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(#53) Pressure Vessel Stress Shells and Failure · Mechanical Engineering Made Simple

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  Get ready design junkies to torque your nuts, harden your
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  springs, and forge that mind into a bulletproof study guide.
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  We are mechanical engineering made Simle.
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  Welcome back. Today we are strapping in for a
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  deep dive into the structures that, I mean, literally hold our
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  industrial world together. Shells of revolution.
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  That's right. We're talking pressure vessels,
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  huge pipelines, complex containers, basically anything
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  built around a central axis. Yeah, cylinders, cones, spheres.
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  And The thing is, the real complexity isn't the shape
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  itself, it's about how that shape behaves under massive
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  loads, either from the inside or the outside, right?
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  So our mission today really is to extract the essential
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  knowledge we need to understand the difference between the
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  simple baseline stresses, what we call membrane stresses, and
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  the much more dangerous bending stresses.
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  Especially when you start joining different shapes
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  together. That's where things get
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  critical. We're going to look at how
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  engineers handle those joints to prevent catastrophic failure.
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  Because to most people a tank is just a tank, but to an engineer,
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  if those stress calculations are off by even a tiny fraction,
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  it's a bomb. It absolutely can be.
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  So let's start with the ideal scenario, the engineering dream,
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  which is the thin shell. What's the actual dividing line?
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  When is something officially thin?
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  OK, so the rule of thumb, the definition we use is that a
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  vessel is thin walled if it's wall thickness, let's call it T
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  is less than about 110th of its smaller principal radius of
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  curvature. The 110th rule.
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  So if you meet that criteria, what does that?
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  What does that buy you analytically?
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  It buys you a massive simplification, A huge one if
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  you're thin and the load is smooth and evenly distributed.
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  You know, axisymmetric, right? You can essentially take a
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  complex 3D stress problem and boil it down to a 2D1.
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  You get to completely ignore the stress acting through the
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  thickness of the wall. OK, so that means we're left
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  with just two main forces to worry about, right?
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  Yeah, pure tension running along the wall.
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  Exactly. We focus on just two membrane
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  stresses. The first is the meridional
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  stress S sub one. Think of that one running along
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  the profile, like from the North Pole to the South Pole of a
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  sphere got. It top to bottom.
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  And the second is the circumference differential or
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  hoop stress as sub two. That's the one acting around the
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  circumference. It's the stress that tries to
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  blow a barrel apart sideways and.
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  Just to be clear, the radial stress as sub three and any
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  bending are what basically 0 for?
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  All practical purposes, yes, they're negligible If everything
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  is perfectly symmetrical, every circle just wants to expand or
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  contract uniformly. There's no internal force trying
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  to bend OR warp the wall. Which is why there are standard
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  off the shelf formulas for these simple shapes.
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  That's membrane theory in a nutshell.
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  It's the beautiful, simple starting point.
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  But of course, the real world is never that simple.
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  You don't build things out of infinite isolated spheres.
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  You definitely do not. And that's where the dream ends
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  and the real engineering begins. The moment you introduce any
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  kind of real world imperfection, membrane theory starts to
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  breakdown. And these significant bending in
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  sheer stresses just pop up out of nowhere.
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  They do and there are a few key triggers for these so-called
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  discontinuity stresses. OK.
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  What are the big culprits? What kills the simplicity?
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  Well, first is any kind of concentrated load.
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  So instead of smooth internal pressure, imagine a heavy
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  support bracket welded to the side, a point load. 2nd, and
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  this is a huge one, any abrupt change in the geometry, a sudden
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  change in wall thickness, or welling a flat plate onto the
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  end of a cylinder, any sharp corner.
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  Third is any geometry where the wall starts to become normal to
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  the main axis, like the very edge of a flat topped tank.
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  And the fourth, which we'll get to is just being too thick in
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  the 1st place, but that. Second one, the joint between
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  two different shapes feels like the most common problem you'd
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  run into. It's unavoidable.
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  Take a standard pressure vessel. You have a cylindrical body and
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  it's capped with a spherical or elliptical head.
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  Why is that joint such a headache?
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  Yeah, even if the pressure in the material thickness are
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  identical. Because under that pressure, the
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  two parts want to grow by different amounts.
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  Think about it. The simple formulas tell us that
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  a cylinder expands radially based on its hoop stress.
