Rotational stiffness calculator
A steel shaft with a shear modulus of 80 GPa, a polar moment of 1 × 10⁻⁸ m⁴ and a length of 0.5 m takes 1600 N·m to twist one radian. Enter G, J and L to get κ = GJ/L for your own drive shaft. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Shear modulus
- 80,000,000,000
- Polar moment
- 0.00000001
- Length
- 0.5
- Rotational stiffness
- 1,600 N·m/rad
- Torsional stiffness
- 1,600
- Kappa
- 1,600
- Equation
- κ = 80000000000 × 1e-8 / 0.5 = 1600 N·m/rad
Rotational stiffness, also called torsional stiffness, is the torque required to produce one radian of elastic twist in a shaft. Machine designers, students, and drivetrain engineers compute that stiffness from three numbers: the shear modulus of the material, the polar moment of inertia of the cross-section, and the length of the shaft. The identity is kappa equals G times J divided by L, and the result is in newton-metres per radian when G is in pascals, J is in metres to the fourth, and L is in metres.
How rotational stiffness follows from twist of a prismatic shaft
Rotational stiffness of a shaft is the torque you must apply to produce one radian of elastic twist. In strength of materials the twist angle theta of a prismatic member under a constant torque T is T L over G J, so the ratio T over theta is G J over L. The polar moment of a solid circular cross-section is pi times the diameter to the fourth divided by thirty-two, or equivalently pi r to the fourth over two; a thin-walled tube uses approximately two pi r cubed t.
Steel shear moduli sit near seventy-nine to eighty gigapascals; aluminium alloys sit near twenty-five to twenty-seven; brass sits near forty. A longer shaft is torsionally softer, a fatter section is stiffer, and a stiffer material raises kappa in the same proportion. This calculator evaluates that linear elastic identity only.
It does not integrate a variable or stepped section, it does not add stress concentrations from keyways or fillets, and it does not convert a linear spring rate k equals F over x. Those belong on dedicated landings so this search intent stays the shaft formula people type when they look for a rotational stiffness calculator.
Where a rotational stiffness calculator earns its keep
Drivetrain and coupling designers use a rotational stiffness calculator before they lock a shaft diameter and unsupported length, so the twist under rated torque stays inside a positioning budget for a servo, a stepper, or a gearbox. Strength-of-materials classrooms auto-grade the GJ over L identity and the pascal-metre-newton unit chain without standing up a spreadsheet that silently mixes millimetres into metres. Control engineers compare two candidate shafts at the same material and immediately see why the shorter, thicker bar is the stiffer plant for a torque loop.
Failed validation does not consume balance. Keep linear spring-constant landings and polar-moment-from-diameter landings separate when those formulas are the search intent, so this page stays the shaft stiffness from G, J, and L.
What you can do with it
Sizing a drive shaft for twist
Convert shear modulus, polar moment, and unsupported length into rotational stiffness in N·m/rad before locking a shaft diameter against a servo or gearbox torque budget.
Classroom torsion homework
Auto-grade the κ = GJ/L identity and the pascal–metre–newton unit chain against a deterministic oracle instead of a rounded spreadsheet.
Coupling and plant-stiffness budgets
Compare two candidate shafts at the same material to see why the shorter, thicker bar is the stiffer plant for a torque control loop.
FAQ
How do I calculate the rotational stiffness of a shaft?
Rotational stiffness in N·m/rad is κ = G J / L.
What happens if the shaft length is zero?
Torsional stiffness κ = G J / L is undefined when the length is zero; those failures are not billed. A negative length is also rejected.
What units does the rotational stiffness calculator use?
G in pascals (N/m²), J in metres to the fourth (m⁴), L in metres. The result is newton-metres per radian (N·m/rad). Convert millimetres and gigapascals before sending.
How do I find J for a circular shaft?
For a solid circular cross-section, J = π d⁴ / 32, or π r⁴ / 2. A thin-walled tube is approximately 2 π r³ t.
What shear modulus should I use for steel, aluminium or brass?
Steel sits near 79 to 80 GPa, aluminium alloys near 25 to 27 GPa and brass near 40 GPa. Enter the value in pascals, so 80 GPa becomes 80 000 000 000 Pa.
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
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/rotational-stiffness \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"shear_modulus":80000000000,"polar_moment":1e-8,"length":0.5}'const res = await fetch("https://api.kit.forhosting.com/phys/rotational-stiffness", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"shear_modulus": 80000000000,
"polar_moment": 1e-8,
"length": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/rotational-stiffness",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"shear_modulus": 80000000000,
"polar_moment": 1e-8,
"length": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/rotational-stiffness", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"shear_modulus":80000000000,"polar_moment":1e-8,"length":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"shear_modulus":80000000000,"polar_moment":1e-8,"length":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/rotational-stiffness", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"shear_modulus": 80000000000,
"polar_moment": 1e-8,
"length": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.rotational_stiffness",
"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
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
This calculator accepts those three values, rejects a zero length as invalid_input because the quotient is undefined, and returns the stiffness together with the formula and a substituted equation.
How rotational stiffness follows from twist of a prismatic shaft
That quotient is the stiffness kappa this endpoint returns, in newton-metres per radian when the shear modulus G is in pascals, the polar moment of inertia J is in metres to the fourth, and the length L is in metres.
Input contract, units, response shape, and error behaviour
If length is zero the task fails, because kappa equals G J over L is undefined when the denominator vanishes; a negative length is also rejected, because a shaft length is a positive distance. Example: G equals eighty gigapascals, J equals one times ten to the minus eight metres to the fourth, and L equals half a metre yields sixteen hundred newton-metres per radian. Send three required finite numbers: shear_modulus, the material shear modulus G in pascals; polar_moment, the polar moment of inertia J of the cross-section in metres to the fourth; and length, the shaft length L in metres. Aliases are accepted: G, modulus, and shear_g for the modulus; J, polar_moment_of_inertia, and Ip for the polar moment; L and shaft_length for the length. Each field may arrive as a JSON number or as a numeric string so form widgets and spreadsheet paste stay convenient. Missing fields, empty strings, NaN, and Infinity fail with invalid_input naming the field. On success the body echoes cleaned shear_modulus, polar_moment, and length; returns rotational_stiffness as the primary result in newton-metres per radian; repeats that value as torsional_stiffness and kappa; and returns unit, formula, and a substituted equation string. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance. Pair the free browser form with the same module the API uses so classroom demos and workshop scripts never diverge.
Where a rotational stiffness calculator earns its keep
The browser tool is free for interactive use; scripted API traffic uses prepaid KIT balance at the published rate without hidden per-field surcharges.
Technical questions
What does the rotational stiffness calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.