Calculate the thermal stress that builds up in a fully constrained material from its Young's modulus, thermal expansion coefficient, and temperature change. Solve for the temperature change that produces a given stress, or the restraint force on a specific cross-section.
Thermal Stress — σ = E·α·ΔT
Stiffness of the material, in gigapascals (steel ≈ 200 GPa).
Linear expansion coefficient per degree Celsius (steel ≈ 12×10⁻⁶/°C).
Positive for heating, negative for cooling.
Solve for temperature change — ΔT = σ / (E·α)
Stiffness of the material, in gigapascals.
Linear expansion coefficient per degree Celsius.
Stress magnitude you want to reach or stay under, in megapascals.
Restraint Force — F = σ·A
Stiffness of the material, in gigapascals.
Linear expansion coefficient per degree Celsius.
Positive for heating, negative for cooling.
Cross-sectional area the restrained member acts over, in square millimetres.
Result
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Enter values above to compute.
4 min read3 steps6 terms3 examples6 FAQsσ = E·α·ΔT
Thermal stress is the internal stress a material develops when temperature change is fighting against something that won't let it move.
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Walk-through
How to Use This Calculator
3 steps▸
1
Pick a tab for what you're solving
Use the Thermal Stress tab when you know the modulus, expansion coefficient, and temperature change and want the resulting stress. Use the Solve tab when you know the modulus, expansion coefficient, and a target stress, and want the temperature change that produces it. Use the Force tab when you also know the cross-sectional area and want the restraint force on the supports.
2
Enter your known values
Young's modulus (E) is in gigapascals (GPa), the thermal expansion coefficient (α) is a per-degree-Celsius value (e.g. 0.000012 for steel), and the temperature change (ΔT) is in degrees Celsius — positive for heating, negative for cooling. On the Force tab, cross-sectional area is in square millimetres (mm²).
3
Read the stress and its sign
The result card shows the stress magnitude in megapascals (MPa) along with whether it's compressive (from heating) or tensile (from cooling). The interpretation line below explains what that means for the restrained member.
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Reference
Formula & Methodology
3 formulas▸
Thermal Stress in a Fully Constrained Member
σ = E·α·ΔT
σ (sigma) is the thermal stress in pascals, E is Young's modulus in pascals, α (alpha) is the material's linear thermal expansion coefficient per degree, and ΔT is the temperature change in degrees. This formula only applies when the member is fully restrained — prevented from expanding or contracting at all. Example: E = 200 GPa (steel), α = 12×10⁻⁶/°C, ΔT = 30°C → σ = 200,000 MPa × 0.000012 × 30 = 72 MPa.
Solving for Temperature Change
ΔT = σ / (E·α)
Rearranging the thermal stress formula lets you find the temperature swing that would produce a specific stress — useful for checking how much heating or cooling a restrained member can tolerate before reaching a maximum allowable stress.
Restraint Force
F = σ·A
Once you know the thermal stress, multiplying by the cross-sectional area (A) gives the total force the restrained member exerts on whatever is holding it in place — the number an engineer needs to size anchors, welds, or expansion joints.
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Glossary
Key Terms Explained
6 terms▸
Thermal Stress ↗The internal stress that develops in a material when it is prevented from freely expanding or contracting as its temperature changes. Unlike ordinary mechanical stress from an applied load, thermal stress arises purely from a restrained temperature change: σ = E·α·ΔT.
Thermal Expansion Coefficient ↗A material property (α, alpha) describing how much a material's length changes per degree of temperature change, per unit length. Measured in 1/°C or 1/°F. Steel is about 12×10⁻⁶/°C; aluminum is roughly double that at about 23×10⁻⁶/°C.
Constrained (Restrained) ↗Describes a member that is physically prevented from changing length — fixed at both ends, embedded in a rigid structure, or otherwise unable to expand or contract freely. Thermal stress only develops in a member that is at least partially constrained; a free-standing rod that can expand unopposed develops zero thermal stress.
Young's Modulus ↗A measure of a material's stiffness, defined as the ratio of stress to strain in the elastic region (E = σ/ε). Steel is about 200 GPa. A stiffer material (higher E) develops more thermal stress for the same temperature change and expansion coefficient.
Temperature Change (ΔT) ↗The difference between a material's current temperature and the temperature at which it was installed or last stress-free, in degrees. A positive ΔT (heating) drives compression in a restrained member; a negative ΔT (cooling) drives tension.
Buckling ↗A sudden, large-scale bending or bowing failure that occurs when compressive thermal stress in a restrained member — such as a length of continuously welded rail on a hot day — exceeds what the member can resist while staying straight. Expansion gaps and pre-tensioning are common ways to prevent it.
