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(#128) Why Textbook Math Fails Structural Designs · Mechanical Engineering Made Simple

0:00
Welcome back design junkies to Mechanical Engineering Made
0:02
Simple. If the podcast brings you value,
0:05
help support the show with a like a follow or leave a comment
0:07
below. So I want you to imagine firing
0:11
a high caliber bullet directly into a massive hanging block of
0:15
wood. Right, the classic example.
0:16
Exactly. And if you remember your high
0:18
school physics class, you probably know that this is
0:21
supposed to be, you know, a perfectly simple, elegant math.
0:24
Problem on paper, Sure. Right on paper.
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But what if I told you that in the real world, relying on that
0:30
exact type of textbook math is actually the number one reason
0:35
multimillion dollar structural designs completely,
0:39
catastrophically fail? Yeah, it's a huge.
0:41
Issue it really is. Today we are going on a deep
0:44
dive into something that frankly keeps mechanical and structural
0:47
engineers awake at night. We're talking about the chaotic,
0:50
unforgiving world of impact dynamics.
0:52
It is definitely not as neat as the textbooks make it seem.
0:55
No, it's messy. So our mission for you today is
0:59
very specific. We are going to break down the
1:01
biggest pain points and the actual real world physical
1:05
issues you run into when designing structures to
1:09
withstand violent collisions. Because you will run into them.
1:12
Oh, inevitably we're going to call it the common, almost
1:15
instinctual misconceptions that just keep screwing up structural
1:18
designs. And then we're going to walk
1:21
through a step by step approach so you don't fall into those
1:24
same traps. Yeah, getting to the practical
1:26
side of things. Exactly.
1:28
We want to show you how to actually optimize your design,
1:31
not just on a pristine chalkboard, but out there in the
1:34
unpredictable real world. It's an incredibly vital topic
1:38
and well, I'm really glad we were tackling it today.
1:40
To make sure we are all starting from the exact same baseline,
1:43
let's firmly define what we mean when we say impact.
1:46
Good call. In its simplest mechanical
1:48
definition, an impact occurs when two or more bodies collide,
1:52
generating relatively large impulse type forces at points of
1:56
contact for extremely short periods of time.
1:58
And I really want to emphasize that last part, the short
2:00
periods of time, that element of time is the crucial
2:04
differentiator here. And impact is not a gradual
2:08
static load that a structure can slowly distribute.
2:12
It's a sudden hyper violent transfer of energy that
2:15
basically forces the material to react before it even has time to
2:19
fully register what's happening. That's a really great
2:22
distinction. It's I guess it's the difference
2:25
between gently setting a 50 LB weight on a glass table versus
2:28
dropping that exact same 50 LB weight onto it from a second
2:31
story window. Exactly same mass, entirely
2:34
different physics. Right, so let's start with the
2:36
most common trap I see engineers fall into.
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I call it the billiard ball misconception.
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I like that name. Because you know, when you first
2:43
learn physics, essentially every impact is treated like a
2:48
perfectly smooth, perfectly solid billiard ball hitting
2:51
another billiard ball on a completely frictionless pool
2:54
table. Right.
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And it's easy to see why we start there.
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I mean that elementary analysis actually traces all the way back
2:59
to an experimental observation made by Sir Isaac Newton.
3:02
Oh wow, really? Yeah, Newton observed that when
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2 bodies collide directly, their relative velocity after the
3:09
impact is in a constant ratio to their relative velocity before
3:12
the impact, just acting in the opposite direction.
3:15
OK, that makes sense. We call this constant ratio the
3:17
coefficient of restitution, which is usually designated in
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the equations by the lowercase letter E.
3:23
Right, the little E. Exactly.
3:25
And when you combine that restitution coefficient with the
3:28
law of conservation of linear momentum, you get the
3:31
foundational classical math for simple rigid impact.
3:35
And just to put that math out there and spoke in terms so we
3:37
can sort of visualize the formula, it looks like this.
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It's U prime minus X prime equals negative E multiplied by
3:44
the difference between U&X. And then for the momentum side
3:48
of the equation, it's simply balancing the scales, right?
