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.

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 but does not check hook stress.

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.

Extension Spring Calculator — Rate, Force & Initial Tension

Spring rate, spring index and the force at a given extension for a helical extension (tension) spring — including the built-in initial tension — with embedded material data. Metric and imperial, no signup.

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).

Worked example

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.

Shear modulus of common spring-wire materials.
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.
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 but does not check hook stress.
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.

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; size them separately.
Initial tension is set during coiling; the body coils do not extend until the load exceeds it.
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.

Related calculators

Compression Spring Calculator — Rate, Wahl-corrected stress, solid height and buckling, with built-in material data.
Spring Rate Calculator — Spring rate (spring constant) from wire size, coil diameter and material.
Spring Constant Calculator — Spring constant from two measured load/deflection points (Hooke’s law).

Extension Spring Calculator

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