Transverse shear stress calculator
Find the shear stress at a chosen fibre of a beam: a shear force of 1000 with a first moment of 20, an inertia of 40 and a width of 5 gives 100. Use one consistent unit system throughout. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Shear force
- 1,000
- First moment
- 20
- Moment of inertia
- 40
- Width
- 5
- Transverse shear stress
- 100
Transverse shear stress is the shear stress that develops across a beam cross-section when an internal shear force tries to slide adjacent longitudinal fibers past each other. The classical mechanics-of-materials shear formula evaluates that stress at a chosen fiber as tau equals V Q divided by I t, where V is the transverse shear force, Q is the first moment of area of the portion of the section beyond that fiber, I is the second moment of area of the whole section about the neutral axis, and t is the width at the fiber of interest. Students checking homework, structural and mechanical engineers sizing preliminary beams, timber and steel designers comparing shear to allowable limits, and automated design pipelines all need that number without opening a finite-element suite.
What transverse shear stress means and when τ = VQ/(I t) applies
A beam that carries transverse load develops an internal shear force V at each cross-section, and that force is not uniformly distributed over the height of the section. Adjacent longitudinal fibers tend to slide, so a shear stress τ appears, usually peaking near the neutral axis and falling to zero at the extreme fibers of a rectangular section. The Jourawski or textbook beam shear formula states that the shear stress at a chosen fiber equals V times Q divided by I times t.
Here Q is the first moment of area of the portion of the cross-section that lies beyond the fiber, taken about the neutral axis; I is the second moment of area of the entire section about the same axis; and t is the actual width of material at that fiber, which may step at a flange–web junction.
Search intent clusters around transverse shear stress calculator, beam shear formula VQ over It, first moment of area Q, and horizontal shear in timber joists—people who already have V, Q, I, and t and want a numeric stress they can compare to allowable shear, a homework key, or a spreadsheet. Keep units consistent: if V is in newtons, Q in metres cubed, I in metres to the fourth, and t in metres, stress comes out in pascals; any other coherent system works the same way.
The formula assumes linear-elastic Euler–Bernoulli kinematics, a prismatic member, and a section that is not so thin-walled that shear lag or warping torsion dominate; plastic redistribution, residual stresses, and detailed web-crippling checks need a different analysis path.
Practical use, limits, pricing, and how this differs from F/A and My/I
Use the browser tool when you want an immediate transverse shear check while grading strength-of-materials homework, comparing a hand calculation of V Q over I t at the neutral axis of a rectangle or I-beam web, verifying a timber joist against horizontal-shear allowable values, or teaching how shear stress varies with Q and width. Results are fully deterministic: identical inputs always produce the same transverse shear stress 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 average shear calculator τ equals F over A, which answers a different search when you only know force and area; that lives as a separate shear-stress tool. It is also not the bending-stress calculator σ equals M y over I, which reports normal stress from flexure rather than shear from V and Q.
For midspan deflection under a center point load, use the separate beam deflection calculator.
What you can do with it
Preliminary beam and joist shear checks
Engineers and timber designers estimate transverse or horizontal shear at the neutral axis or a flange–web junction before opening a full FEA model or selecting a catalog section.
FAQ
What is Q, the first moment of area?
Q is the first moment, about the neutral axis, of the portion of the cross-section that lies beyond the fiber where you want τ. For a rectangle of width b and height h, Q at the neutral axis is b h² / 8. You compute Q from geometry; this tool does not look it up from a shape name.
Is this the same as shear stress τ = F/A?
No. Average shear τ = F/A is a different search and a different formula. This calculator evaluates the beam (Jourawski) distribution τ = VQ/(I t) at a chosen fiber. Use the separate average-shear tool when you only have force and area.
Is the transverse shear stress calculator free?
The browser tool runs free on your device.
Where is shear stress highest in a beam section?
Usually near the neutral axis, falling to zero at the top and bottom fibres of a rectangular section. At an extreme fibre no area lies beyond the cut, so the first moment and the stress are both zero.
What happens if the moment of inertia or the width is zero?
The input is rejected, because the formula divides by the inertia times the width. A zero first moment, on the other hand, is valid and simply gives zero shear stress.
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/transverse-shear-stress \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"shear_force":1000,"first_moment":20,"moment_of_inertia":40,"width":5}'const res = await fetch("https://api.kit.forhosting.com/eng/transverse-shear-stress", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"shear_force": 1000,
"first_moment": 20,
"moment_of_inertia": 40,
"width": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/eng/transverse-shear-stress",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"shear_force": 1000,
"first_moment": 20,
"moment_of_inertia": 40,
"width": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/eng/transverse-shear-stress", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"shear_force":1000,"first_moment":20,"moment_of_inertia":40,"width":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"shear_force":1000,"first_moment":20,"moment_of_inertia":40,"width":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/eng/transverse-shear-stress", 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_force": 1000,
"first_moment": 20,
"moment_of_inertia": 40,
"width": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "eng.transverse_shear_stress",
"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 capability takes the four finite numbers, rejects zero inertia or zero width with a clear invalid_input error, and returns transverse shear stress plus echoed inputs and formula labels as structured JSON.
What transverse shear stress means and when τ = VQ/(I t) applies
Delivering transverse_shear_stress, shear_force, first_moment, moment_of_inertia, width, formula, and method as JSON makes the result easy to paste into notes, pin in automated tests, or reuse across languages without reimplementing the arithmetic.
How this calculator validates input and applies the shear formula
Shear force and first moment may be positive, negative, or zero so sign conventions for shear direction and which side of the fiber you integrate stay under the caller’s control; a zero first moment simply yields zero shear stress, which is the physically correct value at an extreme fiber where no area lies beyond the cut. Extremely small denominator magnitudes that would not solve stably on floating-point hardware are also rejected. There is no average-shear F over A mode, no bending-stress My over I mode, no unit conversion, and no section-property lookup that invents Q from a shape name; this is the pure VQ over I t evaluation only. Send a JSON body with the four required fields shear_force, first_moment, moment_of_inertia, and width. 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. Moment of inertia and width must be non-zero: zero inertia or zero width is explicitly rejected as invalid_input because either would divide by zero in VQ over I t, matching the product contract for this capability. Optional aliases such as V, Q, I, t, shear, force, first_moment_of_area, statical_moment, second_moment, inertia, section_width, thickness, and b are accepted when documented so forms and scripts can use textbook symbols. After validation the engine computes transverse shear stress as shear force times first moment divided by the product of moment of inertia and width, 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 F/A and My/I
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 zero inertia or zero width—over opaque widgets that cannot explain why a bad input was refused.
Integration use cases
Homework and exam verification
Students and graders compute τ = VQ/(I t) for known shear force, first moment of area, section inertia, and width, and get a clear invalid_input error when inertia or width is zero.
Deterministic fixtures for mechanics tests
Pipelines that seed unit tests or interactive beam demos call a pure VQ/(I t) oracle instead of reimplementing the shear formula in every language.
Technical questions
How do I calculate transverse shear stress with this tool?
Send JSON with shear_force (V), first_moment (Q), moment_of_inertia (I), and width (t). The response returns transverse_shear_stress equal to V times Q divided by I times t, plus echoed inputs and formula labels.
What happens if moment of inertia or width is zero?
The request fails with invalid_input. Zero inertia or zero width would divide by zero in τ = VQ/(I t); validation rejects it and the request is not charged.