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So MU times U prime plus MX times X prime equals MEU times U
3:55
plus MX times X. Exactly.
3:57
You are just cleanly balancing the masses and the velocities
4:00
before the collision with the masses and velocities after the
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collision. It's clean, it's elegant, and it
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assumes that the bodies involved are completely, perfectly rigid.
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Which is incredibly tempting to rely on.
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I mean, it's that classic ballistic pendulum model we
4:16
mentioned at the very beginning with the bullet and the wood
4:18
block, right? If you know the mass of the
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bullet and you know the mass of the wood block hanging from the
4:24
string, you really only have to care about the initial velocity,
4:27
the incoming bullet, and the final swinging velocity of the
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combined embedded mass. You basically just skip over the
4:34
middle part, yeah. You get to completely ignore the
4:37
incredibly complicated, chaotic, messy forces happening during
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the fraction of a second when the bullet is actually tearing
4:43
into the wood fibers and stopping.
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Which is very convenient for a math test.
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But here is where I have to push back on the textbook.
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When does this highly convenient, perfectly elegant
4:55
math completely fall apart for you as a real world engineer?
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Because, frankly, nothing in the real world is perfectly rigid.
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Right, it falls apart the exact moment the collision generates
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vibration. What's fascinating here is how
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the Hertz theory of impact explains this exact physical
5:10
boundary. The Hertz theory gives us a
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mathematical way to look at 2 solid elastic spheres colliding.
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It tells us about the localized pressure distribution and the
5:20
exact duration of the contact between them.
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OK, but the theory has a very strict, highly specific
5:26
limitation. It is only mathematically valid
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if the duration of contact is long in comparison with the
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period of the fundamental mode of vibration of either sphere.
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Wait, wait, let me make sure I'm visualizing this fundamental
5:38
mode correctly. You're saying that basically
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every object naturally wants to vibrate at a certain frequency,
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and if the duration of the crash is actually faster than the
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objects natural ability to vibrate, all hell breaks loose.
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Precisely if they hit and stay in contact long enough.
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Meaning it's a relatively slow pushing kind of impact.
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Right exactly. Then the simple rigid body math
6:00
works fine. The energy moves smoothly, but
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if the impact is sharp and fast enough to excite internal
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vibrations in the structure. Which is basically what happens
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constantly in real world engineering.
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Oh, constantly. In that case, we have to throw
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out the rigid math and look at the vibration energy ratio.
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Because if vibration is generated, the kinetic energy of
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the colliding bodies doesn't just stay as simple
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translational energy, cleanly moving the object from point A
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to point B. So where does it go?
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A massively significant portion of that energy gets instantly
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converted into internal vibrational energy.
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The object effectively rings like a struck bell.
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Oh wow. And once your structural beam or
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your chassis is ringing, that simple rigid body math is
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essentially useless. So if real things ring, we
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obviously aren't just dealing with perfectly rigid spheres
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floating in a vacuum. We're dealing with flexible
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steel plates, massive eye beams, actual complex structural
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assemblies. Yeah, the real stuff.
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And this is exactly where I see designs consistently fail.
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It's what our sources call the single degree of freedom trap.
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Yes, the single degree of freedom trap.
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When we move from rigid bodies to something like an elastic
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plate, the physics changes dramatically and, well,
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violently. Imagine a solid steel sphere
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striking an infinitely extended elastic plate.
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The impact doesn't just uniformly push the plate
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downward, it creates high speed flexural waves, and the math
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behind this shows that the half period of these flexural waves
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is exactly equal to the duration of the impact itself.
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The total kinetic energy from that striking sphere is
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aggressively split up. Split up into what?
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It becomes both the kinetic and the potential energy of the wave
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motion physically traveling outward through the plate.
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OK, so the usual analogy here is dropping a stone into a pond,
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seeing the ripples move out. But to really capture the
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violence of this, I think it's more like dropping a massive
7:55
boulder onto a frozen pond. That's a good way to look at it.
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Because the ripples aren't just softly moving water, you know.
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They are shock waves violently bending, twisting, and snapping
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the ice as they travel outward at supersonic speeds.
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That's what's actually happening to your structural steel.
