Thermal Stress and Expansion Calculator
Find the thermal stress that develops in a fully restrained member from a temperature change, alongside the free expansion it would have had if unrestrained.
🌡️ What is Thermal Stress?
Thermal stress is the internal stress that develops in a material when a temperature change occurs but the material is prevented from expanding or contracting freely. In a fully restrained member, no length change at all is permitted, so the stress that would otherwise be relieved by free expansion instead builds up internally, following sigma = E x alpha x deltaT, where E is the elastic modulus and alpha is the coefficient of linear thermal expansion.
Engineers meet this everywhere restrained members cross large temperature ranges: continuous welded rail that heats up in the sun, a pipeline carrying hot fluid between fixed anchor points, a bridge deck restrained between abutments, or a concrete slab poured against rigid formwork. Each case reduces to the same formula once alpha, deltaT, and E are known, and the same reasoning explains why expansion joints and sliding bearings exist.
A common point of confusion is mixing up thermal stress with free thermal expansion. Free expansion (deltaL = alpha x L0 x deltaT) is simply how much longer or shorter an unrestrained member becomes, it produces no internal stress at all. Thermal stress only appears when that natural length change is blocked, and unlike free expansion it does not depend on the member's original length.
This calculator computes both values side by side, the thermal stress that would develop if the member were fully restrained, and the free expansion it would have undergone if it were not, with the sign (tension or compression) determined automatically from whether the temperature change is a heating or a cooling.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Steel Rail Heated in Sunlight
A 10 m steel rail (alpha = 12x10⁻⁶/°C, E = 200 GPa) heated by 40°C
Example 2 — Aluminum Bracket Cooled Overnight
A 5 m aluminum bracket (alpha = 23x10⁻⁶/°C, E = 69 GPa) cooled by 30°C
Example 3 — Steel Pipeline Section Heated by Process Fluid
A 3 m steel pipe section (alpha = 11.7x10⁻⁶/°C, E = 200 GPa) heated by 60°C
❓ Frequently Asked Questions
🔗 Related Calculators
What is thermal stress?
Thermal stress is the internal stress that develops in a material when it is prevented from expanding or contracting freely as its temperature changes. In a fully restrained member it is given by sigma = E x alpha x deltaT, independent of the member's length.
What is the formula for thermal stress in a restrained member?
sigma = E x alpha x deltaT, where E is the elastic modulus, alpha is the coefficient of linear thermal expansion, and deltaT is the temperature change. Heating a restrained member produces compression, cooling produces tension.
Why doesn't length appear in the thermal stress formula?
Thermal strain (alpha x deltaT) is the same fraction of length everywhere along a uniform, fully restrained member, and stress equals modulus times strain (Hooke's law), so length cancels out. Length only matters for the free expansion value, deltaL = alpha x L0 x deltaT.
What is free (unrestrained) thermal expansion?
Free expansion is the change in length a member would undergo if nothing prevented it from expanding or contracting, deltaL = alpha x L0 x deltaT. This calculator shows it alongside the restrained thermal stress for direct comparison.
Why does heating cause compression in a restrained member?
Heating makes the material want to expand. If the ends are fixed and cannot move apart, the material is squeezed back to its original length, which puts it into compression. Cooling has the opposite effect, the material wants to shrink but is held at its original length, putting it into tension.
How do engineers prevent thermal stress from damaging structures?
Expansion joints, sliding bearings, and flexible couplings are commonly used in bridges, pipelines, and railway tracks specifically to let the structure change length freely with temperature, avoiding the buildup of restrained thermal stress.
What units does this calculator use?
Alpha is entered as a value times 10^-6 per degree Celsius (a typical way material coefficients are tabulated), original length in millimeters, temperature change in degrees Celsius, and elastic modulus in gigapascals. Results are shown in megapascals (stress) and millimeters (expansion).
What is a typical coefficient of thermal expansion for steel?
Structural steel has a coefficient of linear thermal expansion of approximately 11.7 to 12 x 10^-6 per degree Celsius, aluminum is roughly twice that, around 22 to 24 x 10^-6 per degree Celsius.
Can this formula be used for a partially restrained member?
Not directly. This formula assumes zero permitted length change (full restraint). A partially restrained member develops a fraction of this stress depending on the stiffness of the restraint, that case requires a compatibility (spring-in-series) analysis rather than this simple formula.
Does thermal stress depend on the cross-sectional area of the member?
No, for a fully restrained axial member the thermal stress sigma = E x alpha x deltaT does not depend on area, since stress is force per unit area and both the induced force and the area scale together. Area only matters when converting stress to an equivalent force (F = sigma x A).