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  A sphere expands uniformly in all directions based on a
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  totally different equation. So the cylinder wants to flare
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  out, say, 100th of an inch, right?
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  And the spherical cap it's welded to only wants to expand
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  by, I don't know, 8 thousandths of an inch.
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  Exactly. They fundamentally disagree on
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  how big the opening should be, and that mismatch, that tiny
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  difference in radial deformation, is what creates the
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  problem. So the material itself has to
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  bend to keep the joint from ripping apart.
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  That's it, perfectly. To force those two edges to stay
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  continuous, the structure generates its own corrective
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  forces, localized bending and shear right at the joint.
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  We call it deformation matching. Which means that the simple
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  membrane stress numbers are useless in that area.
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  You have to account for the bending.
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  I remember seeing a wild example, a toroidal section.
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  That's like a doughnut shape, right?
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  The knuckle region where? A pretty low pressure, just £200
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  per square inch, actually caused a negative change in radius, the
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  part pulled inward by two thousandths of an inch at a
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  certain point. And that tiny, tiny movement is
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  everything. It's the physical manifestation
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  of that massive bending stress. This leads to a really a really
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  powerful analogy that engineers use.
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  OK, the math for analyzing these cylindrical shells is basically
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  identical to solving problems for beams on elastic
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  foundations. Whoa, OK, that sounds like a
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  completely different universe. Yeah.
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  How is a pipe under pressure the same as a beam sitting on a
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  bunch of springs? Well, think about the beam.
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  It's supported along its whole length by some something that
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  resists being pushed down right the springs right in the shell.
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  The beam is the wall itself running lengthwise, and the
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  elastic foundation. The springs is the continuous
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  circumferential stiffness of the rest of the shell.
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  The hoop strength is what's providing the support.
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  To the rest of the pipe is acting like the springs,
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  resisting any local deformation. Precisely, and this is a huge
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  deal because it means engineers can just borrow all the well
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  established solutions from beam theory.
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  You just swap out the beam parameters like beta and EI.
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  The moment of inertia. Yeah, you swap those for the
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  shells parameters, specifically a value called Lambda, the
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  characteristic length and D the flexural rigidity of the shell
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  wall. And what is that flexural
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  rigidity D represent? It's the shells inherent
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  resistance to bending, it's a function of the materials
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  elasticity and crucially the cube of the thickness.
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  It's an incredibly powerful shortcut.
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  That's a clever way to manage the complexity.
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  So let's see this in action. If we're welding, say, a
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  cylinder to a cone, how do you actually calculate those
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  corrective forces? The method is called
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  superosition. You basically solve 2 simler
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  problems an then add the results together.
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  OK, Roblem 1 is the ideal case. You calculate the deformation of
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  each part separately using just the simple membrane formulas for
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  the internal pressure. So you pretend they're not
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  connected and see how they want to flare out on their own.
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  Exactly. Then problem 2 is the corrective
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  part. You add an unknown shear force V
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  naughty and an unknown moment M naughty right at the edge of
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  each piece, and you solve for the V&M that would be required
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  to bend them back so their edges line up perfectly.
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  They have to match in both position and angle.
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  Position and slope. Yes, they have to meet
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  seamlessly, and when you do this, the results can be just
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  staggering. I think I know the case that
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  you're talking about, the vessel with the 24 inch radius cylinder
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  and the 45° conical ends. That's the one under just 300
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  PSI of internal pressure. OK so first what was the simple
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  membrane stress S sub one in the cylinder if you just ignored the
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  cone attached to it? The baseline stress, but only
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  5690 lbs per square inch, a very manageable number.
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  The number you might use if you didn't know any better.
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  The number that would get you fired or worse.
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  Oh, what happened when you added in the bending effects from the
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  junction? The combined stress on the
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  inside surface of the cylinder right near the Weld skyrocketed.
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  It jumped to a tensile stress of 42,260 PSI.
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  Wait. 42,000 compared to 5600, that's what almost 8 times
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  higher? It's a massive stress
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  concentration. The bending stress caused by
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  that joint was monumentally larger than the simple membrane
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  stress. The design would have been
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  dangerously inadequate. And on the outside surface, it
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  was actually in compression, right?
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  Right, A compressive stress of over 30,000 PSI.
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  That huge swing from high tension on the inside to high
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  compression on the outside. That's the signature of a
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  powerful bending moment at work. That is genuinely terrifying.
