# Spring Rate Calculator

Calculate coil spring rate from wire diameter, coil diameter, active coils, and material. Formula: k = Gd⁴/(8D³Na). Supports steel, stainless, bronze.

## What this calculates

The spring rate (stiffness) of a helical coil spring depends on four factors: wire diameter, mean coil diameter, number of active coils, and the wire material's shear modulus. The formula k = Gd^4 / (8D^3Na) lets you predict the force per unit deflection before building a single prototype. This calculator handles common spring materials and gives results in both metric and imperial units.

## Inputs

- **Wire Diameter (d)** (mm) — min 0
- **Mean Coil Diameter (D)** (mm) — min 0 — Mean diameter = outer diameter minus wire diameter.
- **Number of Active Coils (Na)** — min 0.5 — Total coils minus inactive end coils (usually 1-2 per end).
- **Wire Material** — options: Steel / Music Wire (G = 79.3 GPa), Stainless Steel 302 (G = 69 GPa), Phosphor Bronze (G = 41.4 GPa), Beryllium Copper (G = 48.3 GPa), Inconel X-750 (G = 79.3 GPa), Titanium (G = 41.4 GPa), Custom (enter G below)
- **Custom Shear Modulus (G)** (GPa) — min 0 — Only used when material is set to Custom.

## Outputs

- **Spring Rate (k)** (N/mm) — Force per unit deflection
- **Spring Rate** (lb/in) — Spring rate in pounds per inch
- **Spring Index (C)** — D/d ratio (ideal range: 4-12)
- **Force at 10 mm Deflection** (N) — Spring force when compressed/extended 10 mm
- **Natural Frequency (0.1 kg load)** (Hz) — Resonant frequency with a 100 g mass attached

## Details

The fundamental formula for a helical compression or extension spring:

k = G x d^4 / (8 x D^3 x Na)

Where:

  - k = spring rate (N/mm or lb/in)

  - G = shear modulus of the wire material

  - d = wire diameter

  - D = mean coil diameter (OD minus one wire diameter)

  - Na = number of active coils

Notice that wire diameter has a fourth-power effect: doubling wire diameter makes the spring 16 times stiffer. Coil diameter has a cube effect in the denominator: doubling coil diameter makes the spring 8 times softer.

The spring index C = D/d is an important design parameter:

  - C  Very stiff, hard to manufacture, high stress concentration

  - C = 4-12: Ideal manufacturing range

  - C > 12: Prone to tangling and buckling, harder to control tolerances

Active coils are the coils that actually deflect. Closed and ground ends are inactive. For closed ends, subtract 2 from total coils. For closed and ground ends (most common), also subtract 2.

## Frequently Asked Questions

**Q: What is the difference between spring rate and spring constant?**

A: They are the same thing, just different names. Spring rate, spring constant, and spring stiffness all refer to the force per unit deflection (k), measured in N/mm, N/m, or lb/in. Hooke's Law uses the same value: F = kx.

**Q: Why does wire diameter have such a large effect?**

A: Wire diameter appears to the fourth power in the formula because the spring rate depends on the wire's resistance to torsion, which scales with the polar moment of inertia (proportional to d^4). A small increase in wire gauge produces a dramatic increase in stiffness. Going from 2 mm to 3 mm wire increases the spring rate by (3/2)^4 = 5.06 times.

**Q: How do I determine the number of active coils?**

A: Count the total number of coils in the spring, then subtract the inactive end coils. For closed ends (the most common type), subtract 2. For closed and ground ends, also subtract 2. For open ends, subtract 0. If you are designing from scratch, choose the number of active coils to achieve your target spring rate given the other parameters.

**Q: What shear modulus should I use for my material?**

A: For carbon steel and music wire (the most common spring materials), use 79.3 GPa (11,500 ksi). For stainless steel 302/304, use 69 GPa. For phosphor bronze, 41.4 GPa. These values are at room temperature; at elevated temperatures, the modulus drops. Consult the wire manufacturer's data sheet for exact values at your operating temperature.

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Source: https://vastcalc.com/calculators/physics/spring-rate
Category: Physics
Last updated: 2026-04-08