Measurements · Module 10 of 10

Strain Measurement

Strain, and through it stress, is measured with a bonded resistance gauge whose resistance shifts as it stretches. This closing module ties the gauge, the bridge, and material behaviour into one measurement chain.

Readiness check

This closing module measures strain and stress. Tick only what you can do closed-notes.

Multiply a gauge factor by a strain.
Recall that strain is a fractional length change.
Multiply a modulus by a strain for stress.
Recall that a bridge reads a small resistance change.
Work with microstrain (10⁻⁶).

0 or 1 weak itemsContinue and finish the course.

2 weak itemsRevisit the Wheatstone bridge in Module 6.

3 or more weak itemsRevisit stress and strain in Mechanics of Materials.

The core idea

A bonded strain gauge changes resistance in proportion to the strain it feels: ΔR/R = GF × ε, where GF is the gauge factor. The change is tiny, so a Wheatstone bridge reads it. Strain then converts to stress through Hooke's law, σ = Eε.

gauge: ΔR/R = GF × εHooke's law: σ = Eεstrain from output: ε = (ΔR/R) / GF

Strain, the fractional change in length of a loaded part, is one of the most measured mechanical quantities because it leads directly to stress. It is captured by the bonded resistance strain gauge: a fine metal grid glued to the surface so it stretches with the part. Stretching a conductor makes it longer and thinner, raising its resistance, and for a well-designed gauge this change is proportional to strain through the gauge factor, ΔR/R = GF × ε, with GF about 2 for common foil gauges. The catch is scale: engineering strains are often hundreds of microstrain, so ΔR/R is only a fraction of a percent, far too small to read with an ordinary ohmmeter. This is exactly why the Wheatstone bridge of the analog module exists, to null the large baseline resistance and turn the tiny change into a clean voltage, usually with temperature-compensating gauges in the other arms. Once the strain is known, Hooke's law converts it to stress, σ = Eε, where E is the material's modulus, closing the loop from a resistance change to an engineering stress. This single chain, gauge to bridge to Hooke, is how load cells, pressure transducers, and structural tests actually work, and it draws on nearly every earlier module of the course.

The skill works when: you use ΔR/R = GFε for the gauge and σ = Eε for the stress.

The skill breaks down when: the gauge factor is dropped, or the tiny ΔR is read without a bridge.

The concept. A gauge bonded to a stretching surface changes resistance in proportion to strain. A bridge turns that tiny change into a measurable voltage.

The skills, taught in order

Five skills complete the measurement chain from surface to stress.

10.1 Strain and the bonded gauge

Strain is the fractional length change, dimensionless and usually reported in microstrain. A bonded foil gauge glued to the surface deforms with it, converting mechanical strain into a resistance change.

10.2 The gauge factor

The gauge factor relates the fractional resistance change to strain: ΔR/R = GF × ε. It is about 2 for metal foil gauges, so a strain of a few hundred microstrain gives a resistance change of only a fraction of a percent.

10.3 Reading the change with a bridge

Because ΔR/R is so small, a Wheatstone bridge nulls the baseline resistance and amplifies only the change, with dummy or active gauges in the other arms giving temperature compensation. This is the analog front end of the earlier module put to work.

Step	Relation	Turns
Gauge	ΔR/R = GFε	strain into resistance change
Bridge	output ∝ ΔR/R	resistance change into voltage
Hooke	σ = Eε	strain into stress

The full chain: a mechanical strain becomes a voltage, then an engineering stress.

10.4 From strain to stress

In the elastic range, Hooke's law gives σ = Eε, so a measured strain and the material's modulus yield the stress directly. This is what makes strain gauges a stress-measuring tool, not just a displacement one.

10.5 Practical strain measurement

Real installations manage temperature drift with compensation, resolve multi-axis states with rosettes, and place gauges where the strain is highest and most uniform. Good placement and compensation separate a usable reading from noise.

Engineering connection: a load cell is exactly this chain, gauges on a flexure, wired into a bridge, calibrated so its output reads force through ΔR/R = GFε and σ = Eε.

Worked example 1: a strain gauge output

A foil gauge with a gauge factor of 2.0 and a resistance of 120 Ω is bonded to a part strained to 500 microstrain. Find the fractional resistance change and the resistance change.

Figure 1. The gauge factor turns 500 microstrain into a 0.1 percent resistance change, only 0.12 ohms on a 120 ohm gauge.

