Latest / Mechanical Engineering Made Simple / (#138) Why materials snap or hold together

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(#138) Why materials snap or hold together · Mechanical Engineering Made Simple

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Welcome back to Mechanical Engineering Made Simple, the
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podcast that makes you sound smart at parties without needing
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APHD or therapy. I'm your host and today we're
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talking material strength because let's be honest, we all
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want to know what can take a pounding and still hold up.
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So tell me listener, what gets you hard or tensile?
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Is it steel cables, carbon fiber?
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Or that 1X who somehow had infinite compressive strength
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but 0 sheer resistance. Stick around because we're about
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to find out why some things snap like a sheet promise and others
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just keep getting stronger under pressure.
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Yeah, and, you know, it really is true.
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I mean, the craziest part of this, this invisible war
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happening all around us, is just how asymmetrical it is.
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Right. Asymmetrical how?
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Like in what way? Well, take the concrete floor
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you're likely standing on right now.
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If you try to crush it, it's, I mean it is virtually
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indestructible. Yeah, concrete is basically just
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liquid rock that drive. Exactly.
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But if you somehow grab the edges of that exact same
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concrete floor and try to stretch it, you know, pull it
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apart, it would snap and crumble with only about 110th of the
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force required to crush it. Wait, 110th That's that's a
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massive difference. It is, and the built environment
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is completely defined by these hidden physical biases.
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So today, in this deep dive, we're going to look at the
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source material to really reveal the mechanisms behind those
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forces. And honestly, look around
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Wherever you are right now, listener.
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The ceiling above your head, the chair you're sitting in, it
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might look completely still, right?
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Completely static. But it definitely isn't.
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No it's not. It is locked in a high stakes
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microscopic battle. Millions of tiny crystalline
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structures are fighting a constant tug of war against
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gravity, heat and pressure. Yeah, and to understand how that
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war is fought, we really have to look at the simplest weapons in
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the physical arsenal, which are tension and compression.
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Right, the pulling and the crushing.
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Exactly. I mean, tension and compression
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are the absolute bedrock of structural mechanics.
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Engineers refer to these as direct or normal stresses.
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And normal in the math sense, right?
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Not just like everyday normal. Yeah, normal in the mathematical
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sense doesn't mean ordinary or average.
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It means perpendicular. So perpendicular to the
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cross-sectional surface. Right, if you have, say, a
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marble column holding up a portico, the weight of the roof
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is pushing straight down, perfectly perpendicular to the
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flat top of that column. OK, that makes sense.
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That is a normal compressive stress.
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But if you are dangling from a rope bridge, your weight is
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pulling perpendicular to the cross section of the rope, so
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that's a normal tensile stress. The concept of stress is
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fascinating to me because it's not just about the total weight,
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is it? It's about how that weight is
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distributed. Oh, entirely.
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Distribution is everything. Like the sources highlight the
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formula for stress, which is capital P for the total load
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divided by capital A for the cross-sectional area, and that
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immediately made me think of snowshoes.
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Oh, that's a perfect analogy. Right, because if I walk into a
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snowy field wearing heavy boots, I'm going to plunge right
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through the crust up to my knees.
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But if I strap on snowshoes, I can walk right across the
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surface. Yeah, and lost any weight?
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Exactly. I haven't lost a single pound.
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The total force brushing down is exactly the same, but the area
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has massively increased. So by dividing the same force
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over a much larger area, the intensity of the force, the
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actual stress on the snow drops dramatically.
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The snowshoe visual captures the physical mechanism perfectly,
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and you know when you apply that stress to a material, the
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material has to react. And that reaction is what we
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call strain, right? Yes.
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Strain. Strain is essentially a measure
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of deformation. If you take a piece of steel and
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put it under tensile stress by hanging a massive weight from
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it, the steel will stretch. But how do we actually measure
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that strain? It's a simple ratio.
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The strain represented by the Greek letter epsilon is simply
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the extension or the amount it's stretched divided by its
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original length. It's just a ratio of
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deformation. But let's zoom in on that piece
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of hanging steel for a second, because to the naked eye, a
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thick steel cable holding up a heavy engine block doesn't look
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like it's stretching at all. Not even a little bit, right?
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It looks completely rigid. So what is actually happening
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down at the microscopic level? Well, if we could shrink down
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and view that steel under an electron microscope, we would
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see that it isn't a solid, uniform mass.
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It's not, no. It is composed of millions of
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tiny, irregularly shaped crystals packed together in
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random orientations. Wow, OK.
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And these crystals are formed by atoms locked into specific
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geometric lattices. So when you hang that weight and
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apply tension, you are literally pulling those atoms slightly
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further apart. So they don't break, they just
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stretch. Exactly.
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The bonds between them act like unimaginably stiff microscopic
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springs. Which brings us to Robert Hook,
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the famous 17th century scientist who hated Isaac Newton
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but absolutely loved springs. He really did love springs, and
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Hook realized that there is a perfectly linear relationship
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between how hard you pull and how much the material stretches.
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Hook's law. Yes, Hook's law.
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If you double the weight, the steel stretches exactly twice as
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much. Triple the weight, it stretches
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3 times as much. Up to a certain point of course.
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For high strength steel, this straight line relationship holds
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up to about 750 Mega Newtons per square meter.
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That is a lot of Newtons. It is, and in this linear
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elastic range, if you remove the massive engine block from the
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steel cable, those microscopic atomic springs pull everything
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back. The steel wire returns to its
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exact original length down to the micrometer.
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So it has a mechanical memory. It's exactly like snapping a
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rubber band, just scaled up to high strength industrial
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materials. That's exactly what it is, and
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the stiffness of those atomic springs dictates how much the
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material stretches under a given load.
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Engineers call this constant ratio Young's modulus.
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Right, represented by a capital E in the math.
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You got it. Young's modulus is essentially
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the scorecard for stiffness. A material with a high Young's
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modulus, like steel, requires an enormous amount of stress to
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produce even a tiny amount of strain.
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For steel, the sources say it's around 200 giga newtons per
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square meter. Exactly.
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Whereas a material with a low Young's modulus, like rubber,
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requires very little stress to produce massive strain.
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But there is a point of no return, isn't there?
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Because this hooks law only works up to a certain threshold,
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the limit of proportionality. Right, the elastic limit.
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I know this first hand from carrying way too many groceries.
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Well, the plastic bag effect. Yes, if I load a plastic grocery
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bag with heavy cans and pick it up by the loops, the handle
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stretch out. If I set it down, they shrink
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back, but if I add one more can, suddenly the plastic handles
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stretch out rapidly. They turn all thin and white,
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and when I finally set the bag down, those handles stay
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stretched out forever. The memory is gone.
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Exactly, the memory is totally gone.
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What happened there? You've forced the material into
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permanent plastic deformation. When you exceeded that limit of
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proportionality, the atomic bonds couldn't just stretch
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anymore. They gave up, basically, yeah.
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The stress was so high that entire planes of atoms within
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those microscopic crystals literally broke Their bonds,
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slid past one another, and form new bonds in a stretched out
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position. Wow.
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It is a permanent architectural change at the atomic level.
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The material will never recover its original shape.
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Which is terrifying if you are designing a suspension bridge.
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I mean, you don't want the cables permanently stretching
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out and the bridge sagging a few extra feet every single year.
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Definitely. Not.
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But this permanent stretching, this ductility, is actually a
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safety feature in many metals, isn't it?
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It is a massive safety feature and we measure ductility by
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looking at either the percentage reduction in cross-sectional
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area or the percentage increase in length after it breaks.
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So the more it can stretch before snapping, the better.
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Generally, yes. Mild steel and aluminum are
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highly ductile. As an engineer, you want a
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structure to warn you before it fails.
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If a ductile steel beam is overloaded, it will warp.
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It'll sag and visibly bend long before it snaps.
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It gives you time to run away. Exactly.
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It gives people time to evacuate.
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But brittle materials don't give you that warning, do they?
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Beetle materials are treacherous things like cast iron, concrete,
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glass or ceramics. They have almost no plastic
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region. They don't stretch at all.
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Right. Their atomic structure just
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doesn't allow for those planes of atoms to slide gracefully
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past one another. When they reach their ultimate
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stress limit, the bonds just sever.
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They fracture instantly and catastrophically.
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Which brings us back to your concrete floor fact from the
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beginning, concrete is a notoriously brittle material.
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It crushes incredibly well, but snaps under tension, right?
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So I always wondered how modern skyscrapers don't just crack in
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half when the wind bends them. If the concrete is brittle, how
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do they survive the tension? Because when pushing on a
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building creates tension on one side, right?
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It does, and this is where the genius of reinforced concrete
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comes in. Engineers took two materials
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with completely different mechanical properties and
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married them into a composite. So they balance each other out.