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It's this chaotic wave of strain echoing through the material
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matrix. That is a much better way to
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visualize it, and it leads US directly to the biggest, most
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critical pain point highlighted in our deep dive today.
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Let's hear it. There is a fundamental rule that
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engineers ignore at their absolute peril.
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A structure can seldom be treated as a single degree of
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freedom system with any degree of reliability.
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This is huge. It really is.
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Assuming a steel beam just bends in one simple uniform arc, like
8:39
a diving board, bending down under a diver's weight and
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smoothly popping back up is a fatal misconception.
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It's just not how metal behaves under shock.
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Right. Our sources show that even
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relatively small masses moving at relatively low velocities can
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develop suprisingly massive localized bending strains in
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beams, and the reason is precisely because they excite so
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many different modes of vibration simultaneously.
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The beam isn't just bending nicely in the middle, it's
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rippling, it's twisting portionally, and it's vibrating
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in multiple complex harmonic ways, all at the exact same
9:11
time. Right.
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And when you realize that the structure is rippling in all
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these chaotic high frequency modes, you can see why engineers
9:19
immediately look for a quick, intuitive fix.
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Oh, I know this one, yeah. And the most common quick fix is
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always external damping. They look at the design and
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think, well if the structure is vibrating and winging, I'll just
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add external dampeners to absorb the shock.
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Right, I've seen it 100 times. Can't we just slap some massive
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rubber shock absorbers on on the engine mounts or the frame
9:41
connections and call it a day? The rubber will just soak it up.
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You would think so. I mean, it feels intuitively
9:47
correct, yeah. But the facts counter this with
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two incredibly crucial points that prove why that instinct is
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just flat out. Wrong.
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OK, Leia. Mommy.
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First, external damping does almost absolutely nothing to
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reduce deflection and strain in the higher, more complex
10:02
vibration modes. Really.
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Pretty much. A big rubber mount might slow
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down the primary visible bounce of that diving board we talked
10:10
about, but it does virtually 0 to stop the high frequency
10:14
internal ripples that are actually tearing the crystalline
10:17
structure of the metal apart. Hold on, that completely defies
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common sense. I know it does.
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If I put a massive, highly engineered rubber shock absorber
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on a steel mount, you're telling me it does literally nothing to
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stop the metal from tearing? How is that even possible?
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Because of the speed of the wave.
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Which brings us to the second, even more critical point for the
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lifespan of your design. Impacts produce incredibly
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sharp, virtually instantaneous initial rises in strain.
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These spikes happen so fast that they are almost completely
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uninfluenced by external damping.
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By the time the external rubber dampener even physically
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compresses enough to register that an impact has occurred,
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that massive initial spike of flexural strain has already
11:00
blasted completely through your metal structure.
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So the wave outruns the dampener's ability to react.
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So by the time the rubber squishes, the microscopic damage
11:07
inside the steel is already done.
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Yes, the rubber catches the slow aftermath, not the violent
11:13
initial shock. And if this structure is part of
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an industrial machine or a vehicle chassis that gets hit
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repeatedly over its operational life, you have just created a
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massive, highly dangerous and often completely overlooked
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fatigue problem. That's terrifying.
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It is the material is experiencing these violent micro
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strains over and over and over again and you're expensive
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external damping system is completely blind to it.
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So what's the fix? The only thing that could
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actually help make those higher destructive vibration modes
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safely a periodic is internal viscous damping.
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Meaning the inherent property of the material itself to dissipate
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that high frequency energy internal, right?
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So it's turning that microscopic friction into heat rather than
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relying on a bolt on accessory. Precisely, you have to rely on
12:00
the materials internal architecture.
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OK, so if external damping is basically too slow to catch the
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shock waves, elastic waves are incredibly chaotic.
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What happens when we just hit the structure harder?
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Things get worse. Right.
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What happens when the impact is so violent, so severe, that we
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blow right past the elastic limit of the material entirely?
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We are no longer talking about bending waves that eventually
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snap back into place. We are transitioning into
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permanent plastic yielding and this brings us to what I call
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the critical velocity trap. This is where the physics of
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materials gets truly unforgiving.
