Torsion Spring Rate⚠ unverified

Understanding the Parameters

• Wire diameter $d$ — enters as $d^4$ via the wire's bending stiffness (its second moment of area $I = \pi d^4/64$). As with all springs, wire size is the dominant lever.

• Mean coil diameter $D$ and active coils $N$ — together set the active wire length $\ell = \pi D N$. Rate is inversely proportional to this length: a longer, larger, or more-coiled wire is more compliant in bending, so it winds up more easily per unit torque.

• Young's modulus $E$ — the bending stiffness of the material. Using $E$ (not $G$) is the defining feature of torsion springs and a common source of error.

• The constant $10.8$ — Shigley's empirical value. The purely theoretical constant for rate per turn is $\approx 10.2$ (from $64/(2\pi)$); it is raised to $10.8$ to account for friction between coils and end effects.

Registry note: the constant $10.8$ corresponds to Shigley's rate per revolution (turn), i.e. $k'$ in moment per turn — but the output is labelled per radian (N·m/rad). Per-turn and per-radian differ by $2\pi$, so this is a likely unit inconsistency: either the constant should be $\approx 61$–$64$ for a true per-radian rate, or the output unit should be per-revolution. Flagged in Known Issues.

Derivation (Approaching a Proof)

Treat the coiled wire as a curved beam in bending. An applied moment $M$ (about the coil axis) bends the wire; the angular deflection of a beam of length $\ell$ and bending stiffness $EI$ under a constant moment is

$$\theta = \frac{M \ell}{E I}.$$

Substitute the wire length $\ell = \pi D N$ and the round-wire second moment $I = \pi d^4/64$:

$$\theta = \frac{M (\pi D N)}{E(\pi d^4/64)} = \frac{64\, M D N}{E d^4}.$$

The rate per unit radian is therefore

$$k_{\text{rad}} = \frac{M}{\theta} = \frac{E d^4}{64\, D N}.$$

Expressed per turn (multiplying $\theta$ by $2\pi$ to convert), the constant becomes $64/(2\pi) \approx 10.2$; Shigley bumps it to $10.8$ for inter-coil friction and end effects, giving the registry form $k = Ed^4/(10.8\,DN)$. The value therefore represents moment per revolution, which is why the per-radian unit label is suspect (see note).

Dimensional check (per-turn basis). $[k] = \dfrac{\text{Pa}\cdot\text{m}^4}{\text{m}} = (\text{N/m}^2)\,\text{m}^3 = \text{N}\cdot\text{m}$ per turn (turn is dimensionless). ✓

History and Development

Helical torsion springs and their bending-based analysis are covered in Wahl's Mechanical Springs and codified in Shigley with the empirical $10.8$ constant and a bending stress-correction factor for the inner fibre. They are ubiquitous — hinges, counterbalances, clothespins, retractable mechanisms — wherever a return torque rather than a return force is needed.

Related Concepts: Torsion Spring Stress, Helical Spring Rate, Spring Index, Beam Bending Stress, Spring Energy

Notes: Wire bends (uses $E$, not $G$). Watch the per-turn vs per-radian convention (see registry note). Coils tighten under load, so the mean diameter and active length change slightly with deflection.