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Exactly. They pour the concrete around a
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skeleton of ductile steel rebar. When the wind pushes against a
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skyscraper, one side of the building is compressed.
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The concrete handles that beautifully.
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And the other side. The opposite side is stretched
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in tension. The concrete actually cracks
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almost immediately on that side. Wait.
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Really. The concrete actually cracks.
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Yes, but those microscodic cracks don't matter because the
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embedded steel rebar catches the tensile load and holds the
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structure together. It's the Ultimate Team up.
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That's why concrete pillars are great for holding up heavy
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ceilings and compression, but terrible for acting as a diving
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board in tension without steel inside them.
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Precisely. But looking at the literature,
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modern engineering is moving past steel and concrete, right?
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We're getting into aerospace composites, carbon fiber
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reinforced plastics, Kevlar, boron, OH.
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Absolutely. The strength to weight ratios on
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those composites are basically magic compared to steel.
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But here is the massive red flag I saw in the source material.
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It said that high modulus carbon fiber costs about 250 times more
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than standard E glass fiber. It is astronomically expensive.
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And Kevlar is 15 times more expensive.
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If I'm an engineer, I have to walk into a budget meeting and
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justify spending 1/4 of $1,000,000 on a material when I
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could have spent spent 1000. How does that math ever work
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out? Well, it forces structural
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engineering to become an exercise in extreme economics.
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You only pay that premium when the penalty for weight is
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financially catastrophic over time.
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Like in airplanes. Exactly.
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If you are building a commercial airliner, every single ounce of
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weight requires a certain amount of jet fuel to keep it airborne
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over a 20 year lifespan. Making a wing out of ultra
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expensive ultralight carbon fiber will save millions of
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dollars in aviation fuel. So it easily pays back the
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massive upfront cost. Right, but if you are designing
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a static bridge, weight doesn't burn fuel.
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You use steel and concrete because the economic equation
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flips. It's fascinating how the atomic
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lattice of a material directly dictates the financial
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spreadsheet of an airline company.
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It really is all connected. But let's take these concepts of
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pulling and crushing and apply them to a structure that isn't
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just hanging straight down. Think about a diving board or a
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a hotel balcony hanging off the side of a building.
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OK, moving into bending. Yeah, how do these invisible
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forces keep a diving board from just snapping off the wall when
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a heavy swimmer jumps on the very end?
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The diving board introduces us to the cantilever.
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A cantilever is simply any beam that is anchored rigidly at one
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end and completely unsupported at the other.
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Like a springboard. Exactly when our swimmer stands
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on the free end, their downward weight, the lateral load,
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generates 2 internal actions that propagate all the way back
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through the board to the wall, and those are a shearing force
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and a bending moment. OK, the bending moment, that's
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essentially leverage, right? The twisting force.
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Yes, leverage is a great way to think of it.
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We experience this every time we try to carry something heavy.
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Like if I hold a 20 LB bowling ball tightly against my chest,
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it's relatively easy. My spine supports it, sure.
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But if I try to hold that same 20 LB bowling ball straight out
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in front of me with my arms fully locked, my shoulders will
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start screaming. And agony within seconds it.
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Feels infinitely heavier, right? The weight hasn't changed, but
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the bending moment at my shoulder joint has skyrocketed.
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That physical strain in your shoulder perfectly illustrates
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the math. The bending moment represented
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by a capital M is simply the applied load multiplied by the
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distance from the anchor point. So force times distance.
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Exactly. The load is capital W and the
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distance from the wall is the total length capital L minus the
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distance Z. The further out the swimmer
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walks on the diving board, the more massive the twisting force
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becomes at the exact point where the board is bolted to the
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concrete wall. But to truly understand why that
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board doesn't break, we have to look inside the anatomy of the
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bend itself. Right, because the board isn't
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just twisting as a single unit, the material inside is basically
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going to war with itself. The best way to visualize this,
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I think, is to take a really thick paperback book, like a
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massive fantasy novel, and bend it downward in the middle.
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Oh, that's a brilliant visual. You can literally watch the
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mechanics happen. The pages on the top cover pull
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tight. They are stretching, trying to
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cover the outer curve. But the pages on the bottom
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cover bunch up and wrinkle. They are being crushed into the
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inner curve. The paperback book visual hits
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right at the core of the mechanics.
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The material of the diving board acts the exact same way.
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The upper longitudinal fibers of the board are stretching.
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They are in pure tensile stress. Pulling apart.
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Yes, and simultaneously the lower fibers are being
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compressed into a smaller space. Pure compressive stress.
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The material is trying to RIP itself apart at the top and
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crush itself at the bottom. But the board doesn't just bend
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down, does it? The sources detail a secondary
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reaction called the Poisson ratio effect.
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Yes, the Poisson ratio. Which honestly blew my mind
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because it means the diving board actually changes its
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three-dimensional shape. It doesn't just curve downward.
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It's a brilliant quirk of the conservation of volume.
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When you stretch a material longitudinally, you are pulling
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its atoms further apart in One Direction, so the material
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compensates by pulling together laterally.
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So it gets thinner. It gets thinner.
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Think of stretching a piece of dough.
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As it gets longer, gets narrower.
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Conversely, when you compress material, it bulges outward
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laterally. So back to our bending diving
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board. The top fibers are stretching,
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which means they must be getting narrower from side to side.
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And the bottom fibers are compressing, which means they
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must be bulging wider from side to side.
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Exactly. The entire rectangular cross
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section of the diving board physically distorts under the
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load. The top edge pinches in and the
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bottom edge flares out, forming A subtle trapezoid shape.
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That is wild. This three-dimensional
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distortion is happening inside the concrete balconies of high
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rises every time a group of people walks out to look at the
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view. OK, so the top is screaming in
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tension and the bottom is screaming in compression.
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They are experiencing maximum opposite forces.
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Yes, the extremes. Which means somewhere inside
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that board, as you travel from the top to the bottom, the
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pulling has to fade out and the crushing has to fade in.
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What happens in the dead center? Right in the middle of this
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structural violence is the eye of the hurricane.
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Engineers call it the neutral. Axis.
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The neutral axis. Yes, it is a longitudinal plane
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running horizontally right through the center of the beam.
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At this exact plane, the material is neither stretching
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nor compressing. The longitudinal stress passes
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perfectly through 0. Wait, so the material directly
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in the center of the beam is doing absolutely no work against
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the bending, it's just sitting there?
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It is practically useless for resisting the bending moment.
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The extreme top fibers in the extreme bottom fibers are doing
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all the heavy lifting. The material on the neutral axis
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is, from a bending perspective, dead weight.
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Which introduces an incredible design paradox.
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If the center of a solid steel beam isn't doing any work, why
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are we paying for it? Exactly?
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I mean, steel is heavy and expensive.
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If I'm constructing A skyscraper, I don't want
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thousands of tons of useless steel weighing down the
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foundation. And that exact realization
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triggered a revolution in construction materials.
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It gave birth to the universally recognized I-beam, or rolled
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steel joist. It's a classic eye shape.
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Yes, engineers realized they could literally scoop out the
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lazy material from the center of a solid rectangular beam and
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move it to the extreme top and bottom edges where the stress is
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actually happening. So they create the flanges,
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those wide, flat horizontal bars at the top and bottom of the
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eye. That's where all the tension and
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compression fights happen. That's right.
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And they connect those heavy flanges with a relatively thin
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vertical strip of steel down the middle called the web.
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They optimize the geometry to put the strength exactly where
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the physics demands it. The material economy is
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staggering and i-beam can carry the same bending load as a solid
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rectangular beam while weighing a fraction of the amount.
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It completely changed how high we could build.
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But I have a problem with this. No.
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What's the problem? If the I shape is this ultimate
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flawless structural cheat code for bending, why didn't humans
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figure this out earlier with wood?
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Like why don't we see traditional timber framed barns
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built with wooden I beams? Why are old structures always
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built with massive solid rectangular logs?
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You're touching on a fundamental law of engineering here.
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You can never separate the geometry of a structure from the
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microstructure of the material it is made of.
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OK, unpack that. The I-beam geometry works
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perfectly for rolled steel because steel is generally
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isotropic and homogeneous. Meaning it's the same
17:43
everywhere. Exactly.
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It's atomic crystalline structure behaves uniformly no
17:48
matter which direction you apply the force.
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But wood isn't uniform. Far from it.
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Wood is highly anisotropic. It is composed of longitudinal
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cellulose fibers bound by lignin.
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It has a grain. Right, you chop wood along the
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grain. Precisely.
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Its strength along the grain is drastically different from its
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strength across the grain. If you took a massive oak log
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and carved out the sides to make a thin webbed eye shape, you
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would create catastrophic lines of weakness along the grain in
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that thin vertical web. Oh, I see.
