---
title: "Tape standardized in catenary: 23 m span"
subject: "Surveying I"
chapter: 2
key: "semester_3/surveying_i:tape-catenary-span-23m"
source: https://www.bctengineer.com/q/semester_3/surveying_i/tape-catenary-span-23m
site: BCT Engineering
---

# Tape standardized in catenary: 23 m span

Surveying I · Chapter 2: Distance Measurements

- Asked in: 2075 Asoj, 6 marks
- Video: [Common catenary numerical](https://www.youtube.com/watch?v=AvLhrQbVfDQ) — MD EDUCATIONAL INSTITUTE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/tape-catenary-span-23m
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-2

## Question

A tape of nominal length 30 m is standardized in catenary at 50 N tension and found to be 29.8950 m. If the mass of the tape is 0.015 kg/m, calculate the horizontal length of a span recorded as 23 m.

## Short answer

Find true tape length by adding sag at standardisation (0.00974 m) to 29.8950 m, giving 29.9047 m. For the 23 m span, apply tape length correction (-0.0730 m) and sag correction (-0.0044 m), resulting in horizontal length 22.9226 m.

## Answer

**Method.** A tape standardised in catenary has the sag effect included in its stated length. We add this sag back to find the true tape length, then correct the 23 m span for tape length and for its own sag. The field pull is taken equal to the standard pull, 50 N.

#### Data
$w = 0.015\times 9.81 = 0.14715$ N/m, $P = 50$ N, $L = 23$ m, standard length $= 29.8950$ m.

#### 1. Sag at standardisation (30 m)
$$
\frac{0.14715^{2}\times 30^{3}}{24\times 50^{2}} = 0.009744\ \text{m}
$$
True tape length $= 29.8950 + 0.009744 = 29.9047$ m.

#### 2. Tape length correction
$$
C_a = 23\times\frac{29.9047 - 30}{30} = -0.0730\ \text{m}
$$

#### 3. Sag correction for the 23 m span
$$
C_s = -\frac{0.14715^{2}\times 23^{3}}{24\times 50^{2}} = -0.0044\ \text{m}
$$

#### 4. Horizontal length
$$
D = 23 - 0.0730 - 0.0044 = 22.9226\ \text{m}
$$

**Answer: horizontal length of the span $= 22.9226$ m.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is the sag at standardisation added to 29.8950 m?
   A. The stated length already has the sag effect included, so the true length is longer
   B. To correct for temperature
   C. To correct for pull
   D. Because the tape is too short
   Answer: A. Catenary standardisation gives the horizontal span; adding sag gives the true tape length. (Key point: Standardised in catenary)

2. What is the formula for tape length correction on 23 m?
   A. C_a = 23 x (29.8950 - 30)/30 without sag
   B. C_a = 23 x (true tape length - 30)/30
   C. C_a = 23 x 30/29.9047
   D. C_a = (30 - 23) x 0.0097
   Answer: B. The proportional error of the true tape length is applied to the measured length. (Key point: Tape length correction)

3. Which length is used in the sag correction of the field span?
   A. 30 m
   B. The standard length 29.8950 m
   C. The span length 23 m
   D. 23 m divided by 30
   Answer: C. The field span of 23 m sags itself. (Key point: Sag for the span)

4. Both corrections are negative, so the horizontal length is
   A. 23.0774 m
   B. 23.0000 m
   C. 22.9956 m
   D. 22.9226 m, less than 23 m
   Answer: D. Tape is short (reads too long) and sag shortens the horizontal span. (Key point: Final sign)

## Related in Surveying I

- [Tape standardized in catenary: 20 m span](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-catenary-span-20m) — Surveying I, chapter 2, asked 2×
- [Tape standardized in catenary: 24 m span](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-catenary-span-24m) — Surveying I, chapter 2, asked 2×
- [Horizontal length of 30 m span, 50 m tape](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-50m-span-30m-26c) — Surveying I, chapter 2
- [Horizontal length of 16 m, 29.8850 m tape](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-29-8850-16m-fully-supported) — Surveying I, chapter 2
- [Horizontal length for 120.0 m recorded](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-29-995-120m) — Surveying I, chapter 2
- [True length: tape found 2.5 cm too long](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-825-plus-750-m) — Surveying I, chapter 2