This content explains the fundamental principles of pressure vessel design, focusing on how engineers use mechanics to ensure safety by calculating and managing stresses within cylindrical and spherical vessels.
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spray cans, rocket propellant tanks,
subc pipelines,
industrial reactors, and this propane
tank. They all have one thing in common.
Their main purpose is to contain a fluid
that's at a higher pressure than the
environment. These are all pressure vessels.
vessels.
Pressurized fluids can exert enormous
forces on the walls of the vessels that
contain them. In this video, we'll
explore how engineers apply simple
mechanics principles to ensure pressure
vessels can safely contain their
contents without risk of catastrophic failure.
failure.
Most pressure vessels are either
spherical or cylindrical.
One of these shapes handles pressure
much better than the other, even though
it's less commonly used. We'll get to
why that is a bit later on. Let's start
by looking at the cylindrical pressure
vessel. It contains a fluid at high
internal pressure with the outside of
the vessel seeing a lower atmospheric
pressure from the surrounding environment.
environment.
What we're interested in is the
difference between these two pressures.
The net effect of this gauge pressure is
to push outwards on the inside surface
of the cylinder causing stresses to
develop in two main directions within
the vessel wall. in the hoop direction
around the circumference of the cylinder
from the pressure trying to radially
expand the circular cross-section
and in the longitudinal direction from
the pressure pushing against the ends of
the vessel and trying to stretch it
along its length.
Being able to quantify these stresses is
the key to safely designing pressure
vessels. Let's focus on deriving an
expression for the hoop stress first. We
can do this by considering an arbitrary
section of the vessel of length L and
cutting it in half to examine the hoop
Remember that in mechanics stresses
develop in such a way as to maintain equilibrium.
equilibrium.
The pressure load that acts on the
curved inner surface of the cylinder is
balanced by the hoop stress in the
cylinder walls.
The resultant of the internal pressure
is a force with a magnitude equal to the
pressure multiplied by the projected
area over which the pressure acts.
This area is the cylinder diameter D
multiplied by the length L.
The resultant of the hoop stress is a
force that has a magnitude equal to the
hoop stress multiplied by the area over
which it acts. This area is 2 * the wall
The two force terms must be equal to
We can then solve to obtain the famous
A pressure vessel with a diameter of 400
mm, for example, that has a wall
thickness of 10 mm at an internal
pressure of 50 bar, which is 5
megapascals, will develop a 100
megapascal hoop stress in its walls.
Reducing the pressure, reducing the
vessel diameter, or increasing the wall
thickness all result in a lower hoop stress.
[Music]
We can derive an equation for the
longitudinal stress in a similar way by
taking a circumferential slice of the
vessel this time. [Music]
The important insight here is that the
hoop stress is twice as large as a
longitudinal stress.
This means cylindrical pressure vessels
are theoretically more likely to fail by
splitting along their length rather than
around their circumference because the
material reaches its yield point in the
hoop direction long before it does in
The next time you're holding a garden
hose, another type of pressure vessel,
take a close look at the reinforcing
fibers within it, chances are they'll be
oriented at exactly 54.7°
from the axis of the hose. This
carefully selected orientation, called
the magic angle, represents the
theoretically optimal fiber direction
for a scenario where the hoop stress is
exactly twice the magnitude of the
longitudinal stress.
With this angle, the fibers are
perfectly aligned to carry the hoop and
longitudinal loads in the right proportions.
With the hoop stress being the critical
stress, it's the PD over 2T equation
that's used to calculate the minimum
wall thickness required to safely
contain an internal pressure P.
All we need to do is define an allowable
hoop stress, which is often taken to be
in welded vessels. An additional factor,
the joint efficiency factor E is often
applied to account for the potentially
reduced strength at longitudinal welds.
It usually has a value between 7 and 1.
A corrosion allowance is also often
added to the calculated minimum
thickness to account for material loss
over the service life of the vessel.
This is a fundamental equation used in
The equation can also be rearranged to
calculate the maximum allowable pressure
So what about spherical pressure vessels?
vessels?
The curvature of a spherical vessel is
the same in every direction. This means
that at any point in the vessel wall,
the inplane stress is the same in all
directions and there's no distinction
between hoop and longitudinal stress
like there is in cylindrical pressure vessels.
vessels.
This stress is equal to PD over 4T.
This is the huge advantage of spherical
pressure vessels. Their shape
distributes stress evenly in all
directions, resulting in a maximum
stress that's half the maximum stress in
a cylindrical pressure vessel. In
theory, a spherical vessel could be half
the thickness of a cylindrical one for
the same internal pressure.
Despite this, cylindrical vessels are
still far more common in industry. The
reason is simply because they're a lot
less expensive to make. Rolling plate
into cylinders and welding on domed end
caps is much easier than forming a sphere.
Now is probably a good time to mention
that we made two big simplifications
when deriving the PD over 2T and PD over
4T stress equations. The first is that
we treated the hoop stress as a constant
value through the thickness of the
vessel wall.
In reality, it peaks at the inner
surface and gets smaller as you move to
The vessel wall shown here is thin
relative to its diameter. So, the
reduction in hoop stress is small and
the assumption that hoop stress is
constant through the wall thickness is reasonable.
For thick walls though, the variation in
The second simplification is that we
accounted for stresses in the hoop in
longitudinal directions, but we
completely neglected any stresses acting
In reality, on the inner surface of the
pipe wall, there will be a compressive
radial stress equal in magnitude to the
internal pressure to balance the normal
On the outer surface, the radial stress
has a magnitude equal to the external
pressure, which is pretty much zero for
To better understand the significance of
the radial stress, take a look at the
ratio of the hoop stress to the radial
If the vessel is thinwalled, meaning it
has a large d overt ratio, it's clear
the hoop stress will become much larger
than the radial stress. In this case,
it's acceptable to neglect radial
stresses. But if the vessel is
thickwalled, the radial stress becomes
significant relative to the hoop stress
It's common to define thinwalled
pressure vessels as those with a wall
thickness that's less than 5% of the
diameter of the vessel. D over T is
greater than 20 and a thickwalled
pressure vessel for anything thicker
than that.
The PD over 2T and PD over 4T equations
only apply to thinwalled pressure
vessels. For thickwalled vessels, the
hoop, longitudinal, and radial stresses
are given by the slightly more
complicated lame equation for
thickwalled cylinders. The hoop and
radial stresses are functions of the
position within the vessel wall.
To help you keep track of all these
different pressure vessel equations and
when to apply them, I've created a
one-page summary sheet that covers all
of the important information on a single page.
page.
It's part of a growing collection of
engineering summary sheets I've been
developing covering sheer force and
bending moment diagrams, trusses,
torsion, the finite element method,
buckling, and more. There are 14 sheets
in total, and I'm adding more over time.
Each one summarizes key equations and
concepts in a quick review format,
making them ideal as study aids or
reference guides. Great for exams,
interview prep, or for when you just
need to brush up on a particular topic.
All 14 summary sheets are available for
free for anyone who supports the
Efficient Engineer channel on Patreon.
And that includes any new sheets I
release in the future. As a supporter,
you'll also get access to video previews
and behindthe-scenes details showing how
the efficient engineer videos are made.
So head over to efficienineer.com/support
efficienineer.com/support
to join me on Patreon. Get access to the
full set of engineering summary sheets
and help me continue to create videos
and develop resources for the
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If you'd rather just purchase the sheets
directly, you can do that too for $15.
And that's it for this look at pressure
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