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The moment you applied a heavy load, that thin wooden web would
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just split right down the middle, separating the top from
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the bottom. So the geometry is brilliant,
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but you need an artificially uniform material like molten
18:31
steel to actually execute it. Exactly.
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But that brings up an alarming question.
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If you carve out all the middle material to make an I-beam, and
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that thin steel web is just sitting there on the neutral
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axis experiencing 0 pulling and 0 crushing, what is stopping the
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top flange from just shearing off and sliding past the bottom
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flange? And here is where we look
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closer. The web isn't sitting idle while
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the wide flanges are taking all the bending stress.
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That thin vertical web is fighting an entirely different
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hidden war. The shearing force.
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Yes, it is resisting sheer stress represented by the Greek
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letter Tau. OK, we need to clarify shear
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stress because it's different from the pulling and crushing
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we've been talking about. I always picture shear stress
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like a deck of playing cards. The.
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Playing cards analogy is great. If I place a deck of cards flat
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on the table and I push the top card to the right, the cards
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slide past each other. They aren't stretching and they
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are crushing. The horizontal layers are
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grinding past one another. That sliding, grinding force is
19:34
shear. That is the exact mechanism when
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you load a beam, the downward force wants to slice the
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material vertically, just like a pair of scissors, which are
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literally called sheers. Oh.
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Right sheers. And the distribution of this
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shearing force inside the beam is completely counterintuitive
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to everything we just learned about bending.
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Because with bending, the maximum stress was at the
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extreme top and extreme bottom. Exactly, but with sheer stress
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if we look at the internal forces of a rectangular cross
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section. The shear stress is actually
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zero at the very top and 0 at the very bottom.
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Wait, really? Yes, it follows A parabolic
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curve. The sliding, grinding shear
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forces get more and more intense as you move toward the middle of
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the beam, and they hit their absolute maximum peak exactly on
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the neutral axis. The exact spot where the pulling
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and pushing is 0, the grinding slicing force is at its absolute
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maximum. That is a wild physics paradox.
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It's a beautiful inversion because the top half of the beam
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is stretching outward and the bottom half is compressing
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inward. The boundary line between those
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two warring factions, the neutral axis, is experiencing
20:42
the maximum sliding friction. The top half is trying to slide
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one way, the bottom half is trying to slide the other, and
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the material right on the equator is caught in the
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crossfire. So that thin vertical web in
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i-beam isn't just a spacer, it is a dedicated shear wall.
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It doesn't need to be thick to handle tension, but it has to be
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exactly thick enough to stop the top half of the beam from
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cleanly slicing away from the bottom half under maximum shear
21:07
force. The math and the source material
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confirms this. For a narrow rectangular cross
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section, the maximum shear stress on that neutral axis is
21:16
equal to multiplied by capital F / 2, multiplied by lowercase H
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and lowercase T. Meaning it's significantly
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higher in the middle. Exactly.
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The maximum shear stress is basically 50% higher than the
21:29
average shear stress across the whole beam.
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Engineering an i-beam is a master class in balancing these
21:35
conflicting distributions of force.
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You put mass at the top and bottom to fight the bending, and
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you leave just enough mass in the middle to fight the sliding.
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It's just brilliant. OK, so we've perfected the
21:45
straight beam by moving the material to the edges.
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We've conquered the diving board.
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But the real world isn't built out of straight lines.
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No, it's not. When you wrap that metal into a
21:54
cylinder like a submarine hall, an airplane fuselage, or an
21:58
industrial chemical tank, those pulling and sharing forces don't
22:01
disappear. They attack from entirely new 3
22:05
dimensional angles. And this represents a massive
22:07
step up in mathematical complexity.
22:10
We are moving from the one-dimensional mechanics of
22:12
straight lines into the two-dimensional realm of
22:15
surfaces and plates. The sources mentioned using
22:17
small deflection elastic theory for this.
22:19
Yes. To determine how flat, circular
22:22
metal plates and curved cylinders deflect under massive
22:25
pressure, engineers rely on that theory.
22:28
The literature emphasizes a specific concept here called
22:31
flexural rigidity. It dictates how hard it is to
22:34
bend a flat metal plate, and there's a mathematical detail in
22:37
the equation that I found staggering.
