Latest / Mechanical Engineering Made Simple / (#58) Subsurface Shear Failure

Transcript

(#58) Subsurface Shear Failure · Mechanical Engineering Made Simple

0:00
Welcome back design jockeys to Mechanical Engineering Made
0:02
Simple. Today we are taking a deep dive
0:04
into something you rarely think about until it fails.
0:08
The incredibly intense hidden world of contact mechanics.
0:13
If it rolls, slides or connects. From the tiny ball bearings that
0:17
allow your car wheel to spin to the massive steel expansion
0:20
roller supporting a bridge, it's all governed by direct bearing
0:24
and shear stress. Our mission today is simple.
0:27
We want to shortcut decades of engineering texts to understand
0:31
exactly what happens when materials press against each
0:33
other, and, crucially, where and why they ultimately decide to
0:37
fail. And that's a necessary mission
0:39
because the math is unforgiving. We're cutting through some
0:42
foundational work, starting with the late 19th century formulas
0:45
from pioneers like Hertz, because for complex engineering,
0:49
it's not enough just to calculate the crushing force on
0:51
the surface. We need to identify exactly
0:53
where subsurface failures like that flaking you see known as
0:56
spalling actually begin. And you know how engineers use
1:00
decades of empirical testing to dictate the safe loading limits
1:04
for static structures versus highly dynamic high speed
1:07
elements? OK, let's unpack this and start
1:09
with Heiner Kurtz. His mathematical theory from
1:12
around 1880 is really the foundation for contact
1:15
mechanics. Before him, engineers were
1:18
mostly guessing. So what was the core game
1:22
changing insight of Hertz's work?
1:24
Well, Hertz gave us the precise mathematical framework to
1:27
calculate 2 things at once. The deformation and the exact
1:31
size and shape of the tiny contact area that forms when 2
1:34
bodies, say a steel ball and a steel race pressed together.
1:37
His formulas, which you know, they rely on the material being
1:40
homogeneous and elastic, They give you the maximum compressive
1:43
stresses. And critically, these stresses
1:45
are super localized, peeking right at the dead center of that
1:48
small contact area. But if the compression is
1:51
highest right at the surface center, why do engineers spend
1:54
so much time worrying about what happens beneath the surface?
1:57
I remember reading that the Hertz theory, well, brilliant,
2:00
doesn't really predict true bearing fatigue failure.
2:02
That is the essential insight. That's what separates the theory
2:06
from the practice. Hertz's formulas give you the
2:08
maximum squashing force, the compression, but material
2:13
failure, especially fatigue and rolling elements.
2:16
It's almost never caused by pure squashing.
2:19
So what causes it then? It's caused by sheer stress, the
2:22
material trying to slice or tear itself apart.
2:25
Hertz's formulas they don't give you the maximum shear stresses
2:28
which happened in the interior, nor do they give you the maximum
2:31
tensile stress which is right at the boundary of the contact
2:34
area. So failure is more of a tearing
2:36
event. Exactly a shearing event.
2:38
OK. So why is the subsurface peak of
2:40
shear stress inevitable? What's going on there?
2:43
It's because of the unique nature of the Hertzian stress
2:46
field. When you push down hard on a
2:47
small area, that stress isn't just one-dimensional, it's
2:51
triaxial. Triaxial.
2:53
So in all three directions, yes. Imagine squeezing a handful of
2:57
sand. You're applying compressive
2:59
stress in all three directions. The material in the core of that
3:03
contact area is intensely squeezed from the sides as well
3:07
as the top. And this complex stress state
3:10
actually increases the materials apparent strength.
3:12
It prevents surface crushing. Oh, OK, but just a little below
3:16
the surface, maybe 1/4 to 1/3 of the contact with deep.
3:20
The sheer components, the twisting and sliding forces,
3:23
they peak dramatically. So the surface is kind of
3:26
protected by this three-dimensional squeeze, but a
3:28
little deeper that the material is just fighting itself.
3:31
Precisely. Researchers like Bella, Jeff and
3:33
later Thomas and Horsch, they followed up on Hertz work
3:36
specifically to map those subsurface stresses, and they
3:39
confirmed that the failure mode we call spalling, where a piece
3:42
of the surface just flicks up and destroys the bearing, it
3:45
almost always originates at the point of maximum shear stress.
