Helical springs

The Springs module designs cylindrical helical springs of round wire, cold coiled: compression springs after EN 13906-1, extension springs after EN 13906-2 and torsion springs after EN 13906-3. It gives the rate, the forces or moments at the two working positions, the stresses against their limits and, depending on the kind, the solid length, buckling, the hooks, the initial tension and the fatigue safety. The module sits in the Motion and drives group of the navigation.

Springs stand on their own; one project can hold several. The centre of the module draws each spring: a Side view of a compression or extension spring, free and at L₂, coloured by its stress use, or an End view of a torsion spring with its legs at α₁ and α₂. The 3D tab builds the spring at its real wire and coil diameters and coils, with its seats, loops or legs: it rests at its first working position and hovering compresses or extends it to L₂ and back (or winds a torsion spring from α₁ to α₂), coloured by its stress use as it goes. The choice is kept for the module. In 3D, drag to orbit and scroll to zoom; the model stands still and plays its motion once when the pointer moves onto it or it gets the keyboard focus (never with reduced motion). Without WebGL the drawing stays, with a note.

Inputs

Choose + Compression, + Extension or + Torsion at the top of the list. The inspector shows only the groups that apply to the kind.

Spring

Kind: Compression (EN 13906-1), Extension (EN 13906-2) or Torsion (EN 13906-3).
Wire diameter d (mm) and Mean coil diameter D (mm). Their ratio is the spring index $w = D / d$ .
Active coils n: the coils that deflect.
Total coils nₜ: n + 2 for compression springs and n for the others when empty.
Ends (compression only): Closed and ground, solid length $n_{t} \cdot d$ , or Closed, not ground, $(n_{t} + 1.5) \cdot d$ .

Material

Wire: SL, SM, SH, DM or DH (EN 10270-1), or 1.4310 stainless (EN 10270-3). SL, SM and SH are for static duty, DM and DH for dynamic duty.
Tensile strength Rm (MPa): empty takes the wire table's minimum at the wire diameter. Enter the value from the wire certificate when you have it.
Shear modulus G, Young's modulus E (MPa) and Density (kg/dm³): empty takes the wire's values at room temperature.

Lengths (compression and extension)

Free length L₀, Working length L₁ and Working length L₂ (mm). L₂ is the more deflected position: shorter than L₁ for a compression spring, longer for an extension spring. An extension spring's lengths are measured inside the hooks, and it also has Initial tension F₀ (N).

Hooks (extension)

Hook form: German loop (tight bend), English loop (wide bend) or None (not assessed) for screwed-in plugs and other ends. Loop mean radius r₁ (default D/2) and Bend mean radius r₂ (default d for the German loop, D/4 but at least d for the English loop) set the stress factors.

Angles and legs (torsion)

Deflection α₁ and Deflection α₂ (°) from the unloaded position, α₂ the larger. Leg 1 and Leg 2 (mm) add to the deflecting length. Arbor diameter (mm), if the spring is wound on one, is checked against the inside diameter at α₂.

Loading

Load: Static (fewer than about 10⁴ strokes) or Cyclic (fatigue) between L₁ and L₂ (or α₁ and α₂).
Shot peened and Fatigue strength τkO₀ / σqO₀ (lower stress 0) (MPa): cyclic compression and torsion springs. Empty estimates the fatigue strength for 10⁷ strokes as 0.23·Rm for compression springs (0.30·Rm shot peened) and 0.40·Rm for torsion springs (0.50·Rm shot peened). Enter a value from the spring maker or from EN 13906-1's diagrams when you have one.
Seating (buckling) (compression): ν 0.5 both ends fixed, ν 0.7 fixed and pivoting, ν 1 both pivoting, ν 2 one end free.
Bore diameter (compression): the bore the spring works in, checked against the outside diameter at the solid length.

Requirements

Collapsed: Min fatigue safety (1.1 by default) and, for compression springs, Check the stress at the solid length (on by default).

How it is calculated

Compression and extension springs (EN 13906-1 clause 9):

rate $R = \frac{G \cdot d^{4}}{8 \cdot D^{3} \cdot n}$ (9.4, eq. 6);
shear stress $τ = \frac{8 \cdot D \cdot F}{π \cdot d^{3}}$ (9.5, eq. 7), corrected $τ_{k} = k \cdot τ$ with Bergsträsser's $k = \frac{w + 0.5}{w - 0.75}$ (clause 7, eq. 1);
least working length $L_{n} = L_{c} + S_{a}$ with the sum of least gaps $S_{a} = (0.0015 \cdot D^{2} / d + 0.1 \cdot d) \cdot n$ , 1.5 times wider under cyclic load (9.9, eq. 14);
buckling deflection with the seating coefficient ν (9.14, eq. 29), natural frequency (9.12, eq. 20) and the outside diameter's growth at the solid length (9.11, eq. 19).

The static check uses the uncorrected τ. The corrected τk is used for fatigue only, as clause 7 and 10.2 state.

Torsion springs: rate $R = \frac{E \cdot d^{4}}{3667 \cdot D \cdot n_{k}}$ in N·mm/°, with the legs added to n_k, and the bending stress $σ_{q} = q \cdot \frac{32 \cdot M}{π \cdot d^{3}}$ .

