What does "fully constrained" mean?

A fully constrained member is one whose ends are fixed so that no thermal deformation can occur. The restraint forces develop an internal stress equal to σ = E × α × ΔT. Real structures are rarely 100 % fixed, so this gives the upper-bound (worst-case) stress.

Why is stress compressive when the member is heated?

A heated member wants to expand. If the supports prevent expansion, they push back with compressive reaction forces, producing compressive stress inside the member. Cooling produces the opposite: tensile stress.

What units should I use for α?

In SI, CTE is expressed as 1/°C (or 10⁻⁶/°C = µε/°C for metals). In Imperial, use 1/°F. Common steel CTE values: ~11.7 × 10⁻⁶/°C (SI) or ~6.5 × 10⁻⁶/°F (Imperial).

Does the formula change for partial constraint?

Yes. For a partially constrained member, σ = E × α × ΔT × (1 − δ_allowed / δ_free), where δ_free = α × ΔT × L. This tool handles the fully constrained (δ_allowed = 0) case only.

How accurate is the result?

The formula is exact within linear-elastic (Hookean) behavior. Accuracy depends on how well your input values represent the material. For high ΔT, check whether α varies significantly with temperature — if so, use the mean α over the temperature range.

Thermal Stress Calculator

Enter elastic modulus, thermal expansion coefficient, and temperature change to compute thermal stress in a fully constrained member.

Inputs

Unit System

Material Preset (optional)

Elastic Modulus E (GPa)

Coefficient of Thermal Expansion α (× 10⁻⁶ /°C)

Enter the value as-is (e.g. 11.7 for 11.7 × 10⁻⁶/°C).

Temperature Change ΔT (°C)

Positive = temperature rise; negative = temperature drop.

Results

Enter values and click Calculate.

Sign Convention

Heating (ΔT > 0)

Member wants to expand; restraint causes compressive stress (σ < 0).

Cooling (ΔT < 0)

Member wants to contract; restraint causes tensile stress (σ > 0).

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Summary

This thermal stress calculator applies the classical formula σ = E × α × ΔT to determine the stress developed in a member that is fully prevented from expanding or contracting when its temperature changes. Enter the material's elastic (Young's) modulus E, its coefficient of thermal expansion α, and the temperature change ΔT. The tool instantly returns thermal stress σ along with the sign convention (compressive on heating, tensile on cooling). All computation runs client-side.

How it works

Select a unit system: SI (GPa, 1/°C, MPa) or Imperial (Msi, 1/°F, ksi).
Enter the elastic modulus E of the material (or pick a preset).
Enter the coefficient of thermal expansion (CTE) α.
Enter the temperature change ΔT (positive = heating, negative = cooling).
Click Calculate to see thermal stress σ = E × α × ΔT.
Negative σ means compression (member wants to expand but is restrained); positive σ means tension.

Use cases

Checking pipe stress in process piping fixed between two rigid supports.
Evaluating stress in railroad rails constrained by spikes and clips during summer heat.
Estimating thermal stresses in bridge decks and expansion joints.
Verifying mechanics-of-materials homework on statically indeterminate thermal problems.
Comparing thermal stress levels across different materials at the same ΔT.
Sizing expansion loops and bellows to keep stresses within allowable limits.
Teaching Hooke's Law applied to constrained thermal loading.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu