Introduction
Thermal stress is the internal stress that develops when a material wants to expand or contract due to a temperature change but is restrained by supports, fasteners, welds, surrounding parts, or friction.
In the simplest one-dimensional case (a straight member constrained so its length cannot change), the magnitude of thermal stress is proportional to the material stiffness and its thermal expansion.
This calculator implements the classic linear-elastic, fully restrained model used for quick engineering estimates. It is especially useful for comparing scenarios: different materials, different temperature swings, or different allowable stress limits.
Enter any three values and leave exactly one field blank to compute the missing quantity.
The equation on this page assumes 100% axial restraint (no net change in length). If your part can slide, rotate, or deform elsewhere, the actual stress may be much lower.
Use this tool as a conservative first pass unless you have confirmed full restraint.
How to use
- Choose consistent units (recommended: Pa for σ and E, 1/°C for α, and °C for ΔT).
- Fill in three fields and leave one field empty.
- Select Compute Missing Quantity to calculate the blank value.
- Interpret the result as a magnitude. Apply sign based on your convention (heating under restraint is typically compressive; cooling is typically tensile).
Tip: a temperature difference in °C is numerically the same as in K, so you may use kelvin for ΔT as long as you treat it as a difference (not an absolute temperature).
Formula (fully restrained, linear elastic)
For a member that is prevented from changing length (100% axial restraint) and remains in the linear elastic range, the thermal stress magnitude is:
σ = E · α · ΔT
Where:
- σ = thermal stress (Pa)
- E = Young’s modulus (Pa)
- α = coefficient of linear thermal expansion (1/°C or 1/K)
- ΔT = temperature change (°C or K as a difference)
Rearrangements used by the calculator:
The calculator rearranges the same equation to solve for the missing variable.
- E = σ / (α · ΔT)
- α = σ / (E · ΔT)
- ΔT = σ / (E · α)
Where the equation comes from (short derivation)
A uniform temperature change produces a free thermal strain of εth = α·ΔT. If the member is fully restrained, the total axial strain must be approximately zero.
That means the mechanical strain must cancel the thermal strain: εmech = −εth.
In linear elasticity, Hooke’s law gives σ = E·εmech, so the magnitude becomes |σ| = E·α·ΔT.
This is why stiff materials (high E) and high-expansion materials (high α) can generate large stresses under restraint, even for moderate temperature changes.
Worked examples
Example 1: compute thermal stress for a restrained steel bar
Suppose a steel bar is fixed at both ends (no axial expansion allowed). Use typical values:
E = 2.0 × 1011 Pa, α = 12 × 10−6 1/°C, and a temperature rise of ΔT = 40°C.
Then:
σ = E · α · ΔT = (2.0 × 1011) · (12 × 10−6) · 40 ≈ 9.6 × 107 Pa = 96 MPa.
This is a substantial stress level for many designs, which is why expansion joints, sliding supports, or flexible connections are often used in rails, pipelines, and long frames.
Example 2: solve for allowable temperature change
If you know the maximum allowable stress for a component, you can estimate the maximum temperature swing before that limit is reached.
For instance, assume an allowable stress magnitude of σ = 50 MPa for an aluminum part with E = 70 GPa and α = 23 × 10−6 1/°C.
Convert units: 50 MPa = 50×106 Pa and 70 GPa = 70×109 Pa.
ΔT = σ / (E·α) = (50×106) / ((70×109)·(23×10−6)) ≈ 31°C.
Interpretation: if the part is truly fully restrained, a temperature change on the order of a few tens of degrees could already push the stress toward the allowable limit.
If the assembly can slip or flex, the real ΔT capacity may be higher.
Assumptions and limitations
- Full restraint: The equation assumes the member cannot change length. Partial restraint reduces stress and requires stiffness/compatibility analysis.
- Uniform temperature change: The calculation assumes the entire member experiences the same ΔT. Temperature gradients can cause bending, warping, and localized stresses.
- Linear elasticity: Valid when the material remains elastic. At high temperatures or high stress, plasticity, creep, or stress relaxation may occur.
- Constant properties: E and α can vary with temperature; for large ΔT, use temperature-dependent data or a more detailed analysis.
- 1D axial model: Real components may have multi-axial constraints, complex geometry, and stress concentrations near holes, welds, and corners.
If your situation involves nonuniform heating, complex supports, or time-dependent effects, treat this calculator as a first-pass estimate and consider a more advanced method (compatibility analysis, beam theory for thermal curvature, or finite element analysis).