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To understand it, we need to look at longitudinal impacts on
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long bars. When a metal bar is struck and
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remains in an elastic state, the stress waves move through it in
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a very predictable mathematical way.
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The internal stress and the resulting strain are directly
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proportional, and they are related by a constant known as
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Young's modulus of elasticity. Right.
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And the math for that, read naturally, is just Sigma equals
12:58
epsilon multiplied by E, where Sigma represents the stress,
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epsilon represents the strain, and E is Young's modulus.
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It's a perfectly linear, highly predictable relationship.
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You put stress in, you get proportional strain out.
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Yes, it's a clean straight diagonal line on a graph.
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But once the impact force is so high that it forces the material
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to permanently yield, once you crossover that threshold into
13:23
plastic deformation, that straight line completely
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disappears. Young's modulus goes entirely
13:30
out the window and no longer applies.
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And the massive pain point here for structural designers is that
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the wave velocities in the plastic phase plummet
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drastically. Plastic strain waves move much,
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much slower through the material than the elastic stress waves
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do. Wait wait, I'm stuck on this
13:46
idea. Why does a slow plastic wave
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matter so much if the metal is yielding?
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Isn't it just, you know, denting?
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It matters because of the critical velocity of impact.
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Think about the physical variables here.
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OK, the plastic wave velocities approach to 0 as the strain is
14:00
indefinitely increased. But your impact velocity, the
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speed of the object hitting your structure, is an independent
14:06
variable. It can just keep getting faster
14:07
and faster I. Don't see where this is going.
14:09
Right. Because of this mismatch, you
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will eventually reach a mathematical point where the
14:14
incoming impact velocity significantly exceeds the
14:17
velocity of the plastic strain wave trying to carry that energy
14:21
away from the impact site. So it just backs up.
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When that happens, the kinetic energy literally cannot
14:28
physically move away from the point of contact fast enough.
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The material can't distribute the load.
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So what's the physical result of that trapped energy?
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The continuity of the material is violently, instantly broken.
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It doesn't just dent, it doesn't just bend, It ruptures
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completely right at the point of impact.
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OK, I think I have a good way to visualize this.
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Let's hear it. It's like the old parlor trick
14:50
where you try to pull a table clock out from under a fully set
14:52
dining table. Oh yeah.
14:54
If you pull the cloth steadily, the energy moves through the
14:57
fabric and you slowly drag the dishes off the table.
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But if you yank that cloth with an immense sudden violent
15:04
velocity, a velocity that exceeds the critical velocity
15:07
the fabrics weave can transmit, you don't move the cloth at all.
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You just RIP a hole straight through the fabric where your
15:13
hands are holding it. That's a fantastic analogy.
15:16
The energy can't propagate, so it just destroys the localized
15:20
area. And the really terrifying thing
15:21
for an engineer here is that your normal instinct doesn't
15:25
work. If I have a beam that keeps
15:27
breaking, my instinct is to just make it thicker, add more steel,
15:32
but if the impact exceeds the critical velocity, the material
15:36
simply shears or shatters locally.
15:39
It doesn't matter how much heavy seal you throw at the rest of
15:41
the problem, because the rest of the steel never even feels the
15:44
impact before the front completely shatters.
15:46
Precisely. The thickest armor in the world
15:48
won't save you if the localized material rips before the energy
15:52
can spread to the rest of the plate.
15:54
So what do we do? Which naturally raises the most
15:56
important question of our entire discussion, right?
15:59
If you can't just make the structure thicker, and you
16:01
absolutely can't rely on simple rigid math.
16:04
And external rubber dampeners are completely blind to the
16:07
initial shock. How is a design engineer
16:10
supposed to actually survive this?
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We can't just defy physics. Which perfectly sets us up for
16:15
the final leg of our journey today, the step by step
16:17
optimization approach. Good stuff.
16:19
Exactly. Armed with this frankly
16:21
terrifying knowledge of exactly how and why materials break
16:25
under dynamic load, how do you actually optimize your design in
16:28
practice to survive? The solution lies in shifting
16:31
your entire engineering mindset to what is called the energy
16:34
method. The energy method OK.
