---
title: "Combined force and moment measurement"
subject: "Instrumentation and Measurement"
chapter: 7
key: "semester_4/instrumentation_and_measurement:axial-force-and-bending-moment-measurement"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/axial-force-and-bending-moment-measurement
site: BCT Engineering
---

# Combined force and moment measurement

Instrumentation and Measurement · Chapter 7: Common Mechanical Measurement System and Transducers

- Asked in: year not given, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/axial-force-and-bending-moment-measurement
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-7

## Question

A rectangular steel bar (E = 200 GPa), 20 mm wide and 10 mm deep, carries both an axial force and a bending moment about the 20 mm-wide axis. Strain gauges on the top and bottom surfaces read +600 microstrain and -200 microstrain. Find the axial force and the bending moment, and show how the gauges can be wired so that each effect is measured separately.

## Short answer

Top/bottom strains give epsilon_a=200 microstrain and epsilon_b=400 microstrain. Axial force=EA epsilon_a=8 kN tension, bending moment=EZ epsilon_b=26.7 N m. Wire top and bottom gauges adjacent to measure bending only, or opposite arms to measure axial only.

## Answer

A bar loaded by an axial force $F$ and a bending moment $M$ has a surface strain that is the sum of an axial part $\varepsilon_a$ (the same on all faces) and a bending part $\pm\varepsilon_b$ (opposite on the top and bottom).

$$
\varepsilon_{top} = \varepsilon_a + \varepsilon_b, \qquad \varepsilon_{bottom} = \varepsilon_a - \varepsilon_b
$$

Solving:

$$
\varepsilon_a = \frac{\varepsilon_{top} + \varepsilon_{bottom}}{2}, \qquad \varepsilon_b = \frac{\varepsilon_{top} - \varepsilon_{bottom}}{2}
$$

#### Calculation
Given: $\varepsilon_{top} = +600\ \mu\varepsilon$, $\varepsilon_{bottom} = -200\ \mu\varepsilon$, $E = 200$ GPa, $b = 20$ mm, $h = 10$ mm.

$$
\varepsilon_a = \frac{600 - 200}{2} = 200\ \mu\varepsilon, \qquad \varepsilon_b = \frac{600 + 200}{2} = 400\ \mu\varepsilon
$$

Area: $A = 0.02 \times 0.01 = 2\times10^{-4}$ m$^2$.

$$
F = EA\varepsilon_a = 200\times10^{9}\times 2\times10^{-4}\times 200\times10^{-6} = 8000\ \text{N} = 8\ \text{kN}
$$

Section modulus about the bending axis:

$$
Z = \frac{bh^2}{6} = \frac{0.02 \times (0.01)^2}{6} = 3.333\times10^{-7}\ \text{m}^3
$$

$$
M = EZ\varepsilon_b = 200\times10^{9}\times 3.333\times10^{-7}\times 400\times10^{-6} = 26.7\ \text{N m}
$$

**Answer: axial force $= 8$ kN (tension), bending moment $= 26.7$ N m.**

#### Wiring to separate the effects
- **For bending moment only:** put a top gauge and a bottom gauge in adjacent arms of a half bridge (or two on each face in a full bridge). The axial strain is equal on both faces, so it cancels in adjacent arms, while the bending strain adds. Output $= V_{ex}G\varepsilon_b/2$.
- **For axial force only:** put a top gauge and a bottom gauge in opposite arms of the bridge, with dummy gauges or fixed resistors in the other two arms (or use a full bridge with Poisson gauges). The bending strain is opposite in sign and cancels, while the axial strain adds. Output $= V_{ex}G\varepsilon_a/2$.

```text
 Bending bridge:  R_top (arm 1), R_bot (arm 2)  adjacent
 Axial bridge:    R_top (arm 1), R_bot (arm 3)  opposite
 (other arms: fixed resistors or dummy gauges)
```

In practice two gauges are bonded on each face: one top-bottom pair is wired into the bending bridge and the other pair into the axial bridge, each with its own excitation and readout channel (or a switch box connects the same pair to each circuit in turn).

## Check yourself

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

1. With top strain +600 and bottom -200 microstrain, the axial and bending parts are:
   A. Axial 400, bending 200
   B. Axial 200, bending 400
   C. Axial 300, bending 300
   D. Axial 800, bending 400
   Answer: B. ea = (600 - 200)/2 = 200 and eb = (600 + 200)/2 = 400 microstrain. (Key point: Separating strains)

2. The axial force F = E A ea with A = 2e-4 m^2 and ea = 200e-6 is:
   A. 800 N
   B. 80 kN
   C. 8 kN
   D. 16 kN
   Answer: C. F = 200e9 x 2e-4 x 200e-6 = 8000 N. (Key point: Axial force)

3. To measure bending only, how should the top and bottom gauges be wired?
   A. In opposite arms so that axial strain adds
   B. In adjacent arms, so that equal axial strain cancels and the bending strain adds
   C. Both in the same arm
   D. In series with a fixed resistor only
   Answer: B. Axial strain is equal on both faces and cancels in adjacent arms, while bending strain is opposite and so adds. (Key point: Wiring for bending)

## Related in Instrumentation and Measurement

- [Shaft torque measurement using strain gauges](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Ashaft-torque-strain-gauge-numerical) — Instrumentation and Measurement, chapter 7
- [Full-bridge cantilever load cell output](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Acantilever-full-bridge-output-numerical) — Instrumentation and Measurement, chapter 6
- [Wheatstone bridge for strain gauges](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Awheatstone-bridge-strain-gauge-circuits) — Instrumentation and Measurement, chapter 6
- [Elastic transducers and strain gauge load cells](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Aelastic-force-transducers-load-cells) — Instrumentation and Measurement, chapter 7

## Related in other subjects

- [Timber column under axial load and bending](https://www.bctengineer.com/question.md?key=semester_5%2Fdesign_of_timber_and_masonry_structures%3Atimber-column-combined-bending-axial) — Design of Timber and Masonry Structures, chapter 3
- [Strain energy due to bending, rectangular beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Astrain-energy-bending-derivation) — Theory of Structures I, chapter 2, asked 2×
- [Cantilever 3 m: bending stress 25 mm below top](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acantilever-3m-udl-20kn-t-section) — Strength of Materials, chapter 7
- [Stepped bar under axial loads by FEM](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Astepped-bar-displacement-stress) — Finite Element Method, chapter 3