σ = E·α·ΔT = 200,000 MPa × 0.000012 × 30 = 72 MPa of compressive stress. Rail is laid and anchored at a set 'neutral' temperature; a 30°C rise above that on a hot day compresses it internally since it can't expand — this is the mechanism behind rail buckling ('sun kink') on unshaded track.
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Checking a Stress Limit
Solving for the maximum allowable temperature swing
On the Solve tab, ΔT = σ / (E·α) = 150 MPa / (200,000 MPa × 0.000012) = 62.5°C. If a designer's allowable stress is 150 MPa, the restrained member can tolerate up to a 62.5°C swing from its stress-free installation temperature before that limit is reached.
On the Force tab, σ = 72 MPa (from the same inputs as the first example) and F = σ·A = 72 MPa × 500 mm² = 36,000 N. An engineer sizing the anchors or welds holding this member in place needs to design for at least this restraint force.
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Reference
Cite This Calculator
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Use either format to cite this calculator in a paper, report, or resource list.
Thermal stress is the internal stress a material develops when temperature change is fighting against something that won't let it move. It's a routine consideration in rail, pipeline, bridge, and building design — anywhere a long run of material is fixed in place and will see a real temperature swing over its service life.
Why Restraint Matters More Than Temperature Alone
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A material that is completely free to expand or contract with temperature — a loose rod lying on a frictionless surface, say — never develops any thermal stress, no matter how large the temperature swing. Thermal stress only appears when something prevents that natural length change: fixed supports at both ends, a rigid surrounding structure, or a weld holding a section of pipe in place. The formula σ = E·α·ΔT captures this directly — it computes the stress that would be needed to force the material back to its original, unrestrained length after a ΔT temperature change. The larger the temperature swing, the stiffer the material (higher E), and the more it wants to expand per degree (higher α), the larger that stress becomes.
Tension, Compression, and Why Rails Buckle
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Heating a restrained member makes it want to get longer; because it can't, it's effectively being squeezed — compressive stress. Cooling has the opposite effect: the member wants to shrink, and being held at its original length puts it in tension. Compressive thermal stress is the more dramatic failure mode in practice, because slender members under high compression can suddenly buckle sideways rather than simply crushing — this is exactly what causes 'sun kink' in continuously welded rail on a hot day, when track laid and anchored during cooler weather is compressed well beyond its buckling threshold once summer heat arrives.
Why Engineers Use Expansion Gaps and Joints
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Because thermal stress in a fully restrained member scales directly with temperature swing, one of the most effective mitigations is simply not fully restraining it. Expansion joints in bridges, sidewalks, and pipelines give a structure somewhere to move so the full σ = E·α·ΔT stress never develops. Where full restraint is unavoidable — welded rail, embedded reinforcing bar, pressure vessels — designers instead account for the expected temperature range up front, choosing a 'neutral' installation temperature and a stress limit the member must never exceed, which is exactly the calculation the Solve tab performs in reverse.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for thermal stress?+
σ = E·α·ΔT, where E is Young's modulus, α is the material's thermal expansion coefficient, and ΔT is the temperature change. This formula applies specifically to a fully constrained member — one that cannot expand or contract at all.
Why do rails buckle in hot weather?+
Continuously welded rail is anchored at a fixed 'neutral' installation temperature and cannot expand when it heats up. On a hot day, the temperature rise above neutral generates compressive thermal stress (σ = E·α·ΔT); if that stress exceeds the track's buckling resistance, the rail suddenly bows sideways — a failure called 'sun kink.'
Is thermal stress tension or compression?+
It depends on the direction of the temperature change. Heating a restrained member (positive ΔT) creates compressive stress, since the material wants to expand but can't. Cooling it (negative ΔT) creates tensile stress, since the material wants to shrink but is held at its original length.
What units does the calculator use?+
Young's modulus is entered in gigapascals (GPa), the expansion coefficient in 1/°C, and temperature change in degrees Celsius. The result is displayed in megapascals (MPa) for stress and newtons (N) for restraint force on the Force tab.
What if the material is free to expand?+
A material that is completely unrestrained develops zero thermal stress regardless of how large the temperature change is — the σ = E·α·ΔT formula only applies to a member that is prevented from changing length. This is exactly why expansion joints and gaps are used to relieve thermal stress in real structures.
Can I solve for the temperature change instead of the stress?+
Yes — use the Solve tab. Given Young's modulus, the expansion coefficient, and a target stress, it rearranges the formula to ΔT = σ / (E·α), giving you the temperature swing that would produce that stress in a fully restrained member.
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