Torsion Spring Calculator — Spring Rate, Torque, Bending Stress & Diameter Change

Use our torsion spring calculator to discover everything about these springy devices. Keep reading to learn:

• What is a torsion spring?
• What are the quantities defining a torsion spring?
• How do I calculate the stress of a torsion spring?
• The formulas to calculate a torsion spring's force, torque, and spring rate.
• And much more!

You will find this tool of great utility whenever torsional springs appear in your studies or work!

What is a torsion spring?

A torsion spring is a mechanical device capable of storing and releasing rotational energy, usually delivering a defined amount of torque. Torsion springs are everywhere, from mundane applications like clothespins and safety pins to clipboards, garage doors, and mouse traps.

As the name suggests, such devices work thanks to torsion — a rotation. We can define two types of torsion springs, distinguishing them by the place where the torsion happens in the element the spring is made of:

• If the torsion is applied at the axis of a helical-shaped spring created by twisting our element, we are dealing with a helical torsion spring.
• If the torsion is applied on the axis of the element itself, we have a torsion bar.

A helical torsion spring is a wire (usually round in section) coiled in a cylindrical shape. The ends of the spring are straight, poking out of the cylinder at a certain angle. By pushing on one of the ends (and keeping the other fixed), we obtain a response from the spring. Every device has a defined rate, which allows us to calculate the torque provided by the spring when loaded at a certain angle.

Quantities that define a torsion spring

Take a look at a torsion spring from the "base" side: you will see a thick circle where we can identify three main quantities:

• The inner diameter of the spring D_i;
• The outer diameter of the spring D_o; and
• The wire diameter d.

The relationship between these quantities is:

D_o = D_i + 2d

We can also define the average (mean) diameter of the spring, D:

D = D_i + d = (D_i + D_o) / 2

Calculations for a torsion spring: stress

Despite their name, the wire in a helical torsion spring works mainly in bending, not in torsion. The maximum bending stress in the wire is:

σ = 32 · M / (π · d³)

• σ — Maximum bending stress (MPa or psi)
• M — Bending moment / torque applied to the spring (N·mm or lbf·in)
• d — Wire diameter (mm or in)

To keep the spring in the elastic region (so it returns to its original shape after loading), the stress must stay below the material's yield strength.

How to calculate a torsion spring force, torque, and spring rate

The spring rate (or spring constant) of a round-wire helical torsion spring tells you how much torque the spring delivers per unit of angular deflection. The formula is:

k = E · d⁴ / (10.8 · D · n)

• k — Spring rate (torque per revolution)
• E — Young's modulus of the wire material (MPa or psi)
• d — Wire diameter
• D — Mean coil diameter
• n — Number of active coils

The factor 10.8 (instead of the theoretical 10.2) accounts for friction between the coils. Once you know the spring rate, the torque (bending moment) at a deflection angle α (expressed in revolutions, where one full turn = 360°) is simply:

M = k · α

If a straight leg of length L pushes on a load, the force at the tip of the leg is:

F = M / L

How to calculate a torsion spring's diameter changes

When you wind a torsion spring in the direction that tightens the coils, its diameter shrinks (and the body gets longer). The loaded mean diameter is:

D' = n · D / (n + α)

where α is the angular deflection in revolutions. The loaded inner diameter then becomes:

D_i' = D' − d

This matters because the spring must never bind on its shaft (arbor) when fully deflected — leave clearance!

How to use our torsion spring calculator

1. Pick your unit system: metric (mm, MPa) or US/Imperial (in, psi).
2. Select the spring material to auto-fill Young's modulus, or choose Custom to type your own value.
3. Enter the wire diameter, inner diameter and number of active coils.
4. Enter the deflection angle in degrees.
5. Optionally, enter the moment arm (leg length) to also get the force at the leg.
6. Read the spring rate, torque, maximum bending stress and the loaded diameters in the results panel.

How to calculate a torsion spring's torque: an example

Consider a steel torsion spring (E ≈ 207,000 MPa) with a wire diameter d = 2 mm, an inner diameter D_i = 20 mm and n = 5 active coils, deflected by 90°.

1. Mean diameter: D = 20 + 2 = 22 mm.
2. Spring rate: k = 207,000 · 2⁴ / (10.8 · 22 · 5) ≈ 2,788 N·mm per turn.
3. Deflection in turns: α = 90° / 360° = 0.25 turn.
4. Torque: M = 2,788 · 0.25 ≈ 697 N·mm ≈ 0.70 N·m.
5. Stress: σ = 32 · 697 / (π · 2³) ≈ 888 MPa.

FAQs

What is a torsion spring used for?

Torsion springs store rotational energy and return torque. They are used in clothespins, mouse traps, clipboards, garage doors, vehicle suspensions (torsion bars) and countless mechanisms that need a controlled rotational force.

Is a torsion spring loaded in torsion or bending?

Despite the name, the wire of a helical torsion spring is loaded mostly in bending. That is why its stress is computed with the bending formula σ = 32M/(πd³) rather than a shear/torsion formula.

Why does the spring diameter decrease when loaded?

Winding the spring in the closing direction adds coils' worth of angle, tightening the helix. The mean diameter becomes D' = nD/(n + α), so it always shrinks as the deflection α grows — remember to leave clearance on the arbor.

What value of Young's modulus should I use?

Typical values: music wire / steel ≈ 207 GPa (30 × 10⁶ psi), stainless steel 302 ≈ 193 GPa (28 × 10⁶ psi), phosphor bronze ≈ 103 GPa (15 × 10⁶ psi), beryllium copper ≈ 128 GPa (18.5 × 10⁶ psi).