One can build an unpressurized liquid tank to any desired shape, unless it is so large that the depth of liquid inside exerts a significant pressure all by itself. The same is not true of pressurized vessels, even small ones, beyond a “single handful of psi” gage pressure inside it. There are very good reasons these tanks are only made in spherical shapes, or as right circular cylinders with hemispherical or spherical-segment ends.
This document presumes the reader knows what “gage pressure”
means, and what mechanical stresses and
strains are.
Figure 1 shows a cylindrical pressure vessel holding
gage pressure P inside. This could be a
tank, or even a pipe or a tube. The “hoop stress” is given by the formula
shown top right, which is sometimes
referred to as Barlow’s pipe stress formula,
which is a measure of the stress trying to split the cylinder open longitudinally. Technically,
you use the cylinder inside diameter for this formula (a requirement
working with pipe and tubing), but as
long as the material thickness is small compared to the diameter, something true of rocket cases and propellant
tanks, there is little-or-no perceptible
difference between inside, outside, and average diameters.
Bottom left, the cross-section
of the cylinder, or any section through
a sphere, is shown. For a cylinder, the “axial stress” is that which resists the
pressure trying to part one end of the vessel from the other. For a sphere,
it’s just the “membrane stress” that resists splitting the sphere apart, no matter the section orientation. That stress works out to be just half the
hoop stress, as shown.
Figure 1 – Pressure Vessel Stresses In the Vessel Material
For a cylindrical pressure vessel fitted with hemispherical
ends, Figure 2 takes this notion
a bit further, showing as it does the
stress distributions on a tiny patch of material, located on either side of the joint. The axial stresses match, while the hoop direction stresses differ by a
factor of 2. There are (at least
theoretically) no stresses perpendicular to the material itself (the radial
direction from the cylinder axis, or
from the hemisphere’s center).
This hoop stress mismatch at the joint corresponds to a
strain mismatch in the circumferential direction, in turn corresponding to a radial displacement
mismatch between the cylinder material,
and the end (or “head”) material.
The cylinder swells radially under pressure twice as much as the end or
“head” swells radially, as measured at
the joint. This distorts both the
cylinder and the head locally at the joint,
as the materials bend locally, in
order to try to stay joined. This
induces large bending stresses locally,
which add in certain ways to the hoop and axial (or membrane) stresses
already described. That makes the joint
quite vulnerable to local overstress failure.
The ”fix” for this is to locally thicken the cylinder and
head materials at the joint. In
effect, the extra material “sops up” the
extra imposed stresses. For
boilers, there are very specific
guidelines for how much local thickening is needed, and how far “local” extends away from the
joint. Those rules are the ASME boiler
code, which is legally mandatory
everywhere in the country, for designing
and building boilers. Every provision
represents a life lost learning that lesson.
This is serious business!
Figure 2 – There Are Distortions With Extra Stresses At the
Joints
There are choices allowed for how to implement those
localized thickenings at the joints.
Those are depicted in Figure 3,
and also apply to solid rocket motor case designs. You can increase the thickness toward the
inside, or toward the outside, or even some of both, just as long as enough extra material is supplied. Finite-element stress-strain analysis can
refine this further.
The same figure also shows a variation on the spheroidal
end, where only a sphere segment is used
as the end membrane. This requires a
connection ring that resists radial swelling at about the same rate as the end
resists radial swelling at its attachment joint. That way,
no thickening of the membrane is required, the ring supplies that for the membrane. You still need a local thickening of the
cylinder material at its attachment to the ring, because of the radial swelling mismatch.
This spherical segment and ring approach lets one enclose
more volume within a given length,
without making the assembly any heavier than a full hemispherical
end. This is how most solid rocket motor
case closures are designed. It is a
well-proven solution. The flatter the
membrane, the heavier the ring
gets, though. It’s a trade-off.
Figure 3 – Full Hemispherical Ends Vs. Spherical Segment
Ends With Rings
There are many possible reasons for wanting to use other
shapes for one or both cylindrical pressure vessel end heads. As long as membrane stress is insignificant
(meaning very low pressure indeed!), you
can do that! But as soon as the pressure
(and the stresses it induces) become significant, those other shapes rapidly become
infeasible. This is shown in Figure 4, where the two spheroidal options are
depicted, along with elliptical and
conical ends. A flat end fails even
worse than the elliptical, from similar
stresses that are just higher, plus a
sharp corner effect that locally greatly magnifies the local stresses even further,
right at the corner.
There is one positive benefit to a conical end, if an axially-directed load must be carried
from the cone tip into the cylinder.
This is an efficient load path for such a load. But it is still a lousy pressure vessel
choice! It will require a lot of
internal stiffeners to keep it from trying to “go round”. Those are going to add significant
weight, there is no way around that
problem! You must trade off the axial
load path advantage against the big weight gain incurred to make a conical
shape a pressure vessel.
Figure 4 – Which End Shapes Work and Which Do Not, and Why
It is a common belief that an elliptical shape is as good as
the ring and spherical segment. This
is not true from a pressure vessel design standpoint, as the figure shows. The elliptical head will try to “go
round”, inducing severe bending
stresses. It is volumetrically
efficient, which is why many
unpressurized railroad tank cars use elliptical heads on cylindrical
bodies. But these tank cars are not
pressurized! Their evident abundance
is deceptive regarding the pressure vessel issues.
That same effect is why you want to use circular cylinders
as your basic tank body for a pressure vessel,
not some elliptical (or other) cross section shape. Those other shapes will try to “go round”
upon pressurization, leading to enormous
bending stresses and very rapid failure.
If you must put pressurized storage within some oddly-shaped volume, you must fill it with multiple small circular
cylinders! The non-circular cross
section is not, and will never be, a successful pressure vessel design! This is why air mattress floats are made the
way that they are, for example: multiple cylinders, connected at the ends so as to fill together
all at once.
Addendum: Exact Analysis
The exact analysis for right circular cylinders and spheres
is given in Figure 5, along with
the geometries that allow these formulations to be made from very simple
measurements.
Figure 5 – Exact Formulations
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