How It Works
Linear Thermal Expansion
When a material is heated, its atoms vibrate more vigorously and occupy more space, causing the material to expand. The change in length is proportional to the original length and temperature change:
dL = alpha * L * dT
Where:
- dL = Change in length (mm, in)
- alpha = Coefficient of linear thermal expansion (10^-6 /C or /F)
- L = Original length at reference temperature (mm, m, in, ft)
- dT = Temperature change (T_final - T_initial)
Volumetric Thermal Expansion
For isotropic materials (same properties in all directions), volumetric expansion is approximately three times the linear expansion:
dV = beta * V * dT = 3 * alpha * V * dT
Where beta is the volumetric coefficient of thermal expansion. This approximation is valid for small temperature changes where higher-order terms are negligible.
Thermal Stress from Restrained Expansion
When a material is prevented from expanding or contracting freely, thermal stresses develop. For a fully restrained member:
sigma = E * alpha * dT
Where:
- sigma = Thermal stress (MPa, psi)
- E = Elastic modulus (Young's modulus) of the material
- Heating produces compressive stress (material wants to expand but cannot)
- Cooling produces tensile stress (material wants to contract but cannot)
The thermal strain that would occur if free to expand is:
epsilon_thermal = alpha * dT
In a restrained condition, this strain is converted to stress via Hooke's Law.
Practical Applications
- Steam Piping: High-temperature steam lines can expand several inches over their length. Expansion loops, bellows, or slip joints accommodate this movement.
- Bridge Expansion Joints: Long bridges use finger joints or modular expansion joints to prevent buckling in summer and cracking in winter.
- Rail Track: Continuous welded rail (CWR) relies on pre-stressing and controlled restraint. Gaps are provided at switches and crossings.
- Pressure Vessels: Differential expansion between shell and internals requires floating heads or expansion bellows.
Design Considerations
- Safety Factor: Apply 1.25-1.5x to calculated expansion for joint sizing
- Temperature Range: Consider seasonal extremes, not just operating temperature
- Anchor Points: Determine fixed points and direction of expansion
- Guides: Allow axial movement while preventing lateral displacement
- Cyclic Loading: Repeated heating/cooling causes fatigue - consider weld quality and stress risers
Thermal Expansion Calculator
Calculate linear and volumetric thermal expansion, thermal stresses in restrained conditions, and expansion joint sizing for piping and structures.
Volumetric CTE (beta) = 3 * alpha for isotropic materials
Expansion Visualization
Results
Key Formulas
Linear Expansion:
dL = alpha * L * dT
Volumetric Expansion:
dV = 3 * alpha * V * dT
Thermal Strain:
epsilon = alpha * dT
Thermal Stress (restrained):
sigma = E * alpha * dT
Thermal Expansion Coefficient Database
| Material | alpha (10^-6/C) | E (GPa) | Sy (MPa) |
|---|---|---|---|
| Carbon Steel | 12.0 | 200 | 250 |
| SS 304 | 17.3 | 193 | 215 |
| SS 316 | 16.0 | 193 | 205 |
| Aluminum 6061 | 23.1 | 69 | 276 |
| Copper | 16.5 | 117 | 70 |
| Brass | 19.0 | 100 | 200 |
| Titanium Ti-6Al-4V | 8.6 | 114 | 880 |
| Invar 36 | 1.2 | 141 | 276 |
| Cast Iron | 10.8 | 100 | 200 |
| Concrete | 12.0 | 30 | 3 |
| PVC | 70.0 | 3 | 45 |
| HDPE | 120.0 | 1 | 25 |
Typical Expansion Values
Steam Piping (carbon steel): ~1.5 mm per meter per 100C rise
Bridge Deck (steel, 100m): ~80 mm total movement (-30C to +50C)
Rail Track (per km): ~300 mm expansion over 25C rise
Note: Always verify CTE values from material certificates. Values vary with temperature range and alloy composition.