How to use this calculator
- Enter the wire and coil geometry. Enter the wire diameter d and the mean coil diameter D. The spring index C = D/d should be roughly 4–12.
- Enter the active coils. Enter the active (body) coil count Nₐ.
- Pick the material. Select the spring material to set the shear modulus G.
- Enter the initial tension and extension. Enter the built-in initial tension Fᵢ and the working extension x to get the force at that point.
- Read the results. Read the spring rate, the spring index and the force at extension (F = Fᵢ + k·x), with the initial tension echoed for reference.
How it works
The spring rate of an extension spring follows the same formula as a compression spring:
k = G · d⁴ / (8 · D³ · Nₐ)
where G is the material shear modulus, d the wire diameter,
D the mean coil diameter and Nₐ the active coils. The wire
diameter enters to the fourth power, so it is by far the most powerful way to change
stiffness, and the spring index is C = D/d.
An extension spring differs from a compression spring in one important way: it is wound
with a built-in initial tension Fᵢ that holds the coils
closed. The spring carries that load before it opens, so the force at an extension
x is
F = Fᵢ + k·x
and the coils stay shut until the applied load exceeds Fᵢ. The selected
material fills in G. Formulas follow standard spring-design practice (e.g.
Shigley, Mechanical Engineering Design, chapter 10).
For the same body-coil rate without initial tension, compare against the spring rate calculator. For compression springs, use the compression spring calculator because solid height, buckling and Wahl-corrected compression stress matter there. The spring index calculator is useful when you are converting inside, outside and mean coil diameters before checking manufacturability. For the end geometry, use the extension spring hook stress calculator.
Worked example
Verified against the live calculator
Music wire, d = 1.5 mm, mean coil D = 12 mm (so the spring
index C = 8), Nₐ = 10 active coils, with
G = 79.3 GPa and a built-in initial tension Fᵢ = 5 N:
k = 79 300 · 1.5⁴ / (8 · 12³ · 10) ≈ 2.9 N/mm
So every millimetre of extension takes about 2.9 N once the coils open. At
10 mm of extension the force is F = 5 + 2.904 · 10 ≈ 34 N.
Those are exactly the numbers the calculator shows when you load this page.
Spring material data
The shear modulus G for common spring materials — the only material
property the rate depends on. The calculator fills it in when you choose a material;
the table is here so you can audit the source.
| Material | Standard | Shear modulus G — GPa (Mpsi) |
|---|---|---|
| Music wire | ASTM A228 | 79.3 (11.5) |
| Oil-tempered | ASTM A229 | 77.2 (11.2) |
| Hard-drawn | ASTM A227 | 77.2 (11.2) |
| Chrome silicon | ASTM A401 | 77.2 (11.2) |
| Chrome vanadium | ASTM A232 | 77.2 (11.2) |
| Stainless 302/304 | ASTM A313 | 69 (10) |
Source: Standard spring-design references (Shigley, Mechanical Engineering Design, Table 10-5; ASTM wire standards). Verify against the governing standard and supplier data for production design.
Frequently asked questions
How do I calculate the spring rate of an extension spring?
Use k = G·d⁴/(8·D³·Nₐ) — the same rate formula as a compression spring — where G is the shear modulus, d the wire diameter, D the mean coil diameter and Nₐ the active coils. Enter those above and the calculator solves it. Spring rate formula explained →
What is initial tension on an extension spring?
Initial tension Fᵢ is the built-in preload wound into the spring that holds the coils tightly closed. The spring carries that force before it extends at all, so it will not open until the applied load exceeds Fᵢ.
Is this an initial tension extension spring calculator?
Yes. Initial tension is a first-class input and output. The calculator keeps Fᵢ separate from the body-coil spring rate, then solves force at extension with F = Fᵢ + k·x.
How do I find the force at a given extension?
Add the initial tension to the rate times the extension: F = Fᵢ + k·x, where k is the spring rate and x the extension beyond the closed-coil length. Below Fᵢ the spring does not move.
How is an extension spring different from a compression spring?
The rate formula k = G·d⁴/(8·D³·Nₐ) is identical. The differences are that an extension spring carries an initial tension Fᵢ (so F = Fᵢ + k·x), it pulls instead of pushes, and it has hook or loop ends whose stresses usually govern the design.
Where do extension springs fail?
Almost always at the hook or loop ends, where bending and torsion concentrate. The body-coil rate is easy to hit; the end geometry is the highest-stressed region and is the usual failure point. This calculator sizes the rate and force; use the extension spring hook stress calculator for a first-pass end-loop screen.
Does this calculate extension spring hook stress?
No. This page solves body-coil rate, initial tension and force. Hook stress depends on hook style, bend radius, loop opening, machine hooks versus crossover loops, end orientation, residual stress and supplier/manufacturer details; use the dedicated hook stress screen with drawing or catalog geometry.
Does this work in metric and imperial?
Yes — toggle SI/Imperial in the header and every input and result converts. Copy a link to share your exact configuration.
Initial Tension & Hook Stress Workflow
Extension spring searches often mix two separate questions: the body spring rate with initial tension, and the hook or loop stress at the ends. This calculator solves the first one and keeps the second boundary explicit.
| Step | Inputs | Result |
|---|---|---|
| Body-coil rate | Wire diameter, mean coil diameter, active coils and material | k = G*d^4/(8*D^3*Na), same as a compression spring. |
| Initial tension | Built-in preload Fi | The spring does not begin extending until the applied load exceeds Fi. |
| Working force | Initial tension plus extension x | F = Fi + k*x at the entered extension. |
| Hook or loop stress | Hook geometry, loop bend radius, end style and supplier data | Use the dedicated hook stress screen, then confirm with drawing/catalog data. |
Method & assumptions
- Static loading only — fatigue, set and stress relaxation are not modelled.
- The hook or loop ends are the highest-stressed region (and the usual failure point) and are not stress-checked here; use the dedicated hook stress screen and supplier data.
- Initial tension is set during coiling; the body coils do not extend until the load exceeds it.
- The body-coil rate equation matches a compression spring; the end hardware and initial tension are what change the design workflow.
- Material constants are typical published values; for a safety-critical design, verify against the governing standard and your supplier’s certified data, and have a professional engineer review it.