Simply supported beam center load calculator
A 1000 N load at the middle of a 6 m beam puts 500 N on each support and a maximum bending moment of 1500 N·m under the load. Enter the load and the span to get both values. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- Load
- 1,000
- Span
- 6
- Left reaction
- 500
- Right reaction
- 500
- Maximum bending moment
- 1,500
- Reaction formula
- P/2
This simply supported beam center load calculator solves the fundamental symmetric statics case: one concentrated load acts at the exact middle of a beam resting on two simple supports. Enter the point load and support-to-support span to obtain the reaction at each support and the maximum bending moment at midspan. The calculation is deterministic, requires no material or section properties, accepts any consistent unit system, and clearly rejects a zero span rather than returning an undefined or misleading result.
Understand the symmetric center-load beam case
A simply supported beam can rotate at its supports and does not develop a fixing moment there. When one point load is placed exactly halfway between the supports, the geometry and loading are symmetric. Vertical force equilibrium requires the two upward reactions to add to the applied downward load, while moment equilibrium makes those reactions equal.
Each support therefore carries one half of the load. The internal bending moment rises linearly from zero at the left support to its largest value beneath the load, then falls linearly back to zero at the right support. This calculator reports that peak as the load multiplied by the span and divided by four.
The model is intentionally narrow: it represents a single central point load on one simply supported span. It does not represent a load away from center, several point loads, a distributed load, a cantilever, fixed ends, support settlement, or a continuous beam. Keeping that boundary explicit makes the returned reactions and moment easy to audit against the equilibrium equations and avoids implying that one short formula covers every beam arrangement.
Enter load and span with consistent units
Provide load as the concentrated force P and span as the clear support-to-support length L. Span must be greater than zero because a zero span has no meaningful lever arm and is rejected with an invalid-input error. The load may carry a sign, which allows a signed statics convention: a positive load produces positive reactions and moment, while a negative load reverses their signs.
The calculator performs no unit conversion, so choose one coherent system before submitting the values. If load is entered in newtons and span in metres, each reaction is in newtons and maximum bending moment is in newton-metres. Using kilonewtons and metres yields kilonewtons and kilonewton-metres.
The same rule works for pounds-force and feet. Returned load and span values make the unit context traceable, while formula and configuration fields document exactly which case was solved. Extremely large values that overflow ordinary JavaScript arithmetic are rejected rather than exposed as infinity.
Use the results for checks and preliminary design
The reaction values are useful for support sizing, free-body-diagram checks, bearing-force estimates, and comparison with hand calculations. The maximum bending moment is a common starting point for preliminary member selection or a later bending-stress calculation, but it is not by itself a complete structural design. Strength checks also require section properties, material resistance, safety factors, load combinations, and the governing design standard.
Serviceability work may require elastic modulus and second moment of area to calculate deflection, while shear, lateral stability, local effects, connections, and support details can also govern. Use this result as a transparent statics calculation, not as certification that a real beam is safe. Failed validation is returned as an input error.
Because the solver has no network access, external model, random source, or clock dependency, identical inputs always produce identical outputs, which makes the capability suitable for repeatable automated pipelines and test suites.
What you can do with it
Check a free-body diagram
Confirm that a centered point load is shared equally by the two simple supports and that the reactions sum to the applied load.
Start a preliminary beam check
Obtain the peak midspan bending moment before applying section properties, material limits, load factors, and the relevant design standard.
FAQ
What formulas does the calculator use?
For center load P and span L, each support reaction is P/2 and the maximum bending moment at midspan is PL/4.
Why are the two support reactions equal?
The load is at the exact center, so the loading and support geometry are symmetric. Force and moment equilibrium therefore divide the load equally.
Which units should I use?
Use any consistent system. Reactions use the load unit, and maximum bending moment uses the load unit multiplied by the span unit.
Does this calculate beam deflection or prove the beam is safe?
No. It calculates only support reactions and maximum bending moment for this load case. Deflection and design checks require additional properties and criteria.
Can I use this calculator if the load is not in the middle of the beam?
No. It solves only one point load at the exact center of a single span on two simple supports. A load off center, several loads, a spread load, a cantilever or fixed ends need a different calculation.
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/mech/beam-point-load-center \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"load":1000,"span":6}'const res = await fetch("https://api.kit.forhosting.com/mech/beam-point-load-center", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"load": 1000,
"span": 6
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/beam-point-load-center",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"load": 1000,
"span": 6
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/beam-point-load-center", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"load":1000,"span":6}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"load":1000,"span":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/beam-point-load-center", 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": 1000,
"span": 6
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.beam_point_load_center",
"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
Enter load and span with consistent units
Both values must be finite numbers; numeric strings are also accepted for convenient form and API use.
Integration use cases
Build deterministic teaching fixtures
Generate stable reaction and moment results for mechanics lessons, grading tools, spreadsheets, and automated statics tests.
Technical questions
What happens when the span is zero?
The request fails with invalid_input because a zero support-to-support span is not a valid beam geometry.