16:36
Instead of just looking at elastic limits and trying to
16:38
keep a structure perfectly stiff, you have to embrace
16:41
deformation. Engineers must look at the total
16:44
energy of absorption required to cause plastic deformation all
16:48
the way up to the ultimate point of rupture.
16:51
So we wanted to deform, yes. You are no longer designing A
16:54
rigid, immovable wall. You are designing A metallic
16:58
sponge. Your goal is to accurately
17:00
calculate how much total kinetic energy the volume of your
17:04
material can soak up through prominent ductile deformation
17:07
before it finally snaps. Our sources actually have a
17:10
brilliant, highly specific real world example of this that
17:13
perfectly illustrates the math. Let's walk through it slowly so
17:16
we can really see how this optimization works.
17:18
Let's do it. Imagine you have a tension bar
17:20
made of medium steel. The material properties of this
17:23
specific alloy tell us that it can absorb about 500 foot pounds
17:28
of energy per cubic inch of volume before it physically
17:31
ruptures. OK, 500 foot pounds per cubic
17:33
inch. Now let's look at 2 totally
17:35
different dynamic impacts. In the first scenario we have a
17:38
heavy 200 LB body moving at a speed of 80 feet per second all.
17:43
Right. So if you calculate the total
17:44
kinetic energy of that moving body, which is 1/2 mass times
17:48
velocity squared, that object packs exactly 20,000 foot pounds
17:53
of kinetic energy. Right, that's a lot.
17:56
Now let's say your engineered steel bar is relatively small,
17:58
let's call it 10 inches long and half an inch in diameter, Pretty
18:01
small. If you calculate the volume of
18:03
that cylinder, it's only enough physical material to absorb
18:06
about 1000 foot pounds of energy total.
18:09
Yeah, so you have an incoming payload of 20,000 foot pounds of
18:13
energy slamming into a bar that can only soak up 1000 foot
18:16
pounds. The result is obvious.
18:18
The bar is completely and violently ruptured.
18:20
It didn't have enough sponge to soak up the water.
18:23
Exactly. But look at the second scenario
18:25
to see how the variables shift. Suppose the incoming moving body
18:29
only weighs 50 lbs and it is moving much slower at 30 feet
18:33
per second. OK.
18:34
So if you run the kinetic energy calculation on that, the total
18:37
energy payload is only about 700 foot pounds.
18:40
And since our little half inch steel bar has enough volume to
18:44
absorb 1000 foot pounds of energy, it actually survives the
18:48
impact. It will.
18:49
It will deform permanently. It will look completely warped
18:53
and ruined to the naked eye. But mathematically and
18:56
physically it will not break. It holds the load, yes.
19:00
So optimization using the energy method isn't about achieving
19:03
infinite stiffness, it's about meticulously balancing the total
19:07
physical volume of your ductile material against the total
19:10
anticipated kinetic energy of the impact.
19:13
You literally just need enough cubic inches of yielding
19:16
material to safely soak up the energy payload.
19:18
That is the theory, and on paper it works beautifully.
19:21
It's how crumple zones and cars are engineered, for instance.
19:24
Right? But we have to address the fatal
19:26
design flaw that routinely ruins this beautiful optimization in
19:29
the real world. Ah yes, the invisible killers of
19:32
structural integrity. Let's talk about what ruins the
19:36
sponge. The most common real world screw
19:38
up that completely nullifies the energy method is the presence of
19:42
stress concentrators. Stress concentrators.
19:44
Specifically, I'm talking about physical notches, sharp internal
19:48
corners, and poor quality welds. OK, why are those so bad?
19:52
Think back to our perfectly machined steel bar that can
19:56
absorb 500 foot pounds per cubic inch.
19:59
That mathematical assumption completely relies on the entire
20:03
bar stretching uniformly. It assumes every single cubic
20:06
inch of that metal is doing its fair share of the work, sharing
20:10
the stress across its entire geometric length.
20:12
But. What if there is a tiny, almost
20:14
microscopic sharp notch cut into the side of the bar?
20:18
Say a machine has made a mistake or a tool slip during
20:20
manufacturing. What happens to the energy then?
20:23
The physics of the energy flow changes completely.
