How much does an airliner cabin elongate at altitude?

Question

How much (about) is a commercial plane's cabin elongated at cruising altitude, due to interior-exterior pressure difference?

Of course there should be a slight shrinkage because of temp difference, but I estimate that overall a 30m cabin should be elongated by few centimeters.

Answer 1

We can identify three effects that have an effect on cabin length:

• Stretching due to the bulkhead pulling on the end of the fuselage;
• Shortening due to the fuselage expanding in diameter (Poisson effect)
• Shrinkage due to temperature

These are simple solid mechanics problems. Given an airplane with length $l$, diameter $d$, wall thickness $t$ made with a material with Young's modulus $E$ and Poisson ratio $\nu$ and linear expansion coefficient $\alpha$ pressurised to a pressure differential $p$, cooled to a temperature differential $\Delta T$....

$$\Delta L = \dfrac{d p L}{ 4 t E} - \dfrac{\nu d p L}{2 t E} - \alpha \Delta T l$$

$$\Delta L = \dfrac{d p L(1-2\nu)}{ 4 t E} - \alpha \Delta T l$$

Let's have a wall thickness of 3mm, diameter of 4m, length of 40m subjected to 0.5bar and 70°C temperature difference. Aluminium $E=$70GPa, $\nu=0.33$, $\alpha=25\mu m/m°C$. Filling in the numbers, we obtain $$\Delta L=3.2mm-70mm\approx-67mm$$

So thermal expansion definitely has the upper hand here, shrinking the fuselage by a few centimetres in total, whereas pressurising the fuselage can only expand it by a few millimetres. The numbers here are very much approximate but the order of magnitude will not change much with "real" numbers.