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The above diagram is misleading. In reality, both the length and thickness are equal to 8mm. As this is from a past paper (one notorious for awkward questions), I'm not sure if it's a typo (an 8m length would be much more straightforward). Is there any possible outcome/plane in which this pipe would indeed buckle?

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    $\begingroup$ I'm pretty sure this is a typo and should be L=8m. $\endgroup$ – atom44 Dec 22 '16 at 22:10
  • $\begingroup$ Kinda random question, but why would you care about buckling in this example? The question was clearly made for thermal expansion and is low on details to really bother figuring oit other stuff. $\endgroup$ – JMac Dec 23 '16 at 10:56
  • $\begingroup$ Well, thermal expansion of a pipe with constrained ends will induce a compressive stress in the pipe (the constrained ends resist the tendency of the pipe to expand by heat, and so the thermal strain is countered by an equal compressive elastic strain, which in turn requires a compressive stress). Whenever dealing with compression of slender members, buckling should always be kept in mind. $\endgroup$ – Involute Dec 23 '16 at 11:03
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It is not necessarily a typo, but you wouldn't normally call a thin walled cylinder 8mm high and 100mm diameter a "pipe".

However, the formulas for Euler buckling are most definitely not valid for a structure with those dimensions. The appendix of this rather old report http://www.dtic.mil/dtic/tr/fulltext/u2/a801283.pdf gives some basic theory and graphs for "hand calculation". Such problems would now be solved by computer modeling, of course.

In real life, the critical load will be very sensitive to the exact boundary conditions.

Since the OP's diagram looks as it if came from a rather old textbook, most likely it is a typo, but since the length as shown in the diagram is apparently less than the "obvious" typo of 8m it is not very clear what the "correct" dimensions for the numerical part of question should be.

Deriving a general formula for a pipe of length L, diameter d and wall thickness t is a sensible exercise, though.

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