Exercise set

Materials Non-Destructive Evaluation and Defect Detection Exercises

Solved NDE exercises for ultrasonic thickness, crack detection, CT voxel size, porosity, XRD, XRF, eddy current, POD, uncertainty and release gates.

Branch: Materials Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on non-destructive evaluation and defect detection: ultrasonic thickness, corrosion rate, crack detection, CT voxel sizing, porosity, XRD, XRF, eddy-current skin depth, probability of detection, uncertainty and release gates. Mechanical characterization and destructive test data are handled in a separate specialist exercise set.

Use these calculations as method screens. Release evidence still needs qualified procedure, inspector qualification where applicable, calibration blocks, equipment settings, coverage map, detection threshold, uncertainty budget and disposition criteria.

How to use these exercises

Use the set as an inspection-release sequence. Exercises 1 to 5 cover ultrasonic wall thickness, corrosion screening, remaining life, crack threshold and probability of detection. Exercises 6 to 12 move into CT, porosity, XRD and XRF method limits. Exercises 13 to 17 add eddy-current penetration, lift-off, false calls, coverage and sizing uncertainty. Exercise 18 combines the method gates into a release decision.

Before calculating, name the inspection method, material, geometry, accessible surface, reference artifact, defect type, defect orientation and acceptance criterion. A method that detects axial cracks in a smooth coupon may not detect tight corrosion under insulation, lack of fusion near a weld toe, a low-contrast pore in CT or a subsurface eddy-current flaw at the wrong frequency. The engineering comment below each exercise identifies the evidence needed before the number can support disposition.

Release Evidence Notes

NDE evidence should state method, calibration artifact, scan coverage, material condition, surface access, detection threshold, defect orientation sensitivity, probability of detection and acceptance rule. A detected signal is not enough without location, sizing and disposition context.

The evidence package should separate detection, sizing and disposition. Detection proves that a relevant indication can be found under qualified conditions. Sizing estimates the flaw dimension with a stated uncertainty. Disposition compares the conservative flaw size with an engineering limit, repair rule, monitoring interval or rejection criterion. Mixing these steps can make an inspection appear stronger than it is.

Coverage evidence is just as important as sensitivity. A high POD does not release an unscanned area, an inaccessible weld root, a region with poor coupling or a CT volume with reconstruction artifacts. Release records should therefore include both the method capability and a map of where that capability was actually applied.

Engineering Boundary Notes

These exercises use simplified formulas. Real inspection planning must account for geometry, surface condition, attenuation, defect morphology, human factors, scan speed, coupling, reconstruction settings and false-call management. Treat pass results as screens for method qualification, not as automatic proof that the part or structure is acceptable.

The main boundary is defect relevance. Ultrasonic thickness loss, CT porosity, XRD stress, XRF chemistry and eddy-current response answer different questions. A clean result in one method does not clear a different failure mode unless the inspection plan explicitly links method, flaw type, location and acceptance rule. The second boundary is uncertainty: near-limit indications should be handled with guarded sizing, repeat inspection or confirmatory testing.

Common Release Mistakes

using ultrasonic thickness without sound-velocity verification;
claiming CT detection below voxel and contrast capability;
treating POD as certainty for every defect type;
accepting XRF composition without depth or coating effects;
using eddy-current skin depth while ignoring lift-off;
releasing an inspection with incomplete coverage map.

Another common mistake is optimizing only for fewer false calls. Lowering sensitivity may reduce repair burden, but it can also move the detection threshold above the critical defect size. False-call rate should be managed with training, procedure control, reference artifacts and review rules rather than by weakening the method.

Do not confuse indication count with risk. A single large defect in a high-stress location can matter more than many small pores in benign regions. The release decision should connect NDE results to stress state, fracture mechanics, corrosion allowance, fatigue life, leak-before-break policy or repair priority.

Scenario Map

Scenario	Main calculation	Release decision
Ultrasonic inspection	thickness, corrosion and crack margin	Repair, monitor or release.
CT inspection	voxel size, porosity and largest defect	Accept, rescan or reject.
X-ray methods	XRD spacing and XRF composition	Validate phase or alloy call.
Eddy current	skin depth and lift-off	Tune probe or reject setup.
Method release	POD, uncertainty and coverage	Qualify method or hold lot.

Validation Package Checklist

method, procedure revision and operator/equipment state;
calibration block, reference defect and sensitivity setting;
scan coverage map and inaccessible areas;
sizing uncertainty and defect orientation limits;
POD or detection-margin evidence;
false-call handling, repeat-inspection rule and review authority;
guarded acceptance criteria for near-limit indications;
link from defect location to stress, corrosion, fatigue or fracture consequence;
disposition rule and release decision.