3:48
Which is always below the surface, not on it.
3:51
Always below the surface. So that's why when designing
3:53
bearings, we are looking for the maximum shear depth just as much
3:57
as we are for the maximum compressive pressure.
4:00
And it's worth noting the general formulas for curved
4:03
bodies give us the backbone for all of this.
4:05
You know, for spheres, cylinders, all sorts of complex
4:08
shapes. But the calculation is just the
4:10
start. It brings us perfectly to
4:12
application. How do engineers bridge this gap
4:15
between these? I mean massive theoretical
4:17
Hertzian stresses and practical safe design limits for real
4:22
world heavy components? I'm thinking of things like
4:25
rollers supporting massive weight where failure is just not
4:28
an option. They rely heavily on empirical
4:30
data and pretty conservative safety margins.
4:34
Take heavy duty applications like railway infrastructure for
4:36
example. You often have a large steel
4:38
cylinder, the roller on a flat steel plate, right?
4:42
The American Railway Engineering Association or area provides
4:45
really strict allowable loading limits for that.
4:48
If we look at a common structural steel with a tensile
4:51
yield point, let's call it subway, of say £32,000 per
4:54
square inch, you'd think the safe load will be tied to that.
4:56
You would, yeah. But here's the surprise.
4:58
The hurt stress corresponding to the maximum allowable load is
5:02
actually held constant at £76,200 per square inch for
5:07
cylinder diameters up to 25 inches.
5:09
Wait, if the material's yield strength is 32,000, how can they
5:13
safely allow more than double that in theoretical herd stress?
5:16
What kind of material magic is happening?
5:18
There. That's the power of that
5:19
triaxial stress field we just talked about.
5:22
Because the material is so constrained and compressed in
5:24
all three directions in that contact zone, it's effective
5:28
yield strength is temporarily magnified.
5:30
The constraint prevents plastic flow, so that 76,200 PSI number
5:35
is the Max surface stress they allow before they worry about
5:38
that subsurface shear stress getting too high.
5:40
That makes so much more sense. The high surface number isn't
5:43
the true failure limit, it's a proxy for the critical
5:46
subsurface shear stress. Exactly, and the allowance
5:49
changes with geometry. For much larger rollers,
5:52
Wilson's slow rolling tests are the go to reference.
5:55
He found that for large rollers over 120 inches in diameter, the
5:59
empirical load per linear inch P that it took to produce just a
6:02
tiny. Any permanent elongation in the
6:05
plate could be calculated with a formula involving diameter D and
6:08
the yield strength S sub Y and. What did he find about static
6:12
versus slow rolling? That was the crucial part.
6:15
He observed that the bearing capacity under static conditions
6:18
was almost identical to that under slow rolling, though you
6:21
know that slow motion did speed up surface deterioration and
6:24
wear. Now let's talk about spheres,
6:26
where the contact area shrinks to a theoretical point and the
6:28
stress just goes through the roof.
6:30
What happens when you test the actual crushing load of a steel
6:33
ball? The results are staggering.
6:35
SKF, the bearing manufacturer, did exhaustive tests and came up
6:39
with an empirical formula. The crushing load P in pounds
6:42
relates to the diameter D in inches using the relationship P
6:46
= 1960 multiplied by the quantity 810, all raised to the
6:50
power of 1.75. OK.
6:52
And what does that translate to in terms of stress?
6:55
Well, this is what's so fascinating.
6:57
This crushing load corresponds to an incredibly high
7:00
theoretical Hertz stress, a number approaching £4,000,000
7:05
per square inch for small balls. £4,000,000 per square inch
7:08
that's that's almost impossible to conceptualize.
7:10
It just emphasizes the scale of localization to bring it down to
7:14
earth. Let's run a quick example.
7:15
Imagine a 1 1/2 inch diameter steel ball in a bearing race and
7:20
it's carrying a 2000 LB load. OK, pretty substantial.
7:23
When you apply Hertzian principles, the contact area
7:26
isn't a point, it's a tiny ellipse.
