Cantilever UDL moment calculator
A cantilever 4 m long under 10 N per metre carries 40 N in total and a fixed-end moment of 80 N·m, the largest bending moment in the beam. Enter the load intensity and length to get that peak. Free in your browser.
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Worked example
- Load intensity
- 10
- Length
- 4
- Fixed end moment
- 80
- Total load
- 40
- Configuration
- cantilever_full_udl
- Location
- fixed_end
A cantilever beam that is fully restrained at one end and free at the other develops its largest bending moment at the built-in support when a uniformly distributed load acts along the full length. The closed-form statics result is the fixed-end moment M equals w L squared over two, where w is the load intensity and L is the span from the wall to the tip. Students checking homework, balcony and canopy designers, and automated mechanics notebooks all need that peak moment before they compute stress or deflection.
What M = wL²/2 means and when the cantilever UDL formula applies
A cantilever is a prismatic member that is built in, welded, or otherwise fully restrained against rotation and translation at one end and is free at the other. When a uniformly distributed load of intensity w acts along the entire length L, every infinitesimal strip of load w dx contributes a moment about the support equal to its force times its lever arm measured from the wall.
Integrating those contributions from the free tip back to the support produces the textbook closed form used by every strength-of-materials course: the fixed-end bending moment equals w times L squared divided by two. An equivalent and often faster derivation replaces the distributed load with its resultant.
The total force W equals w times L and acts at the centroid of the rectangle of load, which sits a distance L over two from the built-in end, so the support moment is W times L over two, which simplifies again to w L squared over two. That value is the maximum bending moment in the member.
If a section is taken a distance x from the free end, the local moment is only w x squared over two; the moment diagram is therefore a parabola that is zero at the tip and peaks at the wall. Search intent clusters around cantilever udl moment calculator, cantilever beam uniformly distributed load moment, fixed-end moment wL2/2, and balcony canopy cantilever moment — people who already have intensity and length and want the peak number, not a deflection, not a fiber stress, and not a finite-element mesh.
Keep units consistent: if w is in newtons per metre and L is in metres, the moment comes out in newton-metres; kilonewtons per metre with metres yield kilonewton-metres. The formula assumes linear-elastic Euler–Bernoulli kinematics, a prismatic cantilever, a uniformly distributed load over the full span, and small-deflection statics with no extra tip point load or applied couple.
How this calculator validates input and applies wL²/2
Load intensity is the uniform force per unit length w along the full cantilever, not the total resultant and not a concentrated tip force. Length is the distance L from the built-in end to the free tip. Both values must be strictly positive so the geometry and load case stay physically meaningful for this product: a zero-length member is not a cantilever span, and a zero intensity is not a uniformly distributed load.
Negative values are rejected rather than silently flipped, because this landing reports the magnitude of the hogging support moment from downward UDL instead of inventing a sign-convention toggle. There is no simply-supported midspan mode w L squared over eight, no fixed-fixed end moment w L squared over twelve, no deflection w L to the fourth over eight E I, and no flexure-formula stress My over I; this is the pure cantilever full-span UDL support moment only.
Practical use, limits, pricing, and how this differs from wL²/8 and wL²/12
Use the browser tool when you want an immediate fixed-end moment while grading strength-of-materials homework, checking a balcony, canopy, sign-arm, or shelf bracket sketched as a cantilever under self-weight or snow, or teaching why the moment diagram of a uniform cantilever is a parabola that peaks at the wall. Results are fully deterministic: identical inputs always produce the same fixed-end moment, the same total load, and the same formula and method labels.
There is no language model, no external structural analysis service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible course materials. This page is not the simply supported UDL moment calculator, whose midspan peak is w L squared over eight and answers a different support condition. It is also not the fixed–fixed beam end moment, which is w L squared over twelve when both ends are clamped.
For midspan deflection of a simply supported span under a center point load, use the separate beam deflection calculator. For fiber stress once you already have a moment, use the bending-stress tool that evaluates sigma equals M y over I.
