Calculate thermal expansion force or solve for Young’s modulus, expansion coefficient, temperature change, or cross-sectional area with selectable units.
Thermal Expansion Force Formula
The calculator uses the fully restrained thermal expansion force formula. It assumes the material wants to expand or contract because of a temperature change, but that movement is prevented.
F = E \alpha \Delta T A
The same equation can be rearranged to solve for any missing value:
E = \frac{F}{\alpha \Delta T A}\alpha = \frac{F}{E \Delta T A}\Delta T = \frac{F}{E \alpha A}A = \frac{F}{E \alpha \Delta T}- F = thermal expansion force, in lbf or N
- E = Young’s modulus of the material, in psi, Pa, kPa, or MPa
- α = coefficient of thermal expansion, in in/in°F or mm/mm°C
- ΔT = change in temperature, in °F or °C
- A = cross-sectional area, in in², mm², cm², or m²
To calculate thermal expansion force, enter Young’s modulus, thermal expansion coefficient, temperature change, and cross-sectional area. To solve for another value, leave that one field blank and fill in the other four. Internally, the calculator converts values to a consistent unit set, applies the formula, then converts the result back to the unit you selected.
Typical Young’s Modulus and Thermal Expansion Values
Use these approximate values only when a material-specific value is not available. Actual values vary by alloy, grade, temperature range, and material condition.
| Material | Young’s Modulus | Thermal Expansion Coefficient |
|---|---|---|
| Carbon steel | 29,000,000 psi, about 200 GPa | 6.5 × 10-6 in/in°F, about 11.7 × 10-6 /°C |
| Stainless steel | 28,000,000 psi, about 193 GPa | 9.6 × 10-6 in/in°F, about 17.3 × 10-6 /°C |
| Aluminum | 10,000,000 psi, about 69 GPa | 12.8 × 10-6 in/in°F, about 23 × 10-6 /°C |
| Copper | 17,000,000 psi, about 117 GPa | 9.4 × 10-6 in/in°F, about 16.9 × 10-6 /°C |
| Concrete | 3,000,000 to 5,000,000 psi, about 21 to 35 GPa | 5.5 × 10-6 in/in°F, about 10 × 10-6 /°C |
Thermal Expansion Force Calculation Examples
Example 1: Calculate force in a restrained steel bar
A steel bar has a Young’s modulus of 29,000,000 psi, a thermal expansion coefficient of 6.5 × 10-6 in/in°F, a temperature increase of 80°F, and a cross-sectional area of 2 in².
F = 29{,}000{,}000 \times 0.0000065 \times 80 \times 2F = 30{,}160 \text{ lbf}The restrained thermal expansion force is 30,160 lbf.
Example 2: Calculate required cross-sectional area
A restrained member produces 20,000 lbf of force. Its Young’s modulus is 10,000,000 psi, thermal expansion coefficient is 12.8 × 10-6 in/in°F, and temperature change is 100°F.
A = \frac{20{,}000}{10{,}000{,}000 \times 0.0000128 \times 100}A = 1.5625 \text{ in}^2The cross-sectional area is 1.5625 in².
Thermal Expansion Force FAQ
What does thermal expansion force mean?
Thermal expansion force is the force created when a material is prevented from freely expanding or contracting as its temperature changes. If the material is free to move, it expands or contracts with little or no restraint force. If it is fully restrained, thermal strain becomes mechanical stress, which creates force over the cross-sectional area.
Does length affect the thermal expansion force?
For a fully restrained straight member with uniform temperature change, length does not appear in the force equation. A longer member would have more free expansion movement, but if both members are fully restrained, the thermal stress is based on EαΔT, not length. Length can still matter in real structures because supports, bending, buckling, and partial restraint change the actual force.
Why can the calculated force be very large?
The force can be large because even a small thermal strain can create high stress when movement is completely blocked. The formula assumes elastic behavior and full restraint. If the material yields, slips at a support, bends, cracks, or is only partly restrained, the real force may be lower than the ideal calculated value.