7:28
Its major axis might be about 1/4 of an inch, and it's minor
7:32
axis is less than 1/10 of an inch.
7:34
We're talking about an area smaller than a watermelon seed.
7:37
So that entire 2000 lbs of force is focused right there.
7:40
Precisely the maximum compressive stress for that
7:42
setup, Sigma sub C Max, is calculated to be £252,000 per
7:48
square inch. That's why bearing steel is such
7:50
a specialty material. Even at these massive
7:52
theoretical stresses, the steel handles it, at least until that
7:56
subsurface shear stress starts the fatigue clock ticking.
7:59
So we've nailed down rolling elements.
8:02
Now let's see how these principles play out in something
8:04
static, like a riveted joint. These are complex, but engineers
8:09
design them all the time. What are the key simplifying
8:13
assumptions they use to make the design manageable?
8:16
Right engineers have to make some concessions to simplify
8:18
things, and these have been validated over a century of
8:22
testing. There are about 6 fundamental
8:24
assumptions in standard practice.
8:26
First, they assume the load is carried entirely by the rivets,
8:29
completely ignoring any friction between the plates.
8:32
That's a massive simplification, especially since we know
8:35
clamping plates together generates a lot of friction.
8:38
It is, and it's highly conservative, which is good for
8:40
safety. 2nd, if the rivet group is centered on the load, they
8:44
assume the load is distributed equally. 3rd for the rivet
8:47
itself, the shear stress is assumed to be uniform across its
8:51
entire cross section. Uniform shear is definitely
8:54
easier to calculate. What about the plate material?
8:56
4th The bearing stress, the crushing between the plate and
9:00
the rivet is assumed uniform over an area of the rivet
9:03
diameter times the plate thickness 5th.
9:07
Tension in the plate is uniform over the net area and 6th
9:11
compression is uniform over the gross area.
9:14
They're shortcuts, but they work surprisingly well for predicting
9:18
the joints ultimate strength. Now here's where it gets really
9:20
interesting. You mentioned some surprising
9:22
test results about rivets used in tension, which kind of defies
9:26
common sense. Absolutely.
9:27
When rivets are driven hot, they cool and contract, creating a
9:31
huge preload. Often the initial tension is 70%
9:35
or more of the rivet materials yield strength.
9:38
Right, So I would assume that means you're already near the
9:40
breaking point before the load is even applied.
9:42
You'd think so, yet tests on hot driven button head ruins show
9:45
that they develop an ultimate strength in direct tension that
9:48
is actually greater than the strength of the original rod
9:51
they were made from. Really.
9:52
Yes, that high initial tension from cooling does not reduce the
9:56
rivet's ability to resist an additional applied tensile load.
10:00
It's a critical factor in why they're so elective.
10:02
That's fascinating, but what about when the load isn't
10:05
centered? What about eccentric loading?
10:07
When the load P is eccentric, say at a distance E from the
10:11
rivet group centroid G, we separate the problem into two
10:14
effects. We treat it as a primary force P
10:17
prime acting through the centroid plus a rotational
10:20
moment which is P * E The load from that moment is what really
10:25
stresses the fasteners. Assuming all the rivets are the
10:27
same size, the load Q and on any given rivet caused by the moment
10:31
acts perpendicular to the line from the centroid to that rivet.
10:35
And I imagine the ones farthest away take the biggest hit.
10:38
They absolutely do. The formula is Q = P * E * R sub
10:42
one, all divided by the sum of R-squared for all the rivets.
10:46
The fact that distance R is squared in the denominator shows
10:49
just how much that rotation concentrates the load on the
10:52
outermost fasteners. That squared relationship is a
10:55
huge deal. So now let's move from static to
10:57
dynamic. You said static and slow rolling
11:00
capacity are similar, which means the real challenge is high
11:02
speed motion. When you get cyclic loading,
11:05
especially in ball bearings, fatigue becomes the dominant
11:08
failure mode. It's the only failure mode that
11:10
matters in the long run, and this is where the physics gets
11:13
brutal Fatigue. Studies on bearings established
11:16
these critical power laws that govern their expected life.
11:19
The number of cycles to damage is inversely proportional to the
11:23
cube of the applied load for point contact like balls.
