Shaft size calculator
A torque of 1000 N·m with an allowable shear of 40 000 000 Pa needs a solid shaft just over fifty millimetres across. Enter torque and allowable shear stress to get the smallest safe diameter in metres and millimetres, plus the radius. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Torque
- 1,000
- Allowable shear stress
- 40,000,000
- Diameter
- 0.0503079599 m
- Diameter mm
- 50.307959916
- Radius
- 0.02515398
- Equation
- d = ∛(16 × 1000 / (π × 40000000)) = 0.050307959916 m
A rotating machine shaft that transmits torque is sized first by torsion: shear stress peaks at the outer fibre of a circular cross-section, and that peak must stay at or below the material allowable. Designers, students, and maintenance engineers therefore ask a single question before they pick a stock diameter: how small can the shaft be and still carry this torque. For a solid circular section the torsion formula tau equals sixteen T over pi d cubed rearranges to a cube root. The minimum diameter is the cube root of sixteen times torque divided by pi times allowable shear stress, with T in newton-metres, tau in pascals, and d in metres.
How minimum shaft diameter follows from torsion of a solid circle
When a solid circular shaft carries a pure torque T, every cross-section is in torsion. The shear stress at a radius r from the centre is T times r divided by J, where J is the polar moment of inertia of that section. For a solid circle J equals pi times diameter to the fourth divided by thirty-two, and the outer fibre sits at r equals d over two, so the maximum shear simplifies to tau equals sixteen T over pi d cubed.
Units must be coherent SI. Torque in newton-metres and allowable shear in pascals, which are newtons per square metre, yield a diameter in metres; the response also reports millimetres because bar stock, cold-drawn rounds, and catalog shafts are quoted that way. Typical carbon-steel allowables with a modest factor of safety sit in the tens of megapascals; a thousand newton-metre torque at forty megapascals already wants a shaft a little over fifty millimetres.
Aluminium alloys and many polymers sit lower, so the same torque produces a larger minimum diameter. This calculator treats a solid circular shaft under torsion only. It does not add bending from belts, gears, or overhung pulleys, it does not apply ASME shock and fatigue factors to a fluctuating torque, and it does not size a hollow tube from inner and outer radii.
Those belong on dedicated landings so this search intent stays the elementary shaft-size formula people type when they look for a minimum diameter from torque and allowable shear stress.
Where a shaft size calculator earns its keep
Drive-train and coupling designers use a shaft size calculator before they lock a stock diameter, so the peak shear under rated torque stays inside the material budget for steel, aluminium, or a polymer coupling hub. Strength-of-materials classrooms auto-grade the cube-root identity and the pascal-newton-metre unit chain without standing up a spreadsheet that silently mixes millimetres into metres. Maintenance shops convert a measured motor torque and a conservative allowable into a replacement-shaft diameter when the original drawing is missing.
Failed validation does not consume balance. Keep polar-moment-from-diameter landings, rotational-stiffness landings, and shear-stress-from-force-over-area landings separate when those formulas are the search intent, so this page stays the minimum solid-shaft diameter from torque and allowable shear. Pin the rounded millimetre value in tests if you compare against shop drawings that quote bar stock to a tenth of a millimetre.
What you can do with it
Sizing a drive shaft from rated torque
Convert motor or gearbox torque and a steel allowable shear into a minimum solid-shaft diameter before locking a catalog bar size.
Classroom torsion homework
Auto-grade the cube-root identity d = ∛(16 T / π τ) and the pascal–newton-metre unit chain against a deterministic oracle.
Replacement shaft when the drawing is missing
Turn a measured motor torque and a conservative material allowable into a replacement diameter on the shop floor.
FAQ
How do I calculate the minimum shaft diameter from torque and allowable shear?
The minimum solid circular diameter in metres is d = ∛(16 T / (π τ)).
What happens if the allowable shear stress is zero or negative?
Shaft diameter d = ∛(16 T / (π τ)) is undefined when the allowable shear stress is not a positive number; those failures are not billed.
What units does the shaft size calculator use?
Torque in newton-metres (N·m) and allowable shear in pascals (Pa = N/m²). Diameter is returned in metres, with a millimetre companion field. Convert N·mm, MPa, and inches before sending.
Does this include bending, keyways, or fatigue factors?
Bending from belts or gears, keyway stress concentration, and ASME shock or fatigue factors belong on dedicated worksheets.
What allowable shear stress should I use?
Carbon steels with a modest factor of safety usually sit in the tens of megapascals. Aluminium alloys and many polymers sit lower, so the same torque calls for a larger shaft. Enter the value in pascals.
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/shaft-size \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"torque":1000,"allowable_shear_stress":40000000}'const res = await fetch("https://api.kit.forhosting.com/phys/shaft-size", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"torque": 1000,
"allowable_shear_stress": 40000000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/shaft-size",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"torque": 1000,
"allowable_shear_stress": 40000000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/shaft-size", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"torque":1000,"allowable_shear_stress":40000000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"torque":1000,"allowable_shear_stress":40000000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/shaft-size", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"torque": 1000,
"allowable_shear_stress": 40000000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.shaft_size",
"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 two numbers, rejects a zero or negative allowable shear as invalid_input because the cube root is then undefined or non-physical, and returns the minimum diameter together with millimetre, radius, formula, and equation fields.
How minimum shaft diameter follows from torsion of a solid circle
Strength design inverts that identity: the smallest diameter that keeps the peak shear at or below an allowable value tau_allow is the cube root of sixteen T over pi tau_allow. That closed form is exactly what this endpoint evaluates, with no iteration and no lookup table.
Input contract, units, response shape, and error behaviour
A negative torque is rejected because a design load is a magnitude; a zero torque is valid and returns a diameter of zero. Example: T equals one thousand newton-metres and tau equals forty megapascals yields a minimum diameter of about fifty point three millimetres. Send two required finite numbers: torque, the magnitude of torque T to transmit, in newton-metres; and allowable_shear_stress, the material allowable shear tau in pascals. Aliases are accepted: T, t, and torque_nm for the torque; tau, tau_allow, shear_stress, allowable_stress, and tau_max for the allowable. 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. If allowable_shear_stress is zero, negative, or otherwise not a positive number, the task fails, because the cube root of sixteen T over pi tau is undefined when the denominator is not positive, and a negative allowable is not a material property. Those failures are not billed. On success the body echoes cleaned torque and allowable_shear_stress; returns diameter as the primary result in metres; repeats that length as diameter_mm in millimetres and radius in metres; 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 on a cube-root edge case.
Where a shaft size 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 shaft size calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.