Exercise set

Strain Gauge Bridge and Load Cell Sensor Exercises

Solved strain gauge and load cell exercises for bridge output, gauge factor, shunt calibration, lead resistance, self-heating and release checks.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on strain-gauge bridges and load-cell measurement chains. The goal is to connect strain, gauge factor, bridge output, excitation, amplifier range, calibration, wiring effects and release evidence instead of treating the load cell as a black-box number.

Assume small-strain bridge approximations unless an exercise states otherwise. Real installations also need adhesive condition, gauge orientation, transverse sensitivity, thermal compensation, lead routing, bridge completion tolerance, excitation stability, amplifier common-mode range, shunt calibration, creep, overload history and uncertainty evidence.

Release Evidence Notes

A strain-gauge result should identify the load path, gauge location, bridge type, excitation voltage, amplifier gain, calibration method, zero return, temperature state and mechanical boundary condition. A bridge that passes an electrical check can still be wrong if the strain field, adhesive, fixture or load path does not match the model.

Engineering Boundary Notes

Strain-gauge measurements are coupled mechanical-electrical measurements. The electrical bridge does not know whether the strain came from the intended load path, local bending, thermal expansion, preload, adhesive creep, transverse strain or installation damage. Before a measurement is accepted, the engineer should state which structural member, section, gauge orientation and load case the strain is supposed to represent.

For load cells, the most common boundary mistake is assuming the calibrated element sees exactly the applied process load. Eccentric loading, side load, fixture friction, cable restraint, baseplate flexibility and overload history can all change the relationship between applied force and bridge output. A release record should therefore include not only bridge calibration, but also mounting, alignment, zero return, overload review and a check against an independent load or known mass where practical.

Electrical headroom is a separate boundary. A bridge can be mechanically valid while the amplifier saturates, the excitation droops, the ADC clips or the cable injects noise. Keep the mechanical load-path evidence and electronic range evidence separate until both pass.

Common Release Mistakes

Do not release a strain-gauge channel from shunt calibration alone. Shunt calibration proves part of the electrical path, not the mechanical strain transfer. Do not use a load-cell certificate without checking mounting and overload history. Do not ignore zero return after proof loading. Do not compare microstrain results from different gauges unless orientation, temperature state, bridge completion and gauge location are controlled.

Validation Package Checklist

Minimum evidence should include bridge type, excitation, gain, calibration load, zero return, thermal state, load-path sketch and overload review.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Bridge signal	1, 2, 3, 4	quarter, half and full bridge output	Decide whether the raw signal is measurable.
Calibration and scaling	5, 6, 7, 8	shunt calibration, sensitivity and span	Convert voltage into force or strain.
Error sources	9, 10, 11, 12, 13	lead resistance, zero drift, self-heating, transverse strain and nonlinearity	Decide whether installation bias is acceptable.
Release gates	14, 15, 16, 17, 18	overload, creep, uncertainty, amplifier range and final acceptance	Release, derate or retest the channel.

Exercise 1: Quarter-Bridge Output

A foil strain gauge has $GF=2.1$ and strain $\varepsilon=500\ \mu\varepsilon$ . Excitation is $5.0\ \text{V}$ . Estimate quarter-bridge output:

\displaystyle \frac{V_o}{V_{ex}}\approx \frac{GF\varepsilon}{4}

Solution

Convert strain:

\varepsilon=500\times 10^{-6}

Output:

\displaystyle V_o=5.0\frac{2.1(500\times 10^{-6})}{4}=1.31\ \text{mV}

Engineering Comment

This is a millivolt signal. Amplifier offset, bridge balance and lead pickup can be comparable to the measurement.

Plausibility Check

Small strain in a quarter bridge should produce a low millivolt output at $5\ \text{V}$ excitation.

Exercise 2: Amplifier Output from Bridge Signal

The bridge signal from Exercise 1 is amplified with gain $G=500$ . Find amplifier output.

Solution

V_{amp}=G V_o=500(1.31\ \text{mV})=0.656\ \text{V}

Engineering Comment

The output is large enough for an ADC but still below many amplifier rails. Common-mode range must be checked separately.

Plausibility Check

Multiplying a millivolt signal by hundreds should produce a fraction of a volt.

Exercise 3: Half-Bridge Bending Output

Two active gauges in a bending half bridge see equal and opposite strain magnitude $600\ \mu\varepsilon$ . Use:

\displaystyle \frac{V_o}{V_{ex}}\approx \frac{GF\varepsilon}{2}

with $GF=2.0$ and $V_{ex}=5.0\ \text{V}$ .

Solution

\displaystyle V_o=5.0\frac{2.0(600\times 10^{-6})}{2}=3.0\ \text{mV}

Engineering Comment

The half bridge doubles sensitivity relative to a quarter bridge and improves temperature compensation when gauges see the same temperature.

Plausibility Check

The output is larger than the quarter-bridge case, as expected.

