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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.

Pondlife
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George Metaxas
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1 Answers1

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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.

Sanchises
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  • I wonder if someone did actually made a measurement. It's very easy using a laser meter. – George Metaxas Aug 29 '20 at 08:49
  • I'm not sure airline operators will appreciate it if you shine a laser through a cabin full of passengers. – Sanchises Aug 29 '20 at 09:03
  • Interesting... I'll love to know what's the Poisson's effect contribution... – xxavier Aug 29 '20 at 10:51
  • @xxavier To cover that exact question I already wrote the equation to easily show the Poisson effect, namely, the $(1-2\nu)$. So we have about 10mm extension minus about 7mm Poisson contraction. – Sanchises Aug 29 '20 at 11:59
  • Since the elongation of the cabin due to the pressure difference is so small, I wonder how this miniscule difference can cause metal fatigue to the airframe. The length change due to temp difference that is much more impotant, is inherent to the metal, it should not provoke metal fatique. Am I wrong? – George Metaxas Aug 29 '20 at 14:56
  • I realize the math is the math, but 67 mm seems like an awful lot. You generally don't see tension regulators in the elevator and rudder cable runs, only in the wings to accommodate wing flex, and that much change in length would play havoc with cable tensions. – John K Aug 29 '20 at 15:03
  • Do planes cruising at 10,000m use control cables? – George Metaxas Aug 29 '20 at 16:09
  • Aluminium has a very different fatigue behaviour than steel: it will show fatigue at any amount of stress, unlike steel which is "safe" from fatigue below a certain amount of stress. And yes, a B737 for example has mechanical linkages to the control surfaces, with hydraulic augmentation much like power steering in a car. – Sanchises Aug 29 '20 at 18:46
  • @JohnK Related: https://aviation.stackexchange.com/questions/46179/ I'm would think there is some mechanism in the rudder too to take up slack. Alternatively, the cables get cold too? I agree that 7cm is a lot, but I think the math checks out. – Sanchises Aug 29 '20 at 18:57