm kink near 12 m; tape condemned for precise work; section re-measured on [date]." Never erase the original entry - strike through and initial any correction so the audit trail survives.