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  For even more accuracy, there are things called correction
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  terms. Yes, especially for thicker high
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  pressure vessels. The simple models assume the
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  pressure acts on the theoretical center line of the wall.
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  But it doesn't. It acts on the inner surface.
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  Exactly, and when you join two parts with maybe a slightly
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  different thickness, you create a tiny internal step at the
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  Weld. And the pressure pushes on that
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  little step. It does, and that creates tiny
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  radial and axial forces that the basic theory misses.
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  These correction terms account for the real physics of the
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  pressure acting on that internal geometry.
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  They can be critical for safety in extreme applications.
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  The level of detail is just amazing.
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  OK, so let's shift gears. Let's throw out the thin wall
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  assumption entirely. What happens when we get into
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  thick walled vessels? Now you're in a different
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  domain. The biggest change is that the
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  radial stress S sub 3 is no longer zero.
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  You absolutely cannot ignore it. It varies from the full internal
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  pressure on the inside surface down to 0 on the outside.
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  Correct. And the other stresses S sub one
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  and S sub 2 are no longer uniform across the wall either.
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  They vary and this leads to really strange counterintuitive
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  limit for a very thick cylinder. Under extreme pressure, you
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  reach a point of diminishing returns.
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  If your internal pressure Q is already equal to the maximum
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  allowable stress of your material S, you cannot make the
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  vessel safer by adding more thickness.
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  No matter how much thicker you make it, the stress on the inner
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  wall will never drop below S. So you can't just throw more
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  steel at the problem, you need a smarter approach.
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  You need to get creative. You need to induce residual
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  stresses. The classic methods are shrink
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  fitting or a process called auto footage.
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  Threadage. That sounds aggressive.
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  It is controlled damage. You intentionally over
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  pressurize the vessel so much that the inner layers
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  permanently stretch. They enter the plastic range.
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  Then when you release the pressure, the outer layers,
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  which were still elastic, spring back and squeeze the now
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  oversized inner layers. This leaves the inner surface in
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  a state of high residual compression.
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  So when you put it into service, the internal pressure first has
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  to fight through that built in compression before it can even
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  start to put the material in detention.
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  You've got it. It effectively cancels out the
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  highest stress. The shrink fit method we saw
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  required calculating A precise interference fit, something like
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  3 thousandths of an inch. That tiny difference controls
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  the entire vessels capacity. That's all for internal
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  pressure, but you mentioned a completely different failure
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  mode that comes from the outside.
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  And this is so important. External pressure is a whole
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  different beast. It's compressive and compression
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  can lead to buckling or instability failure.
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  The structure just collapses. It collapses catastrophically,
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  often at a stress level far far below the materials actual
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  strength limit. It's not a strength failure,
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  it's a geometric 1, and your standard stress formulas will
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  not predict it. And the example for this was a
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  corrugated tube, something that seems pretty harmless.
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  Exactly. For a tube with 60 corrugations,
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  the calculation showed it could become unstable and squirm.
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  That's the technical term, with an internal pressure of just
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  96.3 PSI. It's nothing.
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  It's. Very low pressure.
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  The geometry, not the material strength dictated when it would
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  fail. It's a huge lesson.
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  It really is. I mean, we've gone from the
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  clean, simple world of membrane stresses, S sub 1S sub two, to
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  the absolute necessity of localized bending analysis at
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  every single joint. And then realizing that just
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  making things thicker has its limits, which forces us into
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  clever pre stressing technique. And finally, acknowledging this
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  lurk making danger of geometric instability, which is a whole
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  separate field of analysis. The amount of work required for
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  a truly safe design is immense. It really shows that mechanical
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  design is all about the deep relationship between stress
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  geometry and deformation. You can't separate them.
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  Which leads to a final thought for our listeners.
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  We know material science gives us a hard limit, a maximum
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  stress any material can take. So given that, what role does
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  intentionally creating controlled damage, inducing
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  residual stress with something like auto frettage play and
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  pushing the boundaries of what's possible?
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  How much more performance in life can we squeeze out of the
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  world's most critical high pressure vessels by essentially
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  pre stressing them at the factory?
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  And maybe more importantly, how much risk are we unknowingly
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  accepting every time we rely on a simple calculation that
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  ignores the complex reality of a joint?
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  That's the deep dive you have to continue on your own.