What is thermal stress?

Thermal stress develops when a material is heated or cooled but cannot freely expand or contract. For a fully constrained member: σ = E × α × ΔT, where E is elastic modulus, α is the coefficient of thermal expansion (CTE), and ΔT is the temperature change.

What is the coefficient of thermal expansion (CTE)?

CTE (α) measures how much a material expands per degree of temperature change. Steel: ~12×10⁻⁶/°C, Aluminum: ~23×10⁻⁶/°C, Copper: ~17×10⁻⁶/°C, Stainless steel: ~17×10⁻⁶/°C.

When does thermal stress occur?

Thermal stress occurs when expansion is constrained: pipes fixed at both ends, rails welded continuously, dissimilar materials bonded together, thick sections with temperature gradients, or any structure restrained from free movement.

How can thermal stress be reduced?

Use expansion joints or loops in piping, allow sliding supports, choose materials with lower CTE, reduce ΔT with insulation, or use flexible connections. In bimetallic systems, match CTE values as closely as possible.

Is thermal stress always compressive when heating?

Yes, for a fully constrained member heated uniformly. The material tries to expand but cannot, so internal compressive stress develops. Cooling causes tensile stress. The sign depends on whether expansion or contraction is restrained.

Free Engineering Tool

Thermal Stress Calculator

Calculate thermal stress σ = E × α × ΔT for a fully constrained member subjected to temperature change.

σ = E·α·ΔTConstrained Member#169

Material Preset

Elastic Modulus E (GPa)

CTE α (×10⁻⁶ /°C)

Temperature Change ΔT (°C)

Quick presets

Results

Thermal Stress σ

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Thermal Strain ε

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Stress Type

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Thermal Stress in Constrained Members

Common CTE Values

Material	E (GPa)	α (×10⁻⁶/°C)
Carbon Steel	200	12
Stainless Steel 304	193	17.3
Aluminum 6061	69	23.6
Copper	117	16.6
Titanium	114	8.6
Brass	100	19

Thermal Stress Calculator | σ = E·α·ΔT | Free Online Tool

Published by Nikolai Shelkovenko on February 15, 2026 February 15, 2026

Results

Thermal Stress in Constrained Members

Common CTE Values