Thermal stress calculator
A steel bar fixed between rigid walls, with the usual modulus of steel, α of 0.000012 per kelvin and a 50 K rise, develops 120 MPa of compressive stress from σ = E·α·ΔT. Cooling gives tension instead. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Youngs modulus
- 200,000,000,000
- Expansion coefficient
- 0.000012
- Temperature change
- 50
- Thermal stress pa
- 120,000,000
- Thermal stress mpa
- 120
- Kind
- compressive
When a straight bar is fully restrained against axial expansion or contraction, a uniform temperature change develops axial thermal stress instead of free length change. In the linear elastic model the magnitude follows the product of Young's modulus E, the linear thermal expansion coefficient alpha, and the temperature change Delta T: sigma equals E times alpha times Delta T. Heating a metal rod fixed between rigid walls produces compression; cooling produces tension. This thermal stress calculator accepts E, alpha, and Delta T, rejects a negative Young's modulus as non-physical for this model, and returns stress in pascals and megapascals plus a compressive, tensile, or zero classification from the sign of the product. Use it when free expansion is prevented and you need the restrained-bar closed form rather than a length-change calculator.
How thermal stress is defined for a constrained bar
Start from free thermal strain: if the bar could expand freely, the axial strain would be alpha times Delta T with no stress. When both ends are fixed, the total axial strain must remain zero, so an equal and opposite mechanical strain is imposed by the supports. Hooke's law in one dimension then gives mechanical stress sigma equal to E times that restrained strain, which is exactly E times alpha times Delta T.
The product is the classic textbook formula for uniform temperature change in a prismatic bar with full axial constraint and no bending. Positive product for ordinary materials with positive alpha and positive E under heating means the free thermal strain would have been expansion, so the supports push back in compression; cooling reverses the sign and develops tension.
Callers who work outside SI must keep units consistent: when E is in pascals and alpha is per kelvin, stress is in pascals; the megapascal field is simply that value divided by one million for compact comparison with yield tables.
Where constrained thermal stress shows up in real work
Introductory strength-of-materials and solid-mechanics courses constantly assign fixed-end bar problems that reduce to sigma equals E alpha Delta T before any factor-of-safety or yield comparison. Piping, rail, and structural steelwork estimate locked-in axial stress from installation temperature to operating temperature when expansion joints or sliding supports are absent. Electronics packaging and composite layups approximate first-order thermal residual stress in constrained layers using the same product when gradients and plasticity are neglected.
Spreadsheet macros often reimplement the multiplication with silent unit mistakes between gigapascals and megapascals or between per-celsius and microstrain coefficients; a single trusted oracle that returns both Pa and MPa and rejects negative modulus reduces copy-paste errors across teams. If you later need free linear expansion length change, mechanical stress from force over area, Young's modulus from stress and strain, or multi-axial thermoelasticity, treat those as separate search intents and landings rather than overloading this constrained-bar thermal stress page.
What you can do with it
Fixed-end bar homework
Compute σ = E·α·ΔT for fully restrained rods so students compare compressive heating cases against tensile cooling cases with shared Pa/MPa output.
Piping and rail temperature locks
Estimate axial thermal stress between installation and service temperatures when expansion loops or joints are not present.
FAQ
What formula does this thermal stress calculator use?
σ = E·α·ΔT for a fully constrained straight bar under uniform temperature change in the linear elastic range. The response reports stress in pascals and megapascals.
What happens if Young's modulus is negative?
Negative E is rejected as non-physical for this model; zero modulus is allowed and yields zero stress.
Is heating compressive or tensile?
For positive α and E ≥ 0, heating (positive ΔT) produces a positive product classified as compressive (expansion prevented). Cooling produces tension.
Does this compute free expansion length change?
No. Free expansion is ΔL = L₀·α·ΔT (a different landing). This page is only for stress when axial expansion is fully prevented.
What does it cost?
The browser tool is free and runs locally.
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/sci/thermal-stress \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"youngs_modulus":200000000000,"expansion_coefficient":0.000012,"temperature_change":50}'const res = await fetch("https://api.kit.forhosting.com/sci/thermal-stress", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"youngs_modulus": 200000000000,
"expansion_coefficient": 0.000012,
"temperature_change": 50
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/sci/thermal-stress",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"youngs_modulus": 200000000000,
"expansion_coefficient": 0.000012,
"temperature_change": 50
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/sci/thermal-stress", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"youngs_modulus":200000000000,"expansion_coefficient":0.000012,"temperature_change":50}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"youngs_modulus":200000000000,"expansion_coefficient":0.000012,"temperature_change":50}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/sci/thermal-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
{
"youngs_modulus": 200000000000,
"expansion_coefficient": 0.000012,
"temperature_change": 50
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "sci.thermal_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
How thermal stress is defined for a constrained bar
This tool validates that Young's modulus, expansion coefficient, and temperature change are finite numbers, rejects a negative modulus with a clear invalid_input message, multiplies the three values, reports stress_pa and stress_mpa, and labels kind as compressive, tensile, or zero. The pipeline is deterministic floating-point arithmetic with no network calls, no random seeds, and no wall-clock timestamps, so identical inputs always produce identical JSON.
What you send and what you get back
All three fields are required. Zero modulus is accepted and yields zero stress. Classic textbook checks include steel with E about two hundred gigapascals, alpha about twelve times ten to the minus six per kelvin, and a fifty kelvin rise, giving one hundred twenty megapascals of compressive thermal stress when fully restrained. Send three finite numbers under the fields youngs_modulus, expansion_coefficient, and temperature_change. Missing fields, empty strings, non-numeric values, NaN, and Infinity fail with invalid_input and are not charged. When Young's modulus is negative the request fails with a message that E must be non-negative for thermal stress sigma equals E alpha Delta T, so clients can correct a sign error instead of receiving a non-physical result. On success the response echoes cleaned inputs, thermal_stress_pa as the product E times alpha times Delta T, thermal_stress_mpa as that value divided by one million, kind as compressive, tensile, or zero, the formula string sigma equals E alpha Delta T, method constrained_thermal_stress, and a units object documenting Pa, one per kelvin, K, Pa, and MPa when you stay in SI. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance. Keep E, alpha, and Delta T in a consistent unit system so the numeric stress is meaningful even if you convert labels offline after the call.
Where constrained thermal stress shows up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Integration use cases
Design tables and unit tests
Generate deterministic thermal-stress fixtures for CAD macros, material tables, and regression tests without reimplementing unit conversion by hand.