What you can do with it
Balcony, canopy, and bracket sketches
Designers estimate the hogging support moment of a short cantilever under self-weight or snow before opening a full FEA model or selecting a catalog section.
FAQ
Why is the formula wL²/2 and not wL²/8 or wL²/12?
wL²/2 is the fixed-end moment of a cantilever under full-span UDL. wL²/8 is the midspan moment of a simply supported beam under the same UDL. wL²/12 is each end moment of a beam fixed at both ends. Those are different support conditions and different landings.
Does this compute deflection or bending stress?
No. This calculator returns only the support moment M = wL²/2 and the resultant W = wL. Tip deflection of a cantilever UDL is wL⁴/(8EI). Fiber stress is σ = My/I once you already have M; use the separate bending-stress tool for that.
Is the cantilever UDL moment calculator free?
The browser tool runs free on your device.
Where is the maximum bending moment on a cantilever?
At the fixed support. The moment is zero at the unsupported tip and grows along a parabola towards the wall, so the built-in end always carries the peak value the calculator returns.
What units does the cantilever moment come out in?
The units follow your inputs: newtons per metre with metres give newton-metres, and kilonewtons per metre with metres give kilonewton-metres. Keep the load intensity and the length in one consistent system.
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/eng/cantilever-udl-moment \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"load_intensity":10,"length":4}'const res = await fetch("https://api.kit.forhosting.com/eng/cantilever-udl-moment", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"load_intensity": 10,
"length": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/eng/cantilever-udl-moment",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"load_intensity": 10,
"length": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/eng/cantilever-udl-moment", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"load_intensity":10,"length":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"load_intensity":10,"length":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/eng/cantilever-udl-moment", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"load_intensity": 10,
"length": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "eng.cantilever_udl_moment",
"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 cantilever UDL moment calculator accepts two finite positive numbers, applies that exact formula, and returns the fixed-end moment together with the total resultant load and stable formula labels as structured JSON. Zero or negative length or intensity fails with invalid_input so a bad geometry is never billed as a success.
What M = wL²/2 means and when the cantilever UDL formula applies
This capability returns fixed_end_moment, load_intensity, length, total_load, formula, configuration, method, and location as JSON so the result is easy to paste into notes, pin in automated tests, or reuse across languages without reimplementing the statics.
How this calculator validates input and applies wL²/2
Send a JSON body with the two required fields load_intensity and length. Every field must be a finite number or a numeric string that parses to one; missing fields, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input before any result is returned, and failed validation is not charged. Optional aliases such as w, udl, intensity, distributed_load, L, span, and beam_length are accepted when documented so forms and scripts can use textbook symbols. After validation the engine computes the total resultant as intensity times length and the fixed-end moment as intensity times length squared divided by two, then rounds to a stable precision so the same input always yields the same JSON across JavaScript engines. The same pure function powers the free in-browser tool and the API path, with no network, no random seed, and no wall clock in the computation path.
Practical use, limits, pricing, and how this differs from wL²/8 and wL²/12
Each successful request is one billable item at $0.002; failed validation does not consume balance. Choose units consistently and stay within the published magnitude bounds so intermediate products remain finite on a Worker isolate. Wire the endpoint into mechanics drills, contest answer checkers, early-stage design worksheets, or preprocessing steps where you prefer explicit domain errors—such as rejecting a zero length—over opaque widgets that cannot explain why a bad input was refused.
Integration use cases
Homework and exam verification
Students and graders compute M = wL²/2 for a known intensity and cantilever length, and get a clear invalid_input error when either value is zero or negative.
Deterministic fixtures for mechanics tests
Pipelines that seed unit tests or interactive beam demos call a pure wL²/2 oracle instead of reimplementing the cantilever statics in every language.
Technical questions
How do I calculate the cantilever UDL moment with this tool?
Send JSON with load_intensity (w) and length (L). The response returns fixed_end_moment equal to w times L squared divided by two, plus the total resultant load W = wL and formula labels.
What happens if length or load intensity is zero or negative?
The request fails with invalid_input. A zero-length member or a non-positive intensity is not this load case; validation rejects it and the request is not charged.