Re: Thick walled spherical pressure vessel

On 21 Oct 2006 08:09:54 -0700, "Edward Green"
<spamspamspam3@xxxxxxxxxxx> wrote:

John C. Polasek wrote:

It's as Mati says. If you apply stress to a rod it stretches
longitudinally by some fraction sigma = dL/L, and the increase in
volume is balanced by "negative side sigmas" dx/x and dy/y that are
each half of dL/L that account for the gain in volume.
For 1% elongation, there would be -1/2% on each of the two sides.
For the sphere, there is nothing to talk about. There are neither ends
nor sides.

Ok. We have three opinions, and one set of equations (more powerful)
that say "Poisson's ratio doesn't count here", but I don't quite buy
it.

Consider a spherical shell of material. Let the shell be compressed by
radial stresses. Focus attention on a cube of material, intially
considered cut from the shell. This cube is compressed along the radial
direction, hence expands along the tangential directions. Now,
consider that the cube of material is not really cut from the shell.
It cannot really expand tangentially. However, its thwarted attempt to
expand tangentially will create compressive stresses (reduce tensile
stresses) in the tangential direction. The extent of this coupling of
radial and tangential stresses will depend on Poisson's ratio.

Maybe by magic this consideration drops out, but it doesn't seem
obvious.

A spherical coordinate system is better. A section of the shell is
given by
V = AT = R^2WT
where W is the steradian measure of the cone circumscribing A. Under
hydrostatic stress there will be minute changes in A and R and T, but
never W, and this geometry applies everywhere. If Poisson's ratio is
..5 then volume is preserved (I'm pretty sure). For steel I looked up
and found it's .29. So dR and dT will be shy of keeping the volume the
same, if that is important, in which case PR is a factor.

Another complication is that linear stress is constant throughout, but
hydrostatic pressure has a constant negative pressure gradient in the
wall from P1 to 0, doesn't it?
I'm pretty sure all you need to know about this problem is in Marks'
Engineering Handbook.
John Polasek
.

Relevant Pages

  • Re: Thick walled spherical pressure vessel
    ... each half of dL/L that account for the gain in volume. ... Consider a spherical shell of material. ... hence expands along the tangential directions. ... expand tangentially will create compressive stresses (reduce tensile ...
    (sci.physics)
  • Re: Thick walled spherical pressure vessel
    ... each half of dL/L that account for the gain in volume. ... Consider a spherical shell of material. ... hence expands along the tangential directions. ... expand tangentially will create compressive stresses (reduce tensile ...
    (sci.physics)
  • Re: Thick walled spherical pressure vessel
    ... each half of dL/L that account for the gain in volume. ... Consider a spherical shell of material. ... Since all three principal stresses are equal at all points in this ... hence expands along the tangential directions. ...
    (sci.physics)
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