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On a scorching summer day, miles of straight railroad tracks can suddenly warp into serpentine zig-zags called sun kinks in a fraction of a second.
When materials heat up, their atoms vibrate more vigorously and push each other apart, creating immense forces if the ends are locked in place.
How do we quantify the crushing internal forces created by a simple temperature rise?

Free vs. Constrained Expansion

If a rod of initial length L0L_0 is unconstrained, a temperature change Ξ”T\Delta T causes a free expansion of Ξ”Lth=Ξ±L0Ξ”T\Delta L_{\text{th}} = \alpha L_0 \Delta T, where Ξ±\alpha is the coefficient of linear expansion (the fractional change in length per degree of temperature change).
To prevent this rod from expanding, rigid walls must push back with a mechanical strain Ξ΅=Ξ”LthL0=Ξ±Ξ”T\varepsilon = \frac{\Delta L_{\text{th}}}{L_0} = \alpha \Delta T, governed by Young's modulus EE (a material's stiffness, defined as the ratio of stress to strain).
By substituting the prevented thermal strain into Hooke's Law (Οƒ=EΞ΅\sigma = E \varepsilon), we find the resulting thermal stress: Οƒ=EΞ±Ξ”T\sigma = E \alpha \Delta T
The total compressive force exerted on the supports is simply F=ΟƒA=EAΞ±Ξ”TF = \sigma A = E A \alpha \Delta T, where AA is the cross-sectional area.
What happens if civil engineers leave a small gap between the beam and the wall to relieve this pressure?

Expansion Joints and Gaps

An expansion joint provides an initial clearance gap Ξ΄\delta. As the temperature rises, the material expands freely without stress until it covers the gap (Ξ”Lth=Ξ΄\Delta L_{\text{th}} = \delta).