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Torsional spring calculator

A spring of 2 mm wire, 16 mm mean coil diameter, 8 active coils and E of 206000 N/mm² has a rate of 402.34375 N·mm per radian. Enter the four values from the drawing to get the rate and the spring index. Free in your browser.

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Worked example

If you enter

Wire diameter: 2
Coil diameter: 16
Active coils: 8
Elastic modulus: 206,000

You get

Spring index: 8 N·mm/rad
Angular stiffness: 402.34375
Spring rate: 402.34375
Kappa: 402.34375

A torsion spring stores energy by bending its round wire as the body coils wind or unwind under an applied moment. Hinge, clip, counterbalance, and ratchet designers therefore need the angular stiffness — moment per radian of wind-up — from the four quantities that appear on a drawing: wire diameter d, mean coil diameter D, number of active coils N, and the elastic (Young) modulus E of the wire alloy. This torsional spring calculator applies the classical bending identity kappa equals E times d to the fourth, divided by sixty-four times D times N, which follows from beam theory once the active length is taken as pi D N and the second moment of the circular wire as pi d to the fourth over sixty-four. Diameters in millimetres and E in newtons per square millimetre yield the rate in newton millimetres per radian.

What torsional-spring angular stiffness means and why κ = E d⁴ / (64 D N)

A torsion spring is a helical coil whose working load is a moment about the coil axis, not an axial force that closes the pitch. Despite the name, the round wire is loaded in bending, not in torsion: each increment of wind-up increases the curvature of the wire the way a cantilever would. For a prismatic beam the rotation theta in radians equals M L over E I.

The second moment of a circular cross-section is pi times d to the fourth over sixty-four. The active length of the body is the mean circumference times the number of working coils, pi D N, when any straight legs have already been folded into the active-coil count the caller supplies. Substituting those two identities cancels pi and yields kappa equals M over theta equals E d to the fourth over sixty-four D N.

Wire diameter sits in the numerator to the fourth power, so a ten-percent change in d moves the rate by about forty-six percent — that is why stock charts list rate against wire size first. Mean coil diameter and active coils sit in the denominator to the first power, so a larger envelope or more working turns is linearly softer.

Elastic modulus E is a material property, not a geometry knob: music wire and hard-drawn steel sit near two hundred and seven thousand newtons per square millimetre, oil-tempered chrome-silicon near two hundred thousand, austenitic stainless near one hundred and ninety-three thousand, and phosphor bronze near one hundred and three thousand.

This calculator does not apply a Wahl-style stress correction, does not add unused end coils on its own, and does not convert millimetres into metres: feed millimetre diameters and E in newtons per square millimetre (megapascals) and the angular stiffness comes back in newton millimetres per radian.

Do not confuse this identity with shaft rotational stiffness kappa equals G J over L, which twists a prismatic bar in shear, or with helical compression spring rate k equals G d to the fourth over eight D cubed Na, which stores energy by twisting the wire under axial load. Those are different search intents and different formulas, and mixing them is a common exam trap.

What you can do with it

Hinge and lid counterbalance sizing

Convert millimetre wire, mean coil, active turns, and handbook E into angular stiffness in N·mm/rad before locking a body diameter against an open-and-close moment budget.

Classroom torsion-spring homework

Auto-grade the κ = E d⁴ / (64 D N) identity and the millimetre–megapascal unit chain against a deterministic oracle instead of a rounded spreadsheet that mixes G with E.

FAQ

What units should I use for d, D, and E?

Millimetres for both diameters and newtons per square millimetre (megapascals) for Young modulus E. That pair yields angular stiffness in newton millimetres per radian, the unit this landing returns.

Is this the same as shaft torsional stiffness or compression spring rate?

No. Shaft stiffness is κ = G J / L with the bar in shear. Helical compression rate is k = G d⁴ / (8 D³ Na) with the wire in torsion. A torsion spring bends the wire; this page uses κ = E d⁴ / (64 D N).

How much does the wire diameter change the spring rate?

A lot, because it enters to the fourth power: wire about ten percent thicker gives a rate roughly forty-six percent higher. Mean coil diameter and active coils act only linearly, so a wider coil or more turns makes the spring proportionally softer.

Can I enter a fractional number of coils?

Yes. Values such as 6.75 are accepted when a drawing quotes a partial turn, as long as the count is above zero. The calculator does not add end coils or legs on its own, so include them in the count if they wind.

How do I get the wind-up angle from the rate?

Divide the moment you apply by the rate. With the moment in newton millimetres, the result is the wind-up in radians. The calculator itself returns only the static angular stiffness, without the angle and without any stress correction.