A complete validation package should make the inspection decision reproducible. Another engineer should be able to see what was inspected, what could not be inspected, what defect size could be detected, how sizing uncertainty was handled and why the final decision was release, repair, monitor, rescan or reject.

Exercise 1: Ultrasonic Thickness

Ultrasonic round-trip time is $6.8\ \mu s$ and sound velocity is $5900\ \text{m/s}$ . Compute wall thickness.

Solution

t=\dfrac{v\Delta t}{2}=\dfrac{5900(6.8\times10^{-6})}{2}=0.0201\ \text{m}=20.1\ \text{mm}

Engineering Comment

Velocity should match the material and temperature; otherwise thickness is biased.

Plausibility Check

Microsecond time of flight in steel-like material gives millimeter-scale thickness.

Exercise 2: Corrosion Rate from Thickness Loss

Original thickness was $24.0\ \text{mm}$ . Current thickness is $20.1\ \text{mm}$ after $6$ years. Estimate corrosion rate.

Solution

r=\dfrac{24.0-20.1}{6}=0.65\ \text{mm/year}

Engineering Comment

Use multiple locations and historical readings before extrapolating remaining life.

Plausibility Check

About $4\ \text{mm}$ loss over six years gives about two thirds millimeter per year.

Exercise 3: Remaining Life Screen

Minimum allowable thickness is $17.0\ \text{mm}$ , current thickness is $20.1\ \text{mm}$ and corrosion rate is $0.65\ \text{mm/year}$ . Estimate remaining life.

Solution

t_{rem}=\dfrac{20.1-17.0}{0.65}=4.77\ \text{years}

Engineering Comment

Inspection interval should include uncertainty, corrosion variability and consequence of failure.

Plausibility Check

About $3\ \text{mm}$ remaining at $0.65\ \text{mm/year}$ gives under five years.

Exercise 4: Critical Crack Detection Margin

Critical crack length is $4.0\ \text{mm}$ . Qualified detection threshold is $2.5\ \text{mm}$ . Compute margin.

Solution

m=4.0-2.5=1.5\ \text{mm}

Engineering Comment

The threshold must apply to crack orientation, location and material condition.

Plausibility Check

The method detects cracks smaller than the critical size.

Exercise 5: Probability of Detection

At the critical defect size, POD is $0.92$ . Estimate missed-defect probability.

Solution

P_{miss}=1-0.92=0.08=8\%

Engineering Comment

An eight percent miss probability may be unacceptable for high-consequence defects.

Plausibility Check

POD and miss probability add to one.

Exercise 6: CT Voxel Size

A CT scan reconstructs a $60\ \text{mm}$ field across $1200$ voxels. Compute voxel size.

Solution

\Delta x=\dfrac{60}{1200}=0.050\ \text{mm}=50\ \mu m

Engineering Comment

Minimum reliable feature size is usually several voxels, not one voxel.

Plausibility Check

Sixty millimeters split into twelve hundred intervals gives fifty micrometers.

Exercise 7: Minimum Resolvable Feature

If the release rule requires at least $3$ voxels across a pore and voxel size is $50\ \mu m$ , estimate minimum resolvable diameter.

Solution

d_{min}=3(50)=150\ \mu m

Engineering Comment

Contrast, reconstruction and thresholding can increase the practical limit.

Plausibility Check

Three voxels is three times the voxel size.

Exercise 8: CT Porosity Fraction

Detected pore volume is $3.6\ \text{mm}^3$ in a part volume of $2400\ \text{mm}^3$ . Compute porosity fraction.

Solution

f=\dfrac{3.6}{2400}=0.0015=0.15\%

Engineering Comment

Porosity fraction should be paired with largest-pore and location criteria.

Plausibility Check

A few cubic millimeters in thousands is a small fraction.

Exercise 9: Largest-Pore Gate

Largest detected pore is $0.42\ \text{mm}$ and release limit is $0.35\ \text{mm}$ . Does it pass?

Solution

0.42>0.35\ \text{mm}

It fails the largest-pore gate.

Engineering Comment

Largest defect can govern fatigue even when total porosity is low.

Plausibility Check

The measured pore exceeds the limit by $0.07\ \text{mm}$ .

Exercise 10: Bragg Spacing

XRD peak uses wavelength $\lambda=0.154\ \text{nm}$ and angle $\theta=22.5^\circ$ for first order. Compute spacing:

d=\dfrac{\lambda}{2\sin\theta}

Solution

d=\dfrac{0.154}{2\sin22.5^\circ}=0.201\ \text{nm}

Engineering Comment

Peak identification should include instrument calibration and phase database fit.

