Springs

§1Geometry and the spring index

Wire diameter d, mean coil diameter D, active coils n — plus one ratio that flavours everything else.

spring index C = Dd — aim for 4 to 12; the sweet spot near 8

Below C ≈ 4 the coil is tight to wind and its inner fibre badly overworked; above 12 it is floppy, tangle-prone and hard to guide. Active coils are those free to twist — closed and ground end coils sit out (§4). The running example throughout this page: d = 4 mm, D = 32 mm (C = 8), n = 8 active coils, spring steel with G = 79.3 GPa.

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§2Rate and deflection

The wire is a torsion bar of length π D n; wind-up at the wire becomes travel at the coil. The rate falls out in one line.

k = G d⁴8 D³ n δ = Fk = 8 F D³ nG d⁴

Example 1 — the running spring’s rate

k = 79 300 × 4⁴ / (8 × 32³ × 8) = 9.68 N/mm; under 180 N it closes δ = 180/9.68 = 18.6 mm. The exponents are the design levers: d⁴ over D³n — thicken the wire and the spring stiffens violently; add coils or diameter and it softens.

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§3Shear stress and the Wahl factor

The basic stress is the torsion formula in disguise; the Wahl factor pays for the coil’s curvature, which crowds the shear onto the inner fibre.

τ = K_w 8 F Dπ d³ K_w = 4C − 14C − 4 + 0.615C

Example 2 — stress under the 180 N load

C = 8 gives K_w = 31/28 + 0.0769 = 1.184. τ = 1.184 × 8 × 180 × 32 / (π × 4³) = 271 MPa — comfortable for static duty in spring steel; a fatigue application would want this held well lower and the surface shot-peened. Note K_w grows as C shrinks: at C = 4 the penalty is 40 %, another vote against tight coils.

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§4Ends, solid height and buckling

End treatment sets how the spring stands and how many coils actually work; two housekeeping checks keep it honest in service.

Ends. Closed-and-ground is the standard for compression springs — square seating, and total coils ≈ active + 2. Plain ends save grinding but seat crookedly. Solid height is total coils × d: the running spring’s 10 × 4 = 40 mm; design so working travel never coils it solid, or the load path becomes metal-to-metal and the rate becomes infinite. Buckling: a slender compression spring bows like the Columns page’s strut — keep free length under about 4 × D unguided, or run it over a rod or in a bore (and accept the friction) beyond that.

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§5Stored energy

U = ½ k δ² = ½ F δ

The running spring at 180 N holds U = ½ × 9.68 × 18.6² = 1.67 J — the triangle under its force–deflection line, the same area picture as the spring-work integral on the Algebra and Equations page. Energy scales with δ², so a soft, long-travel spring out-stores a stiff short one at equal peak force — the design fork between a striker spring and a hold-down.

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§6Extension, torsion and leaf springs

Three cousins, one paragraph each — the helical mathematics carries into the first two nearly unchanged.

Extension springs obey the same k and τ formulas but are wound with initial tension: no deflection occurs until the preload is overcome, and the hooks — bent, stress-raised, unpeenable — are where they fail; generous hook radii matter more than wire grade. Torsion springs load the wire in bending, not shear: rate is a moment per degree, and they should always be worked in the wind-up direction so service load tightens the coil onto its arbor. Leaf springs are stacked cantilever beams — the Beams page in laminated form — trading the helical’s compactness for load capacity and built-in friction damping.

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§7Quick reference

The working core of the page on one card rack.

Rate

k = Gd⁴/8D³n

Stress

τ = K_w·8FD/πd³

K_w = (4C−1)/(4C−4) + 0.615/C

Index

C = D/d, aim 4–12

Housekeeping

solid ht = coils × d

free length ≤ 4D unguided

Energy

U = ½kδ²

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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 Springs. 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 Springs by beginning with the duty, not the component or software command. Convert the key ideas—springs, spring, geometry, index, rate—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.