Shafts

§1The torsion formula

Twist shears the shaft in rings: zero at the axis, maximum at the skin. One formula for stress, one for wind-up.

τ = T rJ = 16 Tπ d³ (solid) θ = T LG J (rad) J_solid = π d⁴32 J_hollow = π (d₀⁴ − dᵢ⁴)32

G is the shear modulus (steel ≈ 80 GPa). The d³ in stress and d⁴ in stiffness are the levers of the whole page: a small diameter increase buys a lot of both.

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§2Power to torque

T = 9549 × P (kW)n (rev/min) N·m (9549 = 60 000 / 2π — derived on the Velocity & Energy page)

The running example: 15 kW at 1440 rev/min is T = 9549 × 15/1440 = 99.5 N·m. The same power at 240 rev/min — after a 6:1 reduction — is 597 N·m: gearboxes multiply the shaft-sizing problem exactly as fast as they multiply torque, which is why the slow shaft is always the fat one.

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§3Design for strength

Example 1 — diameter from allowable shear

99.5 N·m with τ_allow = 40 MPa (a modest figure that quietly covers keyways and shock): d³ = 16T/πτ = 16 × 99 500/(π × 40) = 12 670 mm³, d = 23.3 → Ø25 mm, running at 32.4 MPa. Strength alone says 25. §4 says otherwise.

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§4Design for stiffness

A shaft that winds up ruins timing, chatters gears and stores spring energy for the worst moment. Working guidance for machine drives: hold twist to about ¼–1° per metre.

Example 2 — the same shaft, judged by twist

At Ø25: J = 38 300 mm⁴, so θ = TL/GJ = 99 500 × 1000/(80 000 × 38 300) = 0.0324 rad/m = 1.86°/m — double the 1°/m guideline. Solving J for 1°/m needs 71 200 mm⁴ → d = 29.2 → Ø30 mm. Stiffness added 5 mm the stress calculation never asked for — on transmission shafts it usually does.

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§5Hollow shafts

Torsion barely uses the core, so remove it: a bore costs little stiffness and saves real mass.

Example 3 — matching the Ø30 solid with a tube

Keep the same J with dᵢ/d₀ = 0.6: d₀ = 30/(1 − 0.6⁴)^¼ = 31.1 → Ø32 × Ø19 bore. Cross-section area falls to about 0.69 of the solid’s — roughly 31 % lighter for one millimetre more outside diameter. Where inertia matters (indexing drives, robotics) the saving compounds, since the removed metal was also the slowest to accelerate.

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§6Keyways and combined loading

Real shafts carry bending from belt pulls and gear forces on top of torque, and they are slotted for keys exactly where both peak.

A standard profiled keyway costs a shaft on the order of a quarter of its torsional strength and adds a stress-raiser at its ends — sled-runner ends and generous fillets recover much of it. Combined bending-and-torsion design replaces T with an equivalent torque T_e = √(M² + T²) (and M with an equivalent moment ½(M + T_e)) — the classical shortcut for ductile shafts; a full treatment belongs with the Machine Elements pages. The habit that matters here: place keyways and shoulders away from the bending peak when the layout allows, and radius everything.

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§7Quick reference

The working core of the page on one card rack.

Torsion

τ = 16T/πd³

θ = TL/GJ · J = πd⁴/32

Torque

T = 9549 P(kW)/n

Stiffness guide

¼–1° per metre

usually governs drives

Hollow

J = π(d₀⁴−dᵢ⁴)/32

bore 0.6d₀ ≈ −31 % mass

Combined

T_e = √(M² + T²)

radius every keyway end

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Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Shafts. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to carry the subject from function and assumptions through design evidence, verification and controlled release. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Apply Shafts by beginning with the duty, not the component or software command. Convert the key ideas—design, torsion, shafts, formula, power—into measurable requirements and interfaces. Record operating and non-operating environments, duty cycle, expected life, loads, energy sources, human interaction and reasonably foreseeable abnormal conditions. When a value is not a project requirement or verified supplier datum, identify it as an assumption or illustrative value.

Create a calculation and evidence trail that another competent person can audit. Every input should carry a source, unit, revision and uncertainty or tolerance where relevant. Every model should state its boundary conditions and limitations. Keep nominal capacity separate from design capacity, and keep verification margin separate from an arbitrary safety factor. If a code or standard governs the work, confirm the applicable edition and contractual status rather than copying a number from a secondary summary.

Design for manufacture, assembly, inspection, operation and maintenance at the same time. A technically valid geometry can still fail because it cannot be fixtured, measured, cleaned, guarded, reached or replaced. Review process capability, datum or reference strategy, tolerance accumulation, access, error-proofing and changeover. Where people interact with plant, apply the hierarchy of controls and consult those who will operate, clean, maintain and recover the equipment.

Plan verification before release. Define the characteristic, method, equipment, sample or test condition, acceptance criterion, record and responsible person. Validation then asks a different question: whether the resulting system is effective and suitable in the intended use context. A passed drawing check or analysis does not by itself validate usability, maintainability or production performance.

Step-by-step operating method

Define the duty. Capture the required function, interfaces, operating environment, life, loads and unacceptable outcomes.
Establish the model. Identify governing principles, units, material or process data, assumptions and uncertainty.
Develop alternatives. Compare feasible concepts against performance, manufacturability, safety, maintainability and cost.
Verify the design. Use analysis, test, inspection or demonstration with acceptance criteria defined before execution.
Release and learn. Baseline the design, control changes, retain evidence and feed operating results into the next revision.

Illustrative design review record

Illustrative values only. Build a one-page record with the required function, input sources, assumptions, governing load or process condition, failure consequences, selected concept, verification method and acceptance criterion. Mark every numerical input as project requirement, verified supplier data, measured value, calculation output or assumption. Review the weakest evidence first. If an assumption can change safety, compliance, interchangeability or capacity, it must be resolved before release rather than buried in a calculation note.

Evidence class	Question	Release expectation
Requirement	What must the design do and under which conditions?	Approved and traceable
Input	Where did the load, property, tolerance or process limit come from?	Source, unit and revision recorded
Analysis	Which model and assumptions connect input to result?	Checkable calculation or simulation
Verification	How will conformity be demonstrated?	Method and acceptance criterion agreed
Validation	Will the solution work for intended users and conditions?	Representative use evidence

Common failure modes and recovery actions

1. Watch for

Starting detailed design before interfaces and operating limits are agreed.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Using catalogue or typical values as though they were certified project inputs.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Checking nominal performance while ignoring tolerances, degradation and foreseeable misuse.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Confusing verification of requirements with validation of user need.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Releasing drawings or procedures without configuration, inspection and change controls.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.