Thermal Stress Calculator

Calculate thermal stress in constrained structures from temperature change.

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Prevented Thermal Expansion Creates Mechanical Stress

A material that is free to expand changes length with temperature but develops no thermal stress from that expansion alone. For a uniform temperature change, free thermal strain is εth=αΔT. If expansion is completely prevented in a one-dimensional linear-elastic member, the mechanical strain required to cancel it produces stress of magnitude σ=EαΔT.

Heating under full restraint usually creates compression, while cooling creates tension, depending on the sign convention and support arrangement. Partial restraint, temperature gradients, plasticity, creep, composite materials, or multidimensional constraints require more detailed structural analysis.

εth=αΔT,   fully restrained: σ=−EαΔT
SymbolMeaningWhy it appears / units
αCoefficient of thermal expansion1/K or 1/°C.
ΔTTemperature changeK or °C difference.
EYoung’s modulusPa; linear-elastic stiffness.
σThermal stressPa; sign depends on restraint and convention.

Thermal stress can become large because elastic strain limits are small. Expansion joints, sliding supports, and material matching are used to reduce restraint and avoid cracking, buckling, or excessive loads.

Free expansion is the zero-stress limiting case. If a member can change length without restraint, the thermal strain can occur without the EαΔT stress. Under full restraint, increasing E, α, or |ΔT| should increase the elastic thermal-stress magnitude proportionally.

Worked Examples

Example 1: Steel rail ΔT=50°C, fully constrained
σ=200e9×12e-6×50
Result: 120 MPa compressive
Railway expansion joints prevent this
Example 2: Concrete slab ΔT=30°C, summer heat
σ=30e9×12e-6×30
Result: 10.8 MPa — may crack!
Concrete tensile strength only 2-4 MPa
Example 3: Fully restrained steel
E=200GPa, α=12×10−6/K, ΔT=50K
Result: |σ|=120MPa
A modest temperature rise can create substantial stress when expansion is completely blocked.
Example 4: Free expansion
same material and temperature change, no restraint
Result: εth=600με, σ≈0
Free thermal strain alone does not imply thermal stress.

Common Mistakes

⚠️
Calculating stress for a freely expanding member

Without restraint, the member changes length and ideally develops no axial thermal stress.

⚠️
Mixing ppm/K and decimal strain

A coefficient such as 12ppm/K means 12×10⁻⁶/K.

⚠️
Ignoring yielding or creep at large temperature changes

The linear elastic formula can overpredict stress once plastic deformation, creep, or changing material properties become important.

Frequently Asked Questions

Why expansion joints in bridges?
Steel bridges expand ~12 mm per 10m per 10°C. A 100m bridge changes ~120 mm between winter and summer. Without joints, thermal stress would buckle or crack the structure.
Thermal stress in electronics?
Coefficient of thermal expansion (CTE) mismatch causes solder joint fatigue in PCBs. Silicon: α≈3 ppm/°C. FR4 PCB: α≈14-17 ppm/°C. Mismatch → shear stress → crack over thermal cycles.
Why does heating sometimes create compressive stress?
A heated member wants to expand. If supports prevent that expansion, they push back on the member, creating compressive mechanical strain and stress.
Can Celsius be used for ΔT?
Yes. A temperature difference of 1°C equals a difference of 1K, so either scale gives the same ΔT magnitude.
What is partial restraint?
It means the structure can expand somewhat but not freely. Stress then lies between zero and the fully restrained value and depends on support stiffness and system geometry.
Why are expansion joints used?
They allow controlled thermal movement so temperature changes do not generate excessive stress in long structures such as bridges, rails, pipes, and buildings.
When does thermal expansion create stress instead of free movement?
A uniform temperature change creates strain αΔT, but it creates stress only when expansion or contraction is restrained. For complete one-dimensional restraint in a linear elastic material, the magnitude is approximately Eα|ΔT|. Partial restraint, yielding, creep, gradients, and multiaxial geometry require more detailed analysis.

Formula Explorer connections

Interpretation: This formula tracks heat, temperature, work, entropy or transport in a thermodynamic system. Assumption: Use absolute temperature where required and consistent energy units. Constant properties, equilibrium, ideal gases or negligible losses may be assumed.

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