22:39
Oh, the thickness variable. Yes, the stiffness of the plate
22:42
capital D isn't just tied to how thick it is, it's tied to the
22:45
thickness cubed. It is a massive geometric cheat
22:48
code. The flexural rigidity equals
22:50
capital E multiplied by t ^3, all divided by 12 multiplied by
22:55
the quantity 1 minus v ^2. So the Ted is thickness and it's
22:58
cubed. Right.
22:59
What this means in the physical world is that if you have a
23:03
steel plate and you decide to simply double its thickness, you
23:07
do not make the plate twice as stiff.
23:09
Because I'm cubed. Exactly.
23:11
You make it 2 * 2 * 2 eight times stiffer.
23:14
Increasing the thickness by a factor of 2 increases the
23:17
rigidity by a factor of eight. That explains so much about
23:21
armor plating on tanks and warships.
23:23
It really does. Adding just a fraction of an
23:25
inch of steel to a hull exponentially increases its
23:28
ability to resist buckling under the impact of torpedo or extreme
23:32
ocean pressure. And that resistance becomes
23:35
critical when we take those flat plates and curve them into
23:37
pressurized cylinders. There is a classic engineering
23:40
ractice mentioned in our sources called shrink fitting.
23:43
Right shrink fitting. How does that?
23:44
Work. You take an aluminum alloy
23:46
cylinder, heat it U so it expands, and slip it over a
23:49
slightly smaller steel inner cylinder.
23:52
As the aluminum cools, it shrinks rapidly, locking onto
23:55
the steel with tremendous permanent clamping force.
23:57
Generating internal pressure without any actual gas or
24:01
liquid. Exactly.
24:02
It creates an artificial internal pressure lowercase P.
24:05
Which is the same kind of pressure you get inside a scuba
24:08
tank filled with compressed air, or the cabin of a passenger jet
24:11
flying at 30,000 feet. The pressurized air inside is
24:15
violently pushing outward, trying to blow the walls of the
24:18
cylinder apart. Yes, and to stop that from
24:21
happening, the curved metal walls develop what's called a
24:24
circumferential stress. The internal pressure acts
24:28
radially, pushing outward in all directions from the center, but
24:32
the material of the cylinder wall responds circumferentially.
24:35
It tries to hold itself together in a circle.
24:37
It does. The analogy that immediately
24:39
jumped out at me for circumferential stress is the
24:42
Thanksgiving dinner effect. I like where this is going.
24:44
Imagine you eat a massive 5 course holiday dinner.
24:48
Your stomach is completely full and it is physically pushing
24:52
outward against your abdomen. That is radial internal
24:55
pressure. OK, but your letter belt is
24:57
taking the brunt of that force. The belt isn't pushing inward,
25:01
it is being pulled tightly in a circle around your waist to keep
25:04
your pants from bursting open. That's perfect.
25:06
The tension traveling around the circle of the belt is
25:09
circumferential stress. Engineers actually call it hoop
25:12
stress. The belt analogy perfectly
25:15
translates the mechanism. The internal pressure wants to
25:19
separate the cylinder into two distinct halves, blowing it
25:21
apart. The hoop stress is the tension
25:24
running around the circumference holding those halves together.
25:27
And the math for this is surprisingly simple, isn't it?
25:30
It is surprisingly elegant. The circumferential stress Sigma
25:33
equals capital P divided by the product of 1 and T which when
25:37
you simplify it from the internal load equations, means
25:40
the hoop stress is equal to the internal pressure multiplied by
25:43
the radius divided by the wall thickness.
25:46
Pressure times radius divided by thickness.
25:48
If you're ever designing a submarine or a pressure cooker,
25:51
this is the math that keeps it from exploding.
25:53
Literally. But this isn't just abstract
25:55
math. When engineers misunderstood
25:57
this exact equation, it led to some of the most tragic
26:00
disasters in aviation history. I'm thinking of the de Havilland
26:04
Comet in the 1950's, the world's first commercial pressurized
26:08
jetliner. The Comet It is a foundational
26:12
case study in material stress. The engineers designed a
26:16
pressurized cylinder, the fuselage, to operate at high
26:19
altitudes, but they included square passenger windows.
26:22
Right, square windows. Every time the plane pressurized
26:25
and depressurized, the fuselage expanded and contracted the
26:29
hoop. Stress was supposed to float
26:30
evenly around the circumference of the metal skin.
26:33
But the square windows interrupted that flow.
26:36
Violently interrupted it. The hoop stress piled up
26:39
exponentially at the sharp 90° corners of the windows, creating
26:43
massive stress concentrations. Because it couldn't flow
26:46
smoothly around a sharp angle. Exactly.
