Spring rate
-Extension at F
-Body shear stress at F
-Body length
-All results
| Mean coil diameter D | - |
| Spring index C = D / d | - |
| Wahl factor Kw | - |
| Shear modulus G | - |
| Stress at initial tension | - |
| Wire tensile strength (estimate) | - |
| Allowable static stress | - |
| Maximum static load on the body | - |
| Body stress / allowable | - |
Method and formulas
Standard helical spring theory for round wire, so every figure can be checked by hand.
| Quantity | Formula |
|---|---|
| Mean diameter and index | D = OD − d, C = D / d |
| Spring rate | k = G d4 / (8 D3 Na), Na = Nb |
| Extension | x = (F − Fi) / k for F > Fi, else 0 |
| Wahl factor | Kw = (4C − 1) / (4C − 4) + 0.615 / C |
| Body shear stress | τ = Kw · 8 F D / (π d3) |
| Wire tensile strength | Sut = A / dm (d in mm), A and m by material |
| Allowable static stress | 45 % of Sut (35 % for stainless) |
| Body length | Lb = (Nb + 1) d |
Material constants (G; A and m) are the commonly published values for the four wire types: music wire G 81.7 GPa, A 2211 MPa mmm, m 0.145; hard drawn 79.3 GPa, 1783, 0.190; oil tempered 77.2 GPa, 1855, 0.187; stainless 69.0 GPa, 1867, 0.146. They are estimates for wire in the middle of its size range. The rate ignores the hook flexibility and uses the body coils as the active coils.
Worked example: music wire, 1.0 mm wire, 10 mm OD, 20 body coils, 5 N initial tension, 25 N load
D = 10 − 1 = 9.0 mm, C = 9.0. k = 81 700 × 14 / (8 × 9.03 × 20) = 0.7004 N/mm. Extension = (25 − 5) / 0.7004 = 28.55 mm. Kw = 1.1621, so τ = 1.1621 × 8 × 25 × 9.0 / (π × 13) = 666 MPa. Sut = 2211 / 10.145 = 2211 MPa, allowable 45 % = 995 MPa, so the body is at 67 % of the allowable and the body limit is 37.4 N. Body length = 21 × 1 = 21 mm.
Questions
How do you calculate extension spring rate?
The rate of a round-wire coil spring is k = G d^4 / (8 D^3 Na), where G is the shear modulus of the wire, d the wire diameter, D the mean coil diameter (outside diameter minus wire diameter) and Na the number of active coils. For a close-wound extension spring the active coils are the body coils; hooks add a little flexibility that is ignored here.
What is initial tension in an extension spring?
Extension springs are usually coiled with the turns pressed together, which leaves a built-in preload called initial tension Fi. The body does not start to open until the applied load exceeds Fi. After that the deflection is x = (F - Fi) / k. Initial tension is set by the coiling process and the wire; it cannot be set freely, and typical values depend on the spring index. Ask the spring maker for a realistic range.
How do you find the extension for a given load?
x = (F - Fi) / k, valid for loads above the initial tension. Below Fi the spring does not extend at all. Total length is the free (body) length plus the hook allowance plus x.
What stress is used for an extension spring body?
The body is stressed in torsion like a compression spring: tau = Kw x 8 F D / (pi d^3), with the Wahl correction factor Kw = (4C - 1) / (4C - 4) + 0.615 / C, where C = D / d is the spring index. The result is compared with a fraction of the wire tensile strength. A common static guide for the body is about 45 percent of the tensile strength for carbon steel wire and about 35 percent for stainless.
Why do extension springs usually fail at the hooks?
The hook bend has bending and stress concentration well above the body stress. For fatigue the hook, not the body, is usually the weak point. This calculator checks the body only. Use a full or large-radius transition, a lower stress at the hook, or a different end such as a threaded insert for cycled springs.
What spring index is a good choice?
An index C between about 4 and 12 is practical. Below 4 the wire is hard to coil and the stress concentration is high; above 12 the spring tangles and is hard to hold to tolerance.
What does this calculator not cover?
It covers the coil body: rate, extension, body stress, body length and a static load limit. It does not check hook or end stresses, fatigue life, relaxation, temperature or set. Use the spring maker's data for safety-critical or cycled springs. Verify critical designs.
Engineering aid for guidance. It checks the coil body only and does not cover hooks, fatigue, relaxation, temperature or set. Verify critical designs.