Torsional stiffness calculator
A shaft with a shear modulus of 8 × 10¹⁰ Pa, a polar moment of 10⁻⁸ m⁴ and a length of 0.5 m resists twisting with 1600 N·m per radian. Enter the three values in SI units to get its torsional stiffness. Free in your browser.
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Run — free
Worked example
- Shear modulus
- 80,000,000,000
- Polar moment
- 0.00000001
- Length
- 0.5
- Torsional stiffness
- 1,600 N·m/rad
- Stiffness
- 1,600
Torsional stiffness of a shaft is the torque per radian of twist, written κ equals G times J over L, that links an applied moment to the elastic rotation of a prismatic member. This torsional stiffness calculator takes the shear modulus G in pascals, the polar moment of inertia J in metres to the fourth, and the shaft length L in metres, then returns stiffness in newton-metres per radian. Missing fields, empty strings, NaN, Infinity, and non-numeric text also fail and are not billed.
What the torsional stiffness of a shaft actually is
The torsional stiffness of a prismatic shaft is the elastic torque that appears for one radian of relative twist between the two ends. Hooke's law in torsion writes T equals (G J over L) times θ, so the stiffness κ is G J over L and has SI units of newton-metres per radian when G is in pascals, J is in metres to the fourth, and L is in metres.
Search intent clusters around torsional stiffness calculator, shaft torsion constant, kappa equals GJ over L, polar moment of inertia, and turning a known shear modulus and length into a spring rate for a driveshaft or torsion bar. People want a numeric κ they can drop into a twist-angle check, a coupling specification, or a later pendulum-period step—not only a reminder that the formula exists. Doubling the shear modulus or the polar moment doubles stiffness.
Doubling the length halves it. A short, stout steel bar is torsionally stiff; a long, thin aluminium rod is not. The model is a homogeneous prismatic shaft with uniform G, constant J along the axis, and small-angle linear elasticity.
It does not compute J from a diameter, it does not convert gigapascals or millimetres, and it does not integrate a tapered bar. Convert units first so the newton-metre-per-radian result is honest. Once you have κ, the twist for a known torque is simply θ equals T over κ, which is why this number sits next to a torsion-angle check on a lab bench.
Where a shaft torsional-stiffness calculator earns its keep
Machine-design notebooks use a torsional stiffness calculator so a shaft check can turn a tabulated G, a known polar moment, and a bearing span into newton-metres per radian without rearranging Hooke's law in torsion by hand. Driveshaft and torsion-bar sketches need κ before they convert a known torque into radians of twist, or before they size a coupling that must not be softer than the shaft it joins. Exam generators need a deterministic oracle that rejects a non-positive shaft length instead of leaking Infinity into a gradebook.
Laboratory reports that measure G of a specimen can turn that G, the specimen J, and the gauge length into κ before comparing with a torsion-test reading. Failed validation does not consume balance. Keep requests small, pin the rounded stiffness in tests, and you get reproducible newton-metres per radian for homework, LMS plugins, and edge workers without standing up a strength-of-materials package of your own.
What you can do with it
Driveshaft and coupling checks
Turn a tabulated shear modulus, a known polar moment of inertia, and a bearing span into torsional stiffness in N·m/rad before estimating twist under a design torque.
Torsion-bar and lab specimen rates
Convert a measured G, specimen J, and gauge length into κ so a torsion test or a bar rate can be compared without rearranging GJ/L by hand.
FAQ
What happens if shaft length is zero or negative?
Length must be strictly positive: zero would divide by zero, and a negative L is not physical for a shaft. Invalid input is not billed.
Is this the same as a torsional-pendulum period calculator?
Compute stiffness here, then feed κ into the period tool if you need the oscillation time.
What units does the calculator use?
SI: shear modulus in pascals, polar moment in metres to the fourth, length in metres, stiffness in newton-metres per radian (unit N·m/rad).
How do I get the twist angle for a known torque?
Divide the torque by the stiffness. With the torque in newton-metres, the result is the twist in radians. That is why the stiffness is usually worked out first, before a twist-angle check or the sizing of a coupling.
How do I find the polar moment for a solid round shaft?
Multiply pi by the diameter to the fourth power and divide by 32, with the diameter in metres. The calculator does not work out the polar moment from a diameter, so compute it first and then enter the result.
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/torsional-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/torsional-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/torsional-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/torsional-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/torsional-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.torsional_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.
Limits
max_abs | 1000000000000 |
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
Length must be strictly positive: a zero or negative L is rejected with invalid_input because the model would divide by zero or invent a non-physical shaft.
Input contract, response shape, and error behaviour
Magnitudes above one trillion are rejected so a mistyped exponent cannot blow a downstream plot. Length must be strictly positive: zero length would make κ infinite, and a negative length is not physical for this model. Shear modulus and polar moment must also be strictly positive; a zero G or J would give a zero spring rate that is not a shaft. POST a JSON body with three required finite numbers: shear_modulus, the material shear modulus G in pascals; polar_moment, the polar moment of inertia J in metres to the fourth; and length, the shaft path L in metres between the two sections that twist relative to each other. Each field may arrive as a JSON number or as a numeric string. Aliases G, g, modulus, shear, shear_modulus_pa, and modulus_of_rigidity map onto shear_modulus; aliases J, j, polar_moment_of_inertia, Ip, ip, polar_inertia, and second_polar_moment map onto polar_moment; aliases L, l, shaft_length, and length_m map onto length. Missing values, empty strings, NaN, Infinity, objects, arrays, and non-numeric text fail with invalid_input naming the field, and those failures are not billed. On success the body echoes shear_modulus, polar_moment, and length, returns torsional_stiffness as κ in newton-metres per radian, repeats that value as stiffness for display, restates the identity in formula as κ equals G times J over L, sets unit to N·m/rad, and sets method to shaft_gj_over_l so clients can tell this landing from a linear spring-rate tool. Example: shear_modulus eighty gigapascals, polar_moment one times ten to the minus eight metres to the fourth, and length zero point five metres yield torsional_stiffness one thousand six hundred newton-metres per radian, because eighty billion times one hundred-millionth divided by one half is one thousand six hundred.
Integration use cases
Homework and exam oracles
Graders and LMS plugins confirm κ = GJ/L for given G, J, and L, with a clear invalid_input rejection when shaft length is not positive.
Technical questions
How does the torsional stiffness calculator compute κ?
Send a positive shear modulus G in pascals (field shear_modulus, aliases G, g, modulus), a positive polar moment of inertia J in m⁴ (field polar_moment, aliases J, polar_moment_of_inertia, Ip), and a positive shaft length L in metres (field length, aliases L, shaft_length). The API returns κ = G · J / L in newton-metres per radian.
What does the torsional 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.