Thermal Stress Calculator Formula
Understand the math behind the thermal stress calculator. Each variable explained with a worked example.
Formulas Used
Thermal Stress
thermal_stress = e_mpa * alpha * abs(delta_t)Free Thermal Strain
free_strain = alpha * abs(delta_t)Variables
| Variable | Description | Default |
|---|---|---|
modulus | Elastic Modulus (E)(GPa) | 200 |
alpha | Thermal Expansion Coeff (alpha)(1/deg C) | 0.000012 |
delta_t | Temperature Change (delta T)(deg C) | 80 |
e_mpa | Derived value= modulus * 1000 | calculated |
How It Works
Thermal Stress in Restrained Members
When a material cannot expand or contract freely, temperature changes produce internal stresses.
Formula
sigma_thermal = E alpha delta_T
where E is the elastic modulus, alpha is the coefficient of linear thermal expansion, and delta_T is the temperature change. Heating a restrained member produces compression; cooling produces tension.
Worked Example
A steel rail (E=200 GPa, alpha=12e-6 /deg C) heated by 80 deg C while fully restrained.
- 01Free strain = 12e-6 x 80 = 9.6e-4 = 0.000960
- 02Thermal stress = 200,000 MPa x 0.000960 = 192 MPa
- 03This is 77% of steel yield (250 MPa), highlighting why expansion joints are critical.
Frequently Asked Questions
What are typical thermal expansion coefficients?
Steel: 12e-6 /deg C, Aluminium: 23e-6 /deg C, Copper: 17e-6 /deg C, Concrete: 10-12e-6 /deg C, Invar (nickel alloy): 1.2e-6 /deg C. Aluminium expands nearly twice as much as steel.
What if the member is only partially restrained?
If the member can expand partially, the actual stress is reduced proportionally. For a member with partial fixity, sigma = E alpha delta_T x (1 - ratio_of_free_expansion_allowed).
Why do bridges have expansion joints?
Without joints, a 100 m steel bridge heated by 40 deg C would try to expand by about 48 mm. The resulting compressive stress (96 MPa) could buckle the structure. Expansion joints allow free movement.
Learn More
Guide
Thermal Expansion Guide: Calculating Length, Area, and Volume Changes
Understand thermal expansion in engineering materials. Learn to calculate linear, area, and volumetric expansion, handle expansion joints, and avoid thermal stress failures.
Ready to run the numbers?
Open Thermal Stress Calculator