26:49
The metal fatigued micro cracks formed at the corners and
26:52
eventually the hoop stress exceeded the ultimate tensile
26:55
strength of the aluminum skin. The fuselage literally ripped
26:58
itself apart mid flight. It's chilling.
27:00
It's a devastating lesson in how geometry interacts with internal
27:04
pressure. It really is.
27:06
If the radius is large and the internal pressure is high, the
27:09
required thickness of the wall must compensate perfectly and
27:12
the geometry must allow the stress to flow smoothly without
27:15
concentrating. That is why every airplane
27:17
window you see today is an Oval or a circle.
27:20
It allows the hoop stress to flow smoothly around the opening
27:23
without piling up in sharp corners.
27:25
Exactly. It's written in the history of
27:28
engineering failures. It is, but you know everything
27:30
we have talked about so far. A weight hanging from a wire, a
27:33
person standing on a balcony, A pressurized tank.
27:37
These are all static loads. The weight isn't moving.
27:40
Right stationary forces. But the Internet's most a
27:42
burning question asks about the dynamic world.
27:45
How do engineers calculate a bridge holding a massive
27:49
speeding freight train? And this is where a structural
27:51
mechanics transitions into the realm of dynamic complexity.
27:54
Designing a bridge to hold its own static dead weight is the
27:58
easy part. The real challenge is the
28:00
rolling load. Because the train moves.
28:03
Exactly. A freight train is an
28:05
unimaginably massive point load, and its position on the bridge
28:09
changes every fraction of a second.
28:12
Even if the train is moving slowly enough that we could
28:14
ignore the dynamic impact of its inertia, the sheer changing
28:18
position of its weight constantly alters the
28:21
distribution of tension and compression throughout the
28:23
entire skeletal structure of the bridge.
28:25
You can actually feel this right?
28:27
Like if you ever stand on a pedestrian suspension bridge,
28:29
when someone heavy jogs past you, the bridge doesn't just sag
28:33
where they step, the sway ripples through the cables.
28:36
You feel the wave. Yeah, the tension physically
28:38
moves with them. So how do engineers map a target
28:41
that is constantly instantly moving across 1000 foot span?
28:45
They map it by changing their perspective.
28:47
Earlier we talked about bending moment diagrams, which show the
28:50
stress across the entire length of a beam for one specific
28:53
stationary load. But to handle moving trains,
28:57
engineers invented the influence line.
29:00
An influence line completely flips the axis of the math.
29:04
Instead of looking at the whole bridge at 1 moment in time, an
29:07
influence line tracks the bending moment at one specific
29:10
critical point on the bridge as the train travels from one end
29:13
to the other. I want to make sure I'm
29:14
visualizing this correctly. It's like taking a highly
29:17
sensitive heart rate monitor and bolting it to one single crucial
29:22
steel joint dead in the middle of the bridge.
29:24
OK, Yeah. As the freight train approaches
29:27
from a mile away, the stress monitor on that joint starts to
29:30
twitch and rise. As the massive locomotive rolls
29:33
directly over that exact joint, the monitor spikes to its
29:37
absolute maximum peak. And as the train rolls away
29:40
toward the far side, the stress slowly drops back down to
29:43
baseline. That's it.
29:44
The readout from that monitor over time.
29:46
That graph is the influence line.
29:48
That is exactly what an influence line is.
29:51
Engineers must plot these influence lines for every single
29:54
critical cross section of the bridge.
29:55
They have to map the peaks for the bending moments and they
29:57
have to map the separate peaks for the shear forces.
30:00
So they have thousands of graphs.
30:02
Yes, and by overlaying all these graphs they find the absolute
30:06
worst case scenario that any individual steel girder will
30:10
ever experience as a rolling load crosses the span.
30:13
But a modern suspension bridge isn't just one simple beam.
30:16
An airplane fuselage isn't a simple beam.
30:19
They are nightmarishly complex arrangements of stringers, ribs,
30:23
sheeting cables and rings. Very true.
30:25
Before computers. How did engineers possibly
30:28
calculate the influence lines and shear forces for thousands
30:32
of interconnected parts all moving at once?
30:34
The short answer is with extreme difficulty and massive safety
30:37
factors. Pre computer engineering relied
30:40
on slide rules, terrifyingly complex manual calculus, and
30:45
deliberate oversimplification. Because they just couldn't do
30:47
the math for every piece. Exactly because they couldn't
30:50
calculate the exact stress on every single rivet of a complex
30:53
structure, they made educated approximations and then
30:56
multiplied the required material thickness by a massive safety
30:59
factor just to be sure. So they just overbuilt it.