Plausibility Check

Atomic plane spacings are often tenths of a nanometer.

Exercise 11: XRD Residual-Stress Guard

Measured residual stress is $235\ \text{MPa}$ tensile, uncertainty is $20\ \text{MPa}$ and limit is $250\ \text{MPa}$ . Use guarded rule $\sigma+U\le250$ . Does it pass?

Solution

235+20=255\ \text{MPa}

Since $255>250$ , it fails the guarded rule.

Engineering Comment

Residual-stress acceptance should include sign convention and measurement direction.

Plausibility Check

The measured value is close enough to the limit that uncertainty matters.

Exercise 12: XRF Alloy Screen

Required chromium range is $16.0\%$ to $18.0\%$ . XRF reports $15.6\%$ with uncertainty $0.3\%$ . Does it pass unguarded and guarded lower-limit check?

Solution

Unguarded:

15.6\%<16.0\%

It fails even before guarding.

Guarded lower value:

15.6-0.3=15.3\%

Engineering Comment

XRF screening may require confirmatory chemistry when the result is near a specification boundary.

Plausibility Check

The reported value is below the minimum.

Exercise 13: Eddy-Current Skin Depth

For conductivity $\sigma=3.5\times10^7\ \text{S/m}$ , relative permeability $1$ and frequency $100\ \text{kHz}$ , skin depth is approximately:

\delta=\sqrt{\dfrac{2}{\omega\mu\sigma}}

Use $\mu=4\pi\times10^{-7}\ \text{H/m}$ .

Solution

\omega=2\pi(100000)

\delta=\sqrt{\dfrac{2}{(2\pi100000)(4\pi\times10^{-7})(3.5\times10^7)}}=0.269\ \text{mm}

Engineering Comment

Frequency selection should match expected defect depth.

Plausibility Check

High-frequency eddy current in conductive metal has sub-millimeter skin depth.

Exercise 14: Lift-Off Signal Loss

Probe lift-off reduces signal amplitude by $18\%$ . Reference notch signal is $1.8\ \text{V}$ . Estimate signal after lift-off.

Solution

V=1.8(1-0.18)=1.48\ \text{V}

Engineering Comment

Lift-off control is part of method qualification, not an operator preference.

Plausibility Check

The signal remains below the reference but above one volt.

Exercise 15: False-Call Rate

An inspection finds $9$ false calls in $300$ inspected regions. Compute false-call rate.

Solution

f=\dfrac{9}{300}=3.0\%

Engineering Comment

False calls affect repair burden and trust, but reducing them must not reduce sensitivity.

Plausibility Check

Nine out of three hundred is three per hundred.

Exercise 16: Coverage Map Completion

An inspection plan requires $125$ scan zones. Completed records exist for $118$ zones. Compute coverage.

Solution

C=\dfrac{118}{125}=94.4\%

Engineering Comment

Unscanned zones require disposition, especially if they contain high-stress features.

Plausibility Check

Seven missing zones out of 125 leaves coverage below 95 percent.

Exercise 17: Sizing Uncertainty Gate

Measured defect size is $2.8\ \text{mm}$ , sizing uncertainty is $0.4\ \text{mm}$ and reject limit is $3.0\ \text{mm}$ . Use conservative size $a+U$ . Does it pass?

Solution

a_g=2.8+0.4=3.2\ \text{mm}

Since $3.2>3.0\ \text{mm}$ , it fails the conservative gate.

Engineering Comment

Sizing uncertainty can turn a nominal pass into a reject or retest decision.

Plausibility Check

The measured value is close to the limit.

Exercise 18: NDE Release Gate

An inspection package has POD $0.92$ , largest pore $0.42\ \text{mm}$ against $0.35\ \text{mm}$ limit, XRD guarded stress $255\ \text{MPa}$ against $250\ \text{MPa}$ limit, scan coverage $94.4\%$ against $100\%$ required and conservative defect size $3.2\ \text{mm}$ against $3.0\ \text{mm}$ limit. Decide release status.

Solution

Largest pore fails:

0.42>0.35\ \text{mm}

Guarded residual stress fails:

255>250\ \text{MPa}

Coverage is incomplete:

94.4\%<100\%

Conservative defect size fails:

3.2>3.0\ \text{mm}

Release should be held.

Engineering Comment

NDE release requires detection, sizing, coverage and disposition gates to pass together.

Plausibility Check

Multiple inspection gates fail, so the release decision is negative.

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