Free Torsion Spring Calculator

A torsion spring is a mechanical component that stores rotational energy and releases it as a controlled torque when one arm is rotated relative to the other. These devices appear in everyday items such as clothespins, safety pins, door hinges, and automotive mechanisms. Two basic configurations exist: a helical torsion spring, where the wire is coiled into a helix and twisting occurs about the coil axis, and a torsion bar, where torsion is applied directly along the bar’s own axis. This article focuses on helical torsion springs and explains how a helical torsion spring calculator can be used to determine critical design values — including torque, bending stress, spring rate, and dimensional changes — quickly and accurately.

Key Geometric Parameters

A helical torsion spring is formed from a wire (usually round in cross‑section) that is wound into a cylindrical shape. The three essential diameters are:

• Inner diameter $D_i$
• Outer diameter $D_o$
• Wire diameter $d$

The outer diameter is simply the inner diameter plus twice the wire diameter:

$$D_o = D_i + 2d$$

The mean coil diameter $D$ is the average of the inner and outer diameters:

$$D = \frac{D_i + D_o}{2} = D_i + d$$

A useful dimensionless quantity is the spring index $C$, defined as:

$$C = \frac{D}{d}$$

Spring indices typically range between 4 and 12; lower values indicate a tighter coil and higher stress concentration.

The number of turns in the spring is described with three terms:

• Body turns $N_b$ — the number of full revolutions visible on the coil body, plus a fractional allowance for any incomplete turn.
• Active turns $N_a$ — the turns that actually participate in the deflection.
• End turns $N_e$ — the contribution from the straight ends. The common relationship is $N_a = N_b + N_e$.

These dimensions and turn counts form the input set for any torsional spring rate calculator or torsion spring stress calculator.

Bending Stress in a Torsion Spring

When a torque $M$ is applied to one end of the spring, the wire experiences bending stress. The bending stress $\sigma_b$ for a round‑wire helical torsion spring is given by:

$$\sigma_b = K \cdot \frac{32M}{\pi d^{3}}$$

Here $K$ is a stress correction factor that accounts for curvature. Two commonly used expressions exist:

$$K_i = \frac{4C^{2} - C - 1}{4C(C-1)} \qquad K_o = \frac{4C^{2} + C - 1}{4C(C+1)}$$

$K_i$ (for the inner fibre) is always larger and is typically used for design because the highest stress occurs on the inside of the coil. The torque $M$ can be expressed as the product of a force $F$ applied at a distance $l$ from the spring axis: $M = F \cdot l$. Using a torsion spring stress calculator, you can verify that the computed stress does not exceed the material’s yield strength.

Spring Rate, Torque, and Angular Deflection

The torsional spring rate $k$ relates the applied torque to the resulting angular deflection $\theta$:

$$k = \frac{M}{\theta}$$

The rate can be expressed in torque per radian ($k_{\text{rad}}$) or torque per full turn ($k_{\text{rot}}$). Because one full rotation equals $2\pi$ radians:

$$k_{\text{rot}} = 2\pi \, k_{\text{rad}}$$

For a helical torsion spring, the angular deflection in radians can be derived from the material and geometry using:

$$\theta = \frac{M \cdot 180 \cdot D \cdot N_a}{E \cdot d^{4}}$$

where $E$ is the Young’s modulus of the material (e.g., 200 GPa for steel). This formula allows you to predict how far the spring will wind for a given torque. A torsion spring torque calculator often solves for any unknown quantity when the other parameters are supplied.

Coil Diameter Change Under Load

As a torsion spring is wound, the coil diameter decreases. The new mean diameter $D_{\text{load}}$ relates to the original mean diameter $D$ and the number of body turns:

$$\frac{D_{\text{load}}}{D} = \frac{N_b}{N_b + \frac{\theta}{2\pi}}$$

The loaded inner diameter is therefore:

$$D_{i,\text{load}} = D_{\text{load}} - d$$

This reduction is especially important when the spring must fit over a pivot shaft. If the inner diameter shrinks below the shaft diameter, the spring will bind. Designers routinely check this condition using a helical torsion spring calculator.

Example Calculation

Suppose a steel helical torsion spring has the following characteristics:

• Wire diameter $d = 1\ \text{mm}$
• Mean coil diameter $D = 12\ \text{mm}$
• Number of active turns $N_a = 5$
• Young’s modulus $E = 200\,000\ \text{N/mm}^2$ (200 GPa)
• Required torque $M = 0.05\ \text{N·m}$

First, convert the torque to consistent units: $0.05\ \text{N·m} = 50\ \text{N·mm}$. The angular deflection in radians is:

$$\theta = \frac{50 \times 180 \times 12 \times 5}{200\,000 \times 1^{4}} = \frac{540\,000}{200\,000} = 2.7\ \text{rad}$$

Converting to degrees: $2.7 \times \frac{180}{\pi} \approx 154.7^\circ$. The spring rate in N·mm/rad is:

$$k = \frac{M}{\theta} = \frac{50}{2.7} \approx 18.52\ \text{N·mm/rad}$$

or in N·m/rad: $0.01852\ \text{N·m/rad}$. This example illustrates how a torsional spring rate calculator works in practice.

Using the Tool

The torsion spring calculator presented here accepts any combination of known inputs — such as wire diameter, coil diameter, number of turns, torque, or angular deflection — and computes the remaining values. It also reports bending stress and the loaded coil diameter, helping you verify that your design stays within safe limits. Whether you are selecting an off‑the‑shelf spring or designing a custom one, this tool provides the essential calculations in seconds.