Thermal Stress and Composite Bars

Heating widens the mean spacing of atoms, so an unrestrained body expands by δ = αLΔT. Nothing resists the change, so no stress arises anywhere. Establishing this first matters, because the whole topic is about what happens when it is prevented.

1. 1Imagine releasing one end and letting the bar expand freely by αLΔT.
2. 2Now push it back to its original length — a compressive strain of exactly αΔT.
3. 3Multiply by E: the stress is σ = EαΔT.
4. 4Note what cancelled: the length L is nowhere in the answer.

For steel, Eα ≈ 2.4 MPa per °C. A 50 °C rise in a fully restrained member gives 120 MPa with nothing hung on it. A 75 °C swing reaches most of mild steel's yield stress.

Since length does not appear, a 1 m bar and a 100 m bar reach the same stress. Continuous welded rail is laid pre-tensioned for this reason, and a bridge without expansion joints would tear out its own bearings.

Provide a gap g and the bar expands freely until it closes it. Only the surplus is resisted, so σ = E(αLΔT − g)/L. If the gap exceeds the free expansion, the stress is zero — the bar never touches.

Here L is back in the formula, because the gap is a fixed distance while the free expansion grows with length. That is why joint spacing is a design decision rather than a constant.

Bond two materials with different coefficients of expansion and heat them. Each wants a different final length and they are stuck with each other.

1. 1The higher-α material is held back, ending in compression.
2. 2The lower-α material is dragged out, ending in tension.
3. 3Two conditions solve it: internal forces sum to zero (no external load), and both must finish the same length.

Bond them face to face instead of end to end and the mismatch resolves by bending: the higher-expansion side takes the outside of the curve. Curvature is proportional to (α₁−α₂)ΔT, so tip deflection reads temperature directly. That is a bimetallic strip, and it is how thermostats and mechanical thermometer dials work.

Thermal stress is also self-limiting in a way mechanical stress is not: once the member yields, the strain mismatch is relieved and the stress stops growing. That is why thermal loading is treated differently from dead load in design codes.