20:26
You can think of stress moving through a material almost like
20:28
water flowing through a pipe. I like that.
20:30
When that force hits a sharp notch, the flow is violently
20:34
disrupted and the stress lines bunch up and concentrate
20:37
incredibly heavily at the very tip of that tiny notch.
20:40
Oh man. Because the stress is
20:42
exponentially higher there, the material in the immediate
20:45
neighborhood of that notch yields first, it deforms
20:49
plastically and then quickly reaches its rupture point, while
20:52
the rest of the pristine, untouched steel bar remains
20:55
completely unstressed. So the whole thing fails.
20:58
Yes, because the notch acts as a localized weak point, the
21:02
catastrophic break occurs right there, instantly.
21:05
The bulk of your carefully calculated, beautifully
21:08
engineered volume of steel absorbs exactly 0 energy.
21:12
Because it never even got the chance to.
21:14
Exactly. The chain broke at the weakest
21:16
link before the rest of the links even felt the pull.
21:18
Yeah, and the exact same thing happens with a bad Weld.
21:21
Without a doubt, if you take two unnotched, perfectly optimized
21:25
steel bars and join them with a porous, poorly penetrated butt
21:28
Weld, the rupture will happen entirely at the Weld interface,
21:31
right? The full energy absorption
21:33
capacity of the rest of the bar is totally wasted.
21:36
Your beautiful, theoretically perfect design optimization on
21:39
paper means absolutely nothing if stress concentrators are
21:42
introduced during the actual fabrication process.
21:45
So to synthesize all of these complex technical takeaways and,
21:50
you know, directly answer the biggest question you probably
21:53
have right now, how do I ensure my design actually survives a
21:56
violent impact? The $1,000,000 question.
21:58
The answer is a strict 3 part checklist based on everything
22:01
we've uncovered today. Number one, do not use single
22:05
degree of freedom models for dynamic impacts.
22:08
Your structure is not a simple rigid pendulum.
22:11
It is going to ring and vibrate in wildly complex ways and you
22:15
absolutely must account for high speed flexural waves #2 Do not
22:20
rely on external rubber damping to save you from the initial
22:23
strain spikes. The shockwave is simply too
22:25
fast, external dampeners will be blind to it and you will end up
22:28
with a #3. If you are designing for severe
22:31
impacts that push past the elastic limit, use the energy
22:34
method. You have to balance your total
22:37
ductile volume against the kinetic energy payload.
22:40
But, and this is the big but exactly, and this is the most
22:42
important part, you have to vigorously protect that total
22:45
energy absorption capacity by ruthlessly eliminating any
22:49
notches, sharp geometric corners and bad welds from your
22:54
manufacturing process. It really highlights how deeply
22:57
interconnected theoretical physics and practical shop floor
23:00
manufacturing truly are. You simply cannot have one
23:04
without the other and expect to survive drive a dynamic impact.
23:07
A brilliant mathematical model is totally useless if the welder
23:11
had a bad day. Exactly.
23:13
And I think that dynamic leaves us with a incredibly fascinating
23:16
final lingering thought to Mull over as we wrap up.
23:19
We've learned today that a single microscopic notch or a
23:22
hidden bad Weld can completely, utterly nullify the energy
23:26
absorption capacity of a massive, perfectly calculated
23:29
steel beam. It's sobering.
23:30
It is the entire multi $1,000,000 structure fails
23:34
simply because the immense genetic energy found a single
23:37
localized weak point to exploit. It really makes you wonder.
23:40
You know how? About what?
23:42
Beyond just structural steel, what invisible notches, whether
23:46
they are single points of failure in our digital networks,
23:49
bottlenecks in our supply chains, or even hidden
23:52
communication breakdowns in our organizational structures, are
23:55
lurking around us right now, just waiting for the right
23:58
external impact to cause a catastrophic total system
24:01
failure. That's a great point.
24:02
They're everywhere. It's a question every designer
24:05
in every field needs to be asking themselves.
24:08
If you got something useful out of today's Eisode, hit follow
24:10
and drop a like. It hels grow the show an keeps
24:13
the engineering knowledge flowing.