Exercise 4: Full-Bridge Load Cell Output

A full-bridge load cell sensitivity is $2.0\ \text{mV/V}$ at full scale. Excitation is $10\ \text{V}$ . Find full-scale output.

Solution

\displaystyle V_{FS}=2.0\frac{\text{mV}}{\text{V}}(10\ \text{V})=20\ \text{mV}

Engineering Comment

Full-scale output is still small. Cable noise, excitation accuracy and amplifier input range remain important.

Plausibility Check

Millivolts per volt multiplied by volts gives millivolts.

Exercise 5: Load from Output Fraction

A $20\ \text{kN}$ load cell has $20\ \text{mV}$ full-scale output. The measured output is $8.0\ \text{mV}$ . Estimate load.

Solution

Fraction of full scale:

\displaystyle f=\frac{8.0}{20}=0.40

Load:

F=0.40(20)=8.0\ \text{kN}

Engineering Comment

This assumes linearity and zero correction. Field calibration should confirm both.

\displaystyle e_{\%}=100\frac{0.4}{10.0}=4.0\%

Engineering Comment

A four percent span error is too large for most acceptance measurements and should trigger recalibration or repair.

Plausibility Check

$0.4$ is four hundredths of $10$ .

Exercise 9: Lead Resistance Offset Screen

A quarter bridge uses a $350\ \Omega$ gauge. Lead imbalance adds $0.20\ \Omega$ in one arm. Estimate fractional resistance error.

Solution

\displaystyle \frac{\Delta R}{R}=\frac{0.20}{350}=5.71\times 10^{-4}

Engineering Comment

Lead resistance can look like strain. Three-wire or completion compensation is needed for remote gauges.

Plausibility Check

A small fraction of an ohm on hundreds of ohms creates a few hundred ppm.

Exercise 10: Apparent Strain from Resistance Error

Using the fractional error $5.71\times 10^{-4}$ and $GF=2.0$ , estimate apparent strain.

Solution

\displaystyle \varepsilon_{app}=\frac{\Delta R/R}{GF}

\displaystyle \varepsilon_{app}=\frac{5.71\times 10^{-4}}{2.0}=286\ \mu\varepsilon

Solution

Safe overload:

F_{safe}=1.5(20)=30\ \text{kN}

Margin:

M=30-28=2\ \text{kN}

Engineering Comment

The load cell survives the screen, but repeated overload can still change zero and calibration.

Plausibility Check

The peak is below the safe overload but close to it.

Exercise 15: Creep Error

A load cell reads $5.000\ \text{kN}$ immediately after loading and $5.018\ \text{kN}$ after $30$ minutes. Find creep as percent reading.

Solution

e_c=5.018-5.000=0.018\ \text{kN}

\displaystyle 100\frac{0.018}{5.000}=0.36\%

Engineering Comment

Creep can dominate long-duration weighing or proof-load tests. The timing of the reading must be specified.

Plausibility Check

The change is small compared with $5\ \text{kN}$ but not negligible.

Exercise 16: Uncertainty Guard on Load

A reported load is $9.80\ \text{kN}$ with expanded uncertainty $0.15\ \text{kN}$ . The upper limit is $10.0\ \text{kN}$ . Use conservative guard banding.

Solution

Guarded value:

F_g=9.80+0.15=9.95\ \text{kN}

Since:

9.95<10.0

the result passes with $0.05\ \text{kN}$ guarded margin.

Engineering Comment

The nominal result has more margin than the guarded result. The release record should show both.

Plausibility Check

Adding uncertainty moves the result closer to the limit.

Exercise 17: Amplifier Output Swing Gate

A bridge produces $18\ \text{mV}$ at maximum load. Gain is $200$ . Amplifier output swing limit is $\pm 3.0\ \text{V}$ . Check saturation.

Solution

V_{out}=200(0.018)=3.6\ \text{V}

Since:

3.6>3.0

the amplifier saturates.

Engineering Comment

The sensor may be fine while the interface clips. Reduce gain, excitation or range the ADC differently.

Plausibility Check

Large gain on tens of millivolts produces volts, so saturation is plausible.

Exercise 18: Load Cell Release Gate

A load-cell channel has:

Check	Result	Gate
Span error	$0.8\%$	$\le 1.0\%$
Zero return	$0.12\%$ FS	$\le 0.10\%$ FS
Guarded load margin	$0.05\ \text{kN}$	$>0$
Amplifier headroom	pass	pass
Overload history	one event	none allowed

Decide release status.

Solution

Span error passes, guarded load margin passes and amplifier headroom passes. Zero return fails:

0.12\%>0.10\%

Overload history also fails the gate. The channel should not be released for final acceptance until zero stability and overload impact are reviewed.

Engineering Comment

Load-cell release depends on mechanical history and zero behavior, not only span calibration.

Plausibility Check

Two release gates fail, so a hold decision is consistent.

REF