Torsion Spring Calculator: Rate, Angle and Stress

The spring rate per turn, the angle of deflection and the bending stress of a helical torsion spring.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

A torsion spring is loaded by a moment, and its wire is in bending. The rate per turn is k = d⁴ E / (10.8 D N), where the 10.8 (rather than the theoretical 10.2) allows for friction between coils and the arbor. The angle is the moment divided by the rate.

The bending stress is raised on the inside of the coil by a curvature factor Ki that depends on the spring index C = D / d. A torsion spring should be wound up in the direction that closes the coils, and the stress compared with the wire strength at working and maximum deflection.

Formulas used

  • k = \frac{d^4 E}{10.8\,D\,N}\ \text{per turn}
  • \theta = M / k
  • \sigma = K_i\frac{32 M}{\pi d^3}

Worked example

With the default numbers in the calculator above:

  1. Spring index: 7.5
  2. Spring rate per turn: 3.391 N m/turn
  3. Angle of deflection: 63.7 degrees
  4. Bending stress (inside of the coil): 848.2 MPa

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Which way should a torsion spring wind?

In the direction that tightens the coils. Winding it open puts the stress the wrong way.

Why do the coils need an arbor?

It supports the spring and stops it bowing. The coils get smaller as it winds, so leave a clearance.

What units is the rate in?

Moment per turn. Divide by 360 for per degree.

Related calculators

Further reading: Shigley, Mechanical Engineering Design (mechanical springs).

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