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[SOUND] Okay.

We've come to the last topic in the course for mechanics and materials part four.

And that's to go over again the failures, or theories of failure.

And so the learning outcome is to review the normal outcome stress failure,

which we've talked about in my before, the maximum shear stress theory,

which we've also talked about before,

in fact we talked about these in my mechanics and materials part three course.

The maximum normal stress we talked about all the way back to my mechanics and

materials part one course.

And then finally,

we're going to explain something called the maximum distortion energy theory.

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and so the failure theory, we went all the way back to my mechanical

materials part one course today, simple tension test, and

it was easy to perform and provided very good results for a variety of materials.

So we put a tensal force, we looked at the stress strain curve,

we predicted yield, we predicted fracture, we can find Young's Modulus,

a very nice test, very useful test.

But what about when we have more complex loading conditions,

when it's not just the simple axial load?

By have bi-axial loading or tri-axial loading.

Well, in those cases, the cause of failure may be unknown and

there are several theories that can be used for predicting failure for

these various types of loading.

And I'm going to focus on just three.

The first one we will review is the Maximum Normal Stress Theory, that says,

that when the normal stress in our actual engineering element is greater

than what's defined as our failure stress, then we're going to experience failure.

And it assumes that the material is subject to a combination of loads.

And it fails, can be a combination loads and

it fails when the maximum normal stress at any point exceeds the actual

failure stress as determined by that simple tension test, and

we said this by experiences was generally good for just brittle materials.

It was not good for ductile materials like steel, aluminum, plastics, etc.

For those ductile materials we also looked at the maximum shear stress theory

which is also called Tresca's Yield Crieterion and

this said that failure occurs when the shear stress, the actual shear stress,

is greater than what we define as our failure shear stress.

And this is good for ductile material and

that's because in yield, ductile materials.

Usually the yield is caused by

a slippage of crystal planes along the maximum sheer stress surfaces.

And so here's our simple tensional test.

Here's a stress block with that tension on it.

If we draw Mohr's Circle we see that tau failure is one half of

the normal failure based on the simple tension test.

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So failure will occur in this case when the maximum shear

stress at any point for a complex load reaches the failure shear stress,

which is equal to one half the normal yield stress,

or failure stress, as determined by the simple tension test for the same material.

So for the actual conditions the Mohr's Circle may look different but

this is the condition for failure.

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Finally, another very common and useful theory of failure is

the maximum distortion energy theorem or what's called von Mises Yield Criterion.

Now I'm not going to go into the details of the development of the theory.

You can do that on your own or take a more advanced class.

But this is generally regarded now as the best yield criteria for ductile materials.

Better than the the Tresca criterion in most cases.

And this states that failure occurs when the strain energy of distortion or

the change of shape of the element reaches a critical value.

Now when I talk about strain energy that's a concept that again I haven't

covered in this course.

You would have to cover it in a more advanced course.

But alternatively, you can also think of this theory, and

it gives the same results as yielding occurring when the sheer

stress on what we call an octahedral plane, which is a critical value.

And the octahedral.

Octahedral plains or

any plain whose normal makes equal angles with the three principle axis.

And so if you're interested in this, I'd recommend that you look at more

advanced discussion of the max distortion energy theorem or

the Von Mices Yield Criterion on your own.

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And if you do that you'll find

we're just dealing in this course with 2D plane stress.

We're going to say that the outer plane stress is equal to zero and so

by the maximum distortion energy theory this is equation we'll look at.

The failure stress squared is equal to the principle stresses squared-

the first principle stress times the second principle stress.

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Okay, and so, now let's wrap up by applying these failure

theories to the combined loading example that we did in the last module.

This was the loading condition, this was the stress

at point a and then we looked at more circle to find and orientation and

a prediction of the maximum principles or the principles stresses and

the maximum in plain share stress shown here, and so there are those values.

If we were using the various failure theories, let's see what we would say.

For the maximum normal stress theory, our maximum normal stress in nine point one

two mega pascals in tension and that would be, need to be,

less than or equal to what we defined as the normal stress in failure.

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For the maximum sheer stress theory, in this case we found the maximum

sheer stress for this complex loading was 4.79 mega Pascals, and

that would need to be less than the failure sheer stress as defined earlier.

Or, not earlier, but as defined for whatever material you're using and

then finally for maximum energy, distortion energy

theory I'm going to use that formula that I just showed on the last slide.

I'll put the values in and I find out that it ends up being nine,

there should be an equal sign here, and this is sigma failure

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and it's equal to the square root of these values, and I've put those in.

You find that the calculated stress for this theory must be less than,

which is 9.36 must be less than or equal to whatever the failure.

Our yield stress is, if yield is the definition of failure, for

the material that you're using.

And so, that's a good over view of the theories of failure.

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