Problem — Steel rail

Problem

A steel rail $\ell_0 = 12{,}0$ m long is at $10\,^{\circ}$C and reaches $45\,^{\circ}$C on a summer day. The linear expansion coefficient of steel is $\lambda_{\text{acciaio}} = 12 \cdot 10^{-6}$ K$^{-1}$. By how much does the rail lengthen? What would happen if there were no expansion joints?

The rail of initial length $\ell_0$ expands by $\Delta\ell$ when the temperature rises.

Solution

The temperature change is $\Delta T = 45\,^{\circ}\text{C} - 10\,^{\circ}\text{C} = 35$ K.

The elongation due to linear expansion is: $\Delta\ell = \ell_0 \, \lambda \, \Delta T = 12{,}0 \cdot 12 \cdot 10^{-6} \cdot 35 = 12{,}0 \cdot 4{,}2 \cdot 10^{-4} \approx 5{,}04 \cdot 10^{-3} \ \text{m} \approx 5{,}0 \ \text{mm}$

$\ev{\Delta\ell \approx 5{,}0 \ \text{mm}}$

If there were no expansion joints, the elongation would be prevented: the blocked expansion would generate enormous internal compressive stresses. The rail, unable to lengthen, would buckle sideways, deforming dangerously. This is why rails and bridges are always fitted with joints that leave room for thermal expansion.

Collegamenti

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