Helical Spring Rate⚠ unverified

Understanding the Parameters

• Wire diameter $d$ — enters as $d^4$ because the wire resists the load in torsion, and a wire's torsional stiffness scales with its polar moment of inertia $\propto d^4$. It is by far the most powerful lever on rate.

• Mean coil diameter $D$ — enters as $1/D^3$: a larger coil gives the axial force a longer moment arm ($D/2$) to twist the wire and a longer wire per turn, both softening the spring. Big coils make soft springs.

• Active coils $N$ — only the coils free to deflect count (end coils that are ground or closed do not). Rate is inversely proportional to $N$: coils act like springs in series, so more coils share the deflection and soften the assembly.

• Shear modulus $G$ — the material's resistance to shear (torsion). Use $G$, not Young's modulus $E$, because the wire is twisted, not stretched — a frequent beginner error. For steel $G \approx 79$–$81$ GPa.

Derivation (Approaching a Proof)

Model one coil as a curved wire loaded by the axial force $F$. At any wire cross-section the force produces a torque about the wire axis equal to the force times the coil radius:

$$T = F\cdot\frac{D}{2}.$$

The wire behaves as a torsion bar. Its angle of twist over a length $\ell$ is $\phi = T\ell/(GJ)$, where $J = \pi d^4/32$ is the polar second moment of the round wire. The total active wire length is the circumference times the number of coils, $\ell = \pi D N$. Using strain energy (Castigliano's theorem), the axial deflection is the derivative of the stored torsional energy with respect to $F$:

$$\delta = \frac{\partial}{\partial F}\!\int \frac{T^2}{2GJ}\,\mathrm{d}\ell = \frac{T (D/2)\,\ell}{GJ} = \frac{F (D/2)^2 (\pi D N)}{G(\pi d^4/32)} = \frac{8 F D^3 N}{G d^4}.$$

The spring rate is force over deflection:

$$k = \frac{F}{\delta} = \frac{G d^4}{8 D^3 N}.$$

The derivation neglects the small direct-shear and curvature effects (which matter for stress, via the Wahl factor, but negligibly for rate) and assumes a small helix angle. See Helical Spring Deflection for the companion deflection form.

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

History and Development

The torsion-bar model of a helical spring dates to the 19th century and is presented in Wahl's authoritative Mechanical Springs (1944) and in Shigley's Mechanical Engineering Design. It underlies every spring catalogue and CAD spring wizard. Real springs deviate slightly through end-coil effects, pitch/helix angle, and manufacturing tolerance, so the computed rate is a design estimate refined by test.

Related Concepts: Helical Spring Deflection, Helical Spring Stress, Spring Index, Spring Energy, Spring Natural Frequency, Spring Buckling

Notes: Uses shear modulus $G$ (torsion), not $E$. Count only active coils. Springs in series add compliance ($1/k$); in parallel add rate. For preliminary sizing; verify stress with Helical Spring Stress.