11:26
The cube. And for line contact like
11:29
rollers, it's slightly less severe, but still extreme.
11:32
It's inversely proportional to the 4th power of the load.
11:35
So let's put that into perspective.
11:36
If I designed a ball bearing to last 10 years at a specific
11:40
load, but I accidentally overload it by just 10%, what
11:44
happens to my 10 year lifespan? Well, because of that cubic
11:47
relationship, if you increase the load by 10%, a factor of
11:50
1.1, the life reduction is roughly 1.1 ^3, which is about
11:54
1.33. So a tiny 10% overload reduces
11:58
your component's life by about a third.
12:00
But here's the more extreme way to view it.
12:02
If you translate that load sensitivity back into stress,
12:06
the cycle life is inversely proportional to the 9th power of
12:09
the calculated Hertz stress for balls.
12:11
The 9th. The 9th power and the 8th power
12:14
for rollers, even a tiny variation in stress just
12:18
dramatically eats away at the working life of the bearing.
12:21
That is staggering. It really shows why precision is
12:24
so non negotiable. So what's actually considered
12:27
failure in these systems? Well, there are a couple of
12:30
schools of thought. Early on, some focused on
12:32
permanent deformation, but later experts like Palmgren argue that
12:37
a minor permanent deformation has no significant influence on
12:40
function. The industry standard now is
12:43
visible surface damage through spalling.
12:45
And that spalling starts. It nearly always originates at
12:48
those plates of maximum shear stress below the surface,
12:51
confirming everything we learned from Hertz and Bella Jeff.
12:54
We've seen the complexity in rolling elements.
12:57
Let's briefly pivot to another common connection.
13:00
Screw threads. Are they uniformly loaded?
13:03
Not even close. It's a classic engineering
13:05
assumption. That's just false.
13:07
The load in a screw joint is not transferred uniformly.
13:10
Analysis and testing both show that the maximum load is always
13:13
near the loaded face of the nut, and it falls off rapidly from
13:17
there. So there's a concentration
13:18
factor. A big one.
13:20
The thread load concentration factor H can be 2-3 or even 4.
13:25
A factor of 4 means the first few threads are bearing four
13:28
times the average stress. Exactly.
13:30
The rest of the thread material is essentially wasted in terms
13:33
of carrying the load. This phenomenon severely reduces
13:36
the efficiency of bolted joints, especially under repeated
13:38
loading where efficiency can be as low as 6 or 7%.
13:42
And just briefly, how do other components like gear, teeth or
13:45
keys fit into this? Gear teeth are a combo problem.
13:49
The bending stress is analyzed like a cantilever beam, but the
13:53
localized contact stress, the wear and fatigue where the teeth
13:55
engage. That's calculated using Hertz
13:57
theory for keys. Standard design uses simple
14:00
sheer assumptions, but photo elastic analysis shows sheer
14:04
stress is highest at the corners two to four times the average
14:07
value. So across the board, localized
14:09
non uniform stress is the rule, not the exception.
14:12
So what does this all mean? We've gone from the theoretical
14:15
purity of her scene contact to the messy reality of rivets,
14:19
screws, and gears. The definitive take away for you
14:22
is recognizing this intense exponential localization of
14:26
stress and that critical difference between surface
14:28
crushing and the internal tearing force, the subsurface
14:32
shear which dictates fatigue life.
14:34
It means that while theory provides the foundation, the
14:37
practical design of parts with direct bearing has to rely
14:39
heavily on experience, on manufacturated capacities, and
14:43
on testing charts. Theory alone just can't account
14:46
for wear, fatigue, material imperfections, all of it.
14:49
And that leads us to a final provocative thought built right
14:52
out of our sources. We established that true rolling
14:55
without slip is comparable to static loading, but even slight
14:59
slip dramatically increases the severity leading to wear
15:01
failure. So given that, and considering
15:05
that inverse 9th power law for life, how much more critical is
15:08
the integrity and composition of the lubricant film in high speed
15:11
machinery than the initial yield strength of the metal itself?
15:15
That delicate layer is the last line of defense against
15:18
exponential failure. Something to consider next time
15:21
you hear a bearing start to whine.