31:02
They did. That is why early iron bridges
31:04
and early aircraft were so incredibly heavy and overbuilt.
31:08
They couldn't prove it was safe mathematically, so they just
31:10
threw more metal at the problem. So what changed?
31:13
How do we build modern airplane wings that flex 20 feet in the
31:17
air without snapping using the absolute minimum amount of
31:20
carbon fiber? The revolution was born from the
31:22
marriage of elastic theory and supercomputing.
31:25
It is called Matrix methods of structural analysis.
31:28
Matrix methods. Yes, which laid the foundation
31:31
for finite element analysis, or FEA.
31:34
Finite element analysis. That sounds like we are chopping
31:37
the physical world into tiny solvable pieces.
31:40
That is precisely the mechanism. Instead of looking at an
31:42
airplane wing is 1 impossible complex geometry, the computer
31:46
model mathematically subdivides the entire wing into thousands
31:50
or even millions of simple tiny geometric shapes, usually little
31:54
triangles or pyramids. These are the finite elements.
31:57
OK, so the computer basically chops the wing up into millions
32:00
of virtual microscopic Lego bricks.
32:03
Yes, and it assumes that the entire structure behaves like a
32:07
massive 3D web of elastic springs connecting those bricks.
32:11
We know from Hook's Law at the very beginning of our deep dive
32:14
that the math for a simple spring stretching is linear and
32:17
incredibly easy to solve. Right, pulling on a spring is
32:20
basic math. Exactly.
32:22
The computer calculates the forces required to keep every
32:25
single little brick in equilibrium and its boundaries
32:28
or nodes where it connects to his neighbors.
32:30
But if you push on one node, it pulls on the neighbor, which
32:32
pushes on the next neighbor. It's a massive chain reaction.
32:35
Exactly because every node influences the next, the
32:38
computer generates a massive set of simultaneous equations.
32:41
Gigantic matrix. Doing this math by hand would
32:44
take a team of human calculator centuries, but a modern
32:48
supercomputer can solve millions of simultaneous equations in
32:51
seconds. It uses basic hooky elasticity
32:54
to find the exact tension, compression and shear stress at
32:58
every single joint of the wing under simulated load.
33:00
So we take a problem that is physically impossible to
33:03
visualize, break it down into a million tiny, incredibly easy
33:07
spring math problems, and feed it to a machine.
33:10
That's modern engineering. It allows engineers to strip
33:12
away all the heavy, unnecessary safety factor metal because they
33:16
finally have mathematical proof of exactly how the forces flow
33:20
through the structure. It's breathtaking and it all
33:22
circles back to those tiny microscopic crystal stretching
33:25
like springs. It's a beautiful continuum of
33:28
logic. From the atomic slip of a single
33:30
metal crystal to the matrix calculations of a massive
33:34
suspension bridge holding a freight train, the fundamental
33:37
physics remains perfectly consistent.
33:40
Which brings me to a final thought I want to leave our
33:42
listener with. The floor is yours.
33:43
Every single time you step onto a balcony, drive your car over a
33:46
suspension bridge, or even just lean back too far in your office
33:50
chin, millions of microscopic crystals are stretching,
33:53
compressing, and shearing. They're actively fighting to
33:56
maintain their atomic bonds. They're perfectly balancing
33:58
invisible mathematical equations, the Poisson ratio,
34:02
the parabolic shear curve, the cubic flexural rigidity, just to
34:06
keep you from falling to the floor.
34:08
The built world we take for granted is entirely held
34:10
together by the quiet, constant, desperate tension of these
34:13
materials. Next time you look at a towering
34:16
skyscraper or a long span bridge, don't just look at its
34:19
exterior shape. Try to look through it.
34:21
Try to visualize the invisible, grinding flow of forces
34:24
traveling down its bending beams, shearing through its web
34:27
columns, and ultimately disappearing into the earth.
34:30
Once you see the world as a matrix of invisible forces, you
34:33
will never look at architecture the same way again.
34:35
I know I'm definitely going to think twice before I bend a
34:38
paperback book or overstuff a grocery bag.
34:40
The physics are everywhere. If you got something useful out
34:44
of today's episode, hit follow and drop a like.
34:46
It helps grow the show and keeps the engineering knowledge
34:49
flowing.