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How might one calculate or approximate the negative pressure applied to the outer surface of a cylinder required to expand it a certain distance, when held open rigidly at each end?

The cylinder ID = 20*the wall thickness and the material is very elastic.

Many thanks for your help!!

Oli

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  • $\begingroup$ This is one of the more interesting aspects in engineering. There's an old model, but it doesn't work. Turns out defects (such as a pipe not being perfectly circular) in the pipe will make the pipe buckle significantly earlier than the model predicts. And estimating defects is ... difficult. As a result, we've got the result of a good model, with a big conservative fudge factor on top. May or may not be useful to you. $\endgroup$
    – Mark
    May 18 '18 at 21:47
  • $\begingroup$ @Mark OK anything's definitely better than nothing! what is this model you're talking about? Cheers $\endgroup$ May 18 '18 at 22:35
  • $\begingroup$ Is this a fairly stiff material that only deflects a couple % before rupturing, or is it more elastic, where the axial radius of curvature is less than 100 times greater than the nominal cylinder diameter? For steel and other materials that deform only slightly, see this NASA report on tank stresses and deformations $\endgroup$
    – Phil Sweet
    May 19 '18 at 2:12
  • $\begingroup$ You still haven't adequately described the end constraint. Is there a thick band at the ends that is much stiffer than the middle? Or is there and end plate? and what is their thickness and strength compared to the cylinder? The cylinder's shape depends on both the edge axial shear and edge bending moments. Those usually have to be calculated as well based on the actual physics of the situation. $\endgroup$
    – Phil Sweet
    May 19 '18 at 2:20
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    $\begingroup$ When it inflates, the ends can't slide closer together, the distance between the ends is fixed as well? As in tubing hose-clamped around a two barbs fixed to a base? This is a pretty mean version of the calculus of variations. You first need to verify the modulus and poison's values over the range of stresses in the problem. Hopefully, you can confine the problem to where modulus and poison's ratio are constant. otherwise, you may find yourself looking at a carnival balloon the begins to inflate dramatically at one place, and then the zone of inflation stretches down the length of the balloon. $\endgroup$
    – Phil Sweet
    May 19 '18 at 13:22
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It would be indetrminant structure if we release the axial constraints. In a very crude first estimate one could compare it to a syspended bridge with flexible end supports, or a balloon with built-in geometric constraints. I guess if you don't want to use FEM then, you would want to guess a curve for the expanded shape and solve by energy methods. It would be interesting to start from trig functions for the cross section deformation an see where it goes.

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This is a link to a solved problem similar to your question. inflation of an annular cylinder

In their example the two ends are fixed, no traction of cylinder, meaning the length of cylinder doesn't change.

It is a partial deferential boundary value problem solved as part of their online course. NPTEL is one of India's online university institutes.

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  • $\begingroup$ Ok thanks kamran, this looks very detailed. However they "assume that there is no axial variation in the displacement field". Also, sorry I did not put this in the questions but in this problem, the ID of the cylinder = c.20 * the wall thickness, hence for our purposes we can treat the wall as infinitely thin, and simplify the problem a lot. Do you have a more applicable suggestion considering this? Many thanks $\endgroup$ May 19 '18 at 9:57

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