ProblemFind ΔR/R and ΔR for the gauge in Figure 1.
Given / findGF = 2.0, R = 120 Ω, ε = 500 × 10⁻⁶. Find ΔR/R and ΔR.
AssumptionsGauge fully bonded, strain transferred completely.
ModelΔR/R = GF × ε; ΔR = R × (ΔR/R).
EquationsΔR/R = 2.0 × 500×10⁻⁶ΔR = 120 × 0.001
SolveΔR/R = 0.001 (0.1 percent); ΔR = 0.12 Ω.
Check0.12 Ω on 120 Ω is 0.1 percent, matching the fractional change, and far too small to read without a bridge.
ConclusionThe gauge changes by only 0.12 Ω, which is exactly why a Wheatstone bridge is needed to measure it.

Result. ΔR/R = 0.001; ΔR = 0.12 Ω.

Worked example 2: strain to stress

The part above is steel with a modulus E = 200 GPa. Using the measured strain of 500 microstrain, find the stress.

Figure 2. In the elastic range, stress is the modulus times the strain. Here 500 microstrain in steel is 100 megapascals.

ProblemFind the stress for the strained part in Figure 2.
Given / findE = 200 GPa, ε = 500 × 10⁻⁶. Find σ.
AssumptionsLinear elastic behaviour, uniaxial stress.
ModelHooke's law: σ = Eε.
Equationsσ = 200×10⁹ × 500×10⁻⁶
Solveσ = 10⁸ Pa = 100 MPa.
Check200 GPa times 0.0005 is 0.1 GPa, which is 100 MPa, consistent.
ConclusionThe measured strain corresponds to 100 MPa of stress, the engineering quantity the whole chain was built to deliver.

Result. Stress σ = 100 MPa.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Giving strain units	Strain reported in metres	"Is strain a ratio?"	Strain is dimensionless; use microstrain for scale.
Dropping the gauge factor	Resistance change misread as strain	"Did I divide by GF?"	ε = (ΔR/R)/GF.
Reading ΔR directly	Signal lost in noise	"Is a bridge amplifying it?"	Use a Wheatstone bridge for the tiny change.
Forgetting the modulus	Strain reported as stress	"Did I multiply by E?"	Stress is E times strain in the elastic range.

Practice ladder

Level 1 · Direct skill

A gauge with GF = 2.1 and R = 350 Ω feels 1000 microstrain. Find ΔR.

Show answer

ΔR/R = 2.1 × 1000×10⁻⁶ = 0.0021; ΔR = 0.0021 × 350 = 0.735 Ω.

Level 2 · Mixed concept

Aluminium (E = 70 GPa) is strained to 250 microstrain. Find the stress.

Show answer

σ = 70×10⁹ × 250×10⁻⁶ = 1.75×10⁷ = 17.5 MPa.

Level 3 · Independent problem

A 120 Ω gauge with GF = 2 shows ΔR = 0.048 Ω. Find the strain.

Show answer

ΔR/R = 0.048/120 = 0.0004; ε = 0.0004/2 = 0.0002 = 200 microstrain.

Transfer task | Real engineering

Outline the full chain to measure the stress at the surface of a loaded steel beam with a strain gauge.

What good work looks like

Bond a gauge where the strain is highest and aligned with it, wire it into a Wheatstone bridge with temperature compensation, read the bridge output to get ΔR/R, divide by GF for strain, and multiply by E for stress. A good answer names each stage and the relation it uses.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I used ΔR/R = GFε and then σ = Eε correctly."

"Give me three gauge readings; I will find strain and stress for each."

"Find the stress for me." Work the gauge-to-Hooke chain yourself.

"Why is my signal noisy?" Reason about the bridge and compensation yourself.

Portfolio task

Take a real or specified strain-gauge reading and carry it all the way to a stress, showing every stage of the chain.

Must include: a gauge factor step, a bridge note, and a stress from Hooke's law.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the gauge relation.

ΔR/R = GF × ε.

2. Why use a bridge?

The resistance change is tiny; the bridge nulls the baseline and amplifies it.

3. Write Hooke's law.

σ = Eε.

4. What are the units of strain?

None; it is a ratio, reported in microstrain.

5. How is temperature drift handled?

With compensating gauges in the other bridge arms.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive strain and stress from a gauge reading.

+3 daysWork one full gauge-to-stress chain.

+7 daysReturn to the course home and plan a measurement project.

+30 daysReuse the gauge, bridge, and Hooke chain on any strain task.

Textbook mapping

This module follows Figliola and Beasley, Theory and Design for Mechanical Measurements, 5th edition. Use these references to read further.

Topic in this module	Where to read more
Strain gauges and gauge factor	Figliola and Beasley, Chapter 11, Strain Measurement
Bridge circuits for strain	Figliola and Beasley, Chapter 11, with Section 6.4
Strain, stress, and rosettes	Figliola and Beasley, Chapter 11, Strain gauge configurations

Section numbers refer to Figliola and Beasley, 5th edition. Any edition with the same chapter titles is equivalent for study.