---
title: "Elastic transducers and strain gauge load cells"
subject: "Instrumentation and Measurement"
chapter: 7
key: "semester_4/instrumentation_and_measurement:elastic-force-transducers-load-cells"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/elastic-force-transducers-load-cells
site: BCT Engineering
---

# Elastic transducers and strain gauge load cells

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

- Asked in: year not given, 8 marks
- Video: [Coloumn Type Load Cell](https://www.youtube.com/watch?v=fEBBlvSp250) — ENGINEERING TUTORIAL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/elastic-force-transducers-load-cells
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-7

## Question

Explain how force is measured using elastic force transducers. Describe the proving ring and the strain-gauge column-type load cell, and derive the bridge output of a column load cell with four gauges (two axial, two transverse).

## Short answer

Elastic force transducers measure force through elastic deformation; a proving ring measures diameter change. A column load cell with two axial and two transverse gauges in a full bridge gives Vo = -Vex G (1+nu) F/(2 A E), proportional to force and temperature compensated.

## Answer

**Principle:** an applied force deforms an elastic member (a spring, ring, beam or column) within its elastic limit. The deflection or strain is proportional to the force ($F = kx$ or $\varepsilon = F/AE$) and is measured by a displacement transducer or by strain gauges.

#### Proving ring
A steel ring is loaded along a diameter. The change in diameter is proportional to the force.

```text
        F
        v
      .---.
     /     \    deflection measured by
    |       |   micrometer / LVDT / strain gauges
     \     /
      '---'
        ^
        F
```

Advantages: high accuracy and good linearity; used to calibrate testing machines. Limitations: bulky, and each ring covers only a limited range.

#### Strain-gauge column load cell
A solid or hollow steel column carries the force. Four gauges are bonded: two **axial** (aligned with the load, in compression) and two **transverse** (at $90^\circ$, in tension because of the Poisson effect).

```text
        F
        v
     +-----+   Axial gauges: R1 (front face), R3 (back face)
     | R1  |   Transverse gauges: R2 (front face), R4 (back face)
     | R2  |
     +-----+
        ^
        F (reaction)
```

#### Derivation of the bridge output
Axial strain (compressive): $\varepsilon_a = -\dfrac{F}{AE}$. Transverse strain: $\varepsilon_t = +\nu\dfrac{F}{AE}$.

Connect the axial gauges $R_1$, $R_3$ in opposite arms and the transverse gauges $R_2$, $R_4$ in the other two arms. Then

$$
V_o = \frac{V_{ex}G}{4}\left(\varepsilon_1 - \varepsilon_2 + \varepsilon_3 - \varepsilon_4\right) = \frac{V_{ex}G}{4}\left[2\varepsilon_a - 2\varepsilon_t\right]
$$

$$
V_o = \frac{V_{ex}G}{2}\left(-\frac{F}{AE} - \nu\frac{F}{AE}\right) = -\frac{V_{ex}\,G\,(1+\nu)}{2AE}\,F
$$

The output magnitude is proportional to $F$ and is $(1+\nu)/2$ times that of a full bridge with all four gauges at the axial strain. The arrangement also compensates for temperature (all four gauges change alike) and cancels bending (gauges of each pair are on opposite faces).

Example sensitivity: with $G = 2$, $\nu = 0.3$, $\varepsilon_a = 1000\ \mu\varepsilon$ and $V_{ex}=10$ V, $V_o = 13$ mV (1.3 mV/V).

**Other elastic members:** cantilever beam (bending), S-type, shear-beam and ring-type cells, all with strain gauges wired as a bridge.

## Check yourself

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

1. What is the principle of elastic force transducers?
   A. Force changes the fluid temperature
   B. Force deforms an elastic member within its elastic limit, and the deflection or strain is proportional to force
   C. Force generates a magnetic field directly
   D. Force produces permanent deformation
   Answer: B. Within the elastic limit F = kx or strain = F/AE, so measuring deflection or strain gives force. (Key point: Principle)

2. In the column load cell, why do the transverse gauges read tensile strain under a compressive load?
   A. Because of Poisson's effect
   B. Because they are glued upside down
   C. Because of temperature
   D. Because the column buckles
   Answer: A. Axial compression -F/AE causes lateral expansion +v F/AE, which the 90° gauges sense. (Key point: Column strains)

3. What is the bridge output magnitude of the column load cell?
   A. Vex G F / (A E)
   B. Vex G (1 + v) F / (2 A E)
   C. Vex G (1 - v) F / (4 A E)
   D. Vex G v F / (A E)
   Answer: B. Vo = (Vex G/4)(2 ea - 2 et) = (Vex G/2)(F/AE)(1 + v), which is (1+v)/2 times the plain full-bridge value. (Key point: Bridge output)

4. Why does the four-gauge column arrangement compensate temperature and bending?
   A. All four gauges see the same temperature change, which cancels in adjacent arms, and gauges on opposite sides cancel bending
   B. Temperature has no effect on gauges
   C. Bending only affects the transverse gauges
   D. The column is insulated
   Answer: A. Common-mode effects cancel because adjacent arms subtract. (Key point: Compensation)

## Related in Instrumentation and Measurement

- [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
- [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
- [Gauge resistance change and quarter-bridge output](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Agauge-resistance-change-quarter-bridge) — Instrumentation and Measurement, chapter 6
- [Hydraulic and pneumatic load cells](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Ahydraulic-pneumatic-load-cells) — Instrumentation and Measurement, chapter 7
- [Derivation of strain gauge factor](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Astrain-gauge-factor-derivation) — Instrumentation and Measurement, chapter 6

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