Kind	Static limit	Other limits
Compression	τ ≤ 0.5·Rm	τc ≤ 0.56·Rm at the solid length (10.1.1)
Extension	τ ≤ 0.45·Rm	hook bending ≤ 0.7·Rm, bend torsion ≤ 0.45·Rm
Torsion	σq ≤ 0.7·Rm

Fatigue (cyclic compression and torsion springs) runs on a Goodman-type line from the fatigue strength at a lower stress of 0 to 0.5·Rm (compression) or 0.7·Rm (torsion), in place of the standard's diagrams. The fatigue safety is the permissible upper stress over the actual upper stress, at the actual lower stress.

Results

The results column shows the checks and the key results: Rate, Shear stress τ at L₂ (torsion: Bending stress σq at α₂) with its share of the limit, and where they apply Fatigue safety, Hook bending σb at F₂ and Least working length Lₙ.

The Details card lists Spring index / correction, Tensile strength Rm, the stresses, Stroke, At the solid length, Buckling, Initial tension, Hooks, Body length, Diameters, Natural frequency, Fatigue (10⁷ strokes), Mass and Notes.

Checks

Check	Kind	Passes when
Static stress	all	τ (σq for torsion) at L₂ or α₂ is within the static limit
Solid stress	compression	τc ≤ 0.56·Rm, unless switched off
Coil clash	compression	$L_{2} \geq L_{n}$
Buckling	compression	the spring does not buckle before s₂
Bore	compression	the outside diameter at solid length fits the bore
Hook bending, hook torsion	extension	within 0.7·Rm and 0.45·Rm at F₂
Initial tension	extension	τ₀ is not above the range that can be coiled
Fatigue	cyclic compression, torsion	safety ≥ Min fatigue safety
Arbor	torsion	the inside diameter at α₂ clears the arbor

The template

The Springs template holds three springs, all in DH wire:

Spring	Key results
SP1 valve return spring, compression, d 2.5 × D 20 mm, n 8.5, cyclic, shot peened	rate 5.852 N/mm; F₁ 58.52 N, F₂ 128.7 N; τ 419.7 MPa against 950.8 MPa; Lₙ 32.5 mm against L₂ 38 mm; τc 643.8 MPa against 1065 MPa; cannot buckle; 266.6 Hz; fatigue safety 1.341
SP2 door extension spring, d 2 × D 16 mm, n 20, F₀ 18 N, English hooks	rate 1.99 N/mm; F₂ 107.5 N; τ 547.7 MPa against 891 MPa; τ₀ 91.67 MPa within 77.43–122 MPa; hook bending 1242 MPa against 1386 MPa (90 %)
SP3 lid torsion spring, d 2 × D 16 mm, n 6, legs 25 mm, arbor 11 mm	rate 8.873 N·mm/°; 798.5 N·mm at 90°; σq 1132 MPa against 1386 MPa; inside diameter 13.36 mm at α₂

All checks pass. For SP1 the Rm of 1902 MPa is the wire table's minimum at 2.5 mm, and the fatigue strength is estimated as 30 % of Rm (570.5 MPa); a note in the results says so.

Fixing a failing check

Static stress: a thicker wire or a smaller mean diameter, a stronger wire grade, or less deflection.
Solid stress: more active coils (the same stroke at a lower stress), or a free length closer to L₂. Switch the check off only when the spring can never be pressed solid; EN 13906-1 recommends that cold coiled compression springs can be.
Coil clash: fewer active coils or a thinner wire (shorter solid length), or a longer L₂.
Buckling: guide the spring in a bore or on a mandrel, fix both ends (ν 0.5), or reduce the slenderness L₀/D.
Fatigue: shot peening, a DH wire, a smaller stroke, or a lower mean stress.
Hooks: the English loop or a larger bend radius r₂, or screwed-in plugs.
Arbor: a smaller arbor; a torsion spring's coils close in as it winds up.

Standards and scope

Compression springs after EN 13906-1:2013, the edition consulted for this page. Extension and torsion springs follow EN 13906-2 and EN 13906-3 as described in the method notes.
Cold coiled springs of round wire only. Hot coiled, conical, barrel-shaped and disc springs, and wire of other cross-sections, are not covered.
Static limits are the usual design values above. EN 13906-1 limits static stress by the relaxation allowed (10.2); relaxation is not computed.
Material values are at room temperature. The temperature dependence of G and E (8.2), transverse loading (9.13) and impact loading (9.15) are not applied.
Fatigue uses a Goodman-type line instead of the standard's fatigue diagrams.
The fatigue of extension springs is not assessed, because it depends on the hooks.
The initial tension range is the spring makers' preferred range, in place of the standard's diagram. Wire strengths are typical minimum values of EN 10270-1 and EN 10270-3, interpolated by wire diameter.

From the command line

axm new springs.axm.json --template spring-set
axm spring update springs.axm.json SP1 --patch '{"activeCoils":9.5}'
axm solve springs.axm.json

axm guide springs lists every field.