Notes on interpretation and design checks
Thermal stress can be compressive (heating under restraint) or tensile (cooling under restraint). Many engineering references use a sign convention where compressive stress is negative.
This calculator reports the computed value directly from the algebra using your inputs; if you want a signed result, apply your project’s sign convention consistently.
In design work, the computed thermal stress is rarely the only check. Depending on the component and loading, you may also need to evaluate:
- Yielding: compare stress to yield strength (and consider temperature-dependent strength).
- Buckling: compressive thermal stress in slender members can trigger instability at stresses below yield.
- Fatigue: repeated thermal cycles can cause fatigue damage even when peak stress is moderate.
- Brittle fracture: tensile thermal stress at low temperatures can be critical for brittle materials.
- Joint behavior: bolts, welds, adhesives, and solder joints may fail before the base material does.
A practical workflow is: (1) compute a conservative thermal stress using full restraint, (2) compare to allowable limits, and (3) if the margin is small, refine the model to include partial restraint, contact conditions, and realistic temperature distributions.
Reference material values (illustrative)
The table below shows how different combinations of stiffness (E) and expansion (α) affect thermal stress for the same temperature rise.
Values are approximate and depend on alloy, heat treatment, and temperature. Use datasheets or standards for final design.
Example thermal stress values for a 50°C temperature rise under full restraint.
| Material |
Young's Modulus (GPa) |
α (10⁻⁶/°C) |
Stress for ΔT=50°C (MPa) |
| Steel |
200 |
12 |
120 |
| Aluminum |
70 |
23 |
80.5 |
| Brass |
100 |
19 |
95 |
| Invar |
141 |
1 |
7.05 |
Notice how Invar, with its exceptionally low expansion coefficient, develops minimal stress compared with more common metals.
Conversely, aluminum’s high expansion coefficient can generate significant stress even though its modulus is lower than steel’s.
This is why mixed-material assemblies (for example, aluminum parts bolted to steel frames) often require careful allowance for differential expansion.
FAQ: practical questions
Does the formula apply to plates, pipes, or complex shapes?
The calculator is based on a one-dimensional axial model. For plates and shells, restraint can occur in multiple directions and the stress state can be biaxial or triaxial.
Pipes and pressure vessels may also experience thermal gradients through the wall thickness, producing bending stresses.
You can still use this tool for a quick estimate along a dominant restrained direction, but detailed design typically uses more complete theory or finite element analysis.
What if the part is only partially restrained?
Partial restraint is common: sliding supports, flexible mounts, gasketed joints, and long bolted connections all allow some movement.
In those cases, the thermal stress is reduced because some of the thermal strain is accommodated by displacement.
A common approach is to model the member and its supports as springs and solve compatibility: the stiffer the restraint relative to the member, the closer you get to the full-restraint stress.
Should I use °C or K for ΔT?
Use either, as long as it is a difference. A change of 30°C equals a change of 30 K.
Do not enter absolute temperature (like 300 K) unless you truly mean a 300 K change.
Why does the calculator ask me to leave exactly one field blank?
The underlying equation has four variables. If you provide three, the fourth is uniquely determined.
If you leave two blank, there are infinitely many solutions; if you fill all four, the values may be inconsistent.
The script checks for exactly one missing value to keep the result unambiguous.
What about sign (tension vs compression)?
The magnitude from σ = E·α·ΔT is always positive if you enter positive values.
In many conventions, heating under full restraint produces compressive stress (negative), while cooling produces tensile stress (positive).
Apply the sign that matches your analysis method and boundary conditions.
Can thermal stress exceed yield strength?
Yes. If the computed stress exceeds yield, the material may plastically deform, which can reduce stress (stress relaxation) but may permanently distort the part.
At elevated temperatures, creep can further change the stress over time.
If you suspect yielding or creep, the linear-elastic model is no longer sufficient, but it remains a useful indicator that you are in a high-risk regime.
Summary
Thermal stress is a predictable consequence of restrained thermal expansion or contraction.
With the linear-elastic, fully restrained assumption, the relationship σ = E·α·ΔT provides a fast estimate that helps you screen designs, compare materials, and understand sensitivity to temperature swings.
Use the calculator below to solve for stress, modulus, expansion coefficient, or temperature change, and then interpret the result in the context of restraint, geometry, and failure modes.
Editorial review by: JJ Ben-Joseph