Related tools

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/phys/torsional-spring

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/phys/torsional-spring \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"wire_diameter":2,"coil_diameter":16,"active_coils":8,"elastic_modulus":206000}'

const res = await fetch("https://api.kit.forhosting.com/phys/torsional-spring", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "wire_diameter": 2,
    "coil_diameter": 16,
    "active_coils": 8,
    "elastic_modulus": 206000
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/phys/torsional-spring",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "wire_diameter": 2,
    "coil_diameter": 16,
    "active_coils": 8,
    "elastic_modulus": 206000
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/torsional-spring", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"wire_diameter":2,"coil_diameter":16,"active_coils":8,"elastic_modulus":206000}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"wire_diameter":2,"coil_diameter":16,"active_coils":8,"elastic_modulus":206000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/torsional-spring", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "wire_diameter": 2,
  "coil_diameter": 16,
  "active_coils": 8,
  "elastic_modulus": 206000
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "phys.torsional_spring",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Technical notes

A coil count that is not strictly positive fails with invalid_input and is not billed.

Input contract, millimetre units, response shape, and error behaviour

Numerics are rounded to twelve decimal places for stable goldens. Example: wire two millimetres, mean coil sixteen millimetres, eight active coils, and E of two hundred and six thousand newtons per square millimetre yields spring index eight and rate 402.34375 newton millimetres per radian, because two hundred and six thousand times sixteen divided by sixty-four times sixteen times eight is exact. Pair the rate with your own moment if you need wind-up angle; this capability solves static angular stiffness only. POST a JSON body with four required finite numbers: wire_diameter (aliases d, wire_d, wire, diameter_wire), the strictly positive wire diameter in millimetres; coil_diameter (aliases D, mean_coil_diameter, mean_d, mean_diameter), the strictly positive mean coil diameter in millimetres, which must exceed the wire so the inside diameter D minus d stays positive; active_coils (aliases n, N, Na, n_active, coils), the strictly positive count of coils that actually wind — fractional values such as 6.75 are valid when a drawing quotes a partial turn; and elastic_modulus (aliases E, youngs_modulus, young_modulus, modulus, young_e), the strictly positive Young modulus of the wire in newtons per square millimetre. That millimetre and megapascal pair is the contract that makes angular_stiffness come back in newton millimetres per radian without a hidden unit conversion. Fields may arrive as JSON numbers or numeric strings so form widgets and spreadsheet paste stay convenient. Missing fields, empty strings, NaN, Infinity, non-positive diameters or modulus, a coil no larger than the wire, or magnitudes above one trillion fail with invalid_input naming the field. If active_coils is zero or negative the task fails, because kappa equals E d to the fourth over sixty-four D N is undefined when the coil count is not positive; that is the documented error of this landing. Those failures are not billed. On success the body echoes the four rounded inputs, reports spring_index as D divided by d, returns angular_stiffness as the primary result in newton millimetres per radian, repeats that value as spring_rate and kappa, and includes unit, formula, a substituted equation, method torsion_spring_bending, and a short human-readable message.

Where a torsional spring calculator earns its keep

Hinge and lid designers use a torsional spring calculator before they lock a wire size and body diameter, so the moment at the open and closed angles stays inside a hand-feel or damper budget. Machine-element classrooms auto-grade the E d to the fourth over sixty-four D N identity and the millimetre–megapascal–newton-millimetre unit chain without standing up a spreadsheet that silently mixes metres into millimetres or uses shear modulus G in place of Young modulus E. Failed validation does not consume balance. Keep shaft rotational-stiffness landings and helical compression-spring-rate landings separate when those formulas are the search intent, so this page stays the torsion-spring angular rate from d, D, N, and E. Clip, mousetrap, and ratchet mechanism scripts can swap only elastic_modulus to compare music-wire and stainless candidates, or swap only active_coils to see how an extra working turn softens the wind-up. The browser tool is free for interactive use; scripted API traffic uses prepaid KIT balance at the published rate without hidden per-field surcharges.

Integration use cases

Clip, ratchet, and fixture scripts

Pin angular_stiffness from caliper wire, measured mean coil, counted active turns, and listed E in CI using the same pure module as the free browser widget.

Technical questions

How do I calculate torsional spring angular stiffness?

Send positive wire_diameter d in millimetres, coil_diameter D with D > d, a strictly positive active_coils N, and elastic_modulus E in N/mm². The API returns κ = E d⁴ / (64 D N) as angular_stiffness in N·mm/rad.

Why must the number of active coils be positive?

The identity divides by N. A coil count of zero or less is not a working torsion spring, so the task fails with invalid_input and is not billed.

What does the torsional spring calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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