And now we can take our strain transformation, excuse me,

stress transformation equation for sigma sub n, the normal stress,

and we can substitute in cosine 2 theta sub P.

So this should be theta sub P now.

And sine 2 theta sub P, which will end up being theta sub P now.

So these, now we're talking about specific angles,

so these will be theta sub P and theta sub P.

And this is the result I get by substituting those in.

Okay.

Plus or minus.

We can, this value and this value is the same, so we can factor that out.

And we have these two terms times this entire value here.

We see that we have sigma sub x minus sigma sub y over 2 squared,

and then the square root of that.

So that leaves a square root in the numerator.

Okay?

And so here is our expression boiled down.

And that now gives us the normal stresses on these principal planes.

We have two of them, and

I've labeled them principal stress number one and principal stress number two.

Here it is again.

So, at some angle, theta, we rotate our block.

And at that angle theta we end up with what's called principal stresses.

And I've shown them.

They can be both positive.

They can be both negative.

One could be positive, one could be negative.

But you'll note in this algebraic,

in this development, I've considered max and mins to be algebraic quantities.

And so, as I said, they could both be positive.

I might have plus 1,500 and plus 500.

Or I might have plus 800 and minus 200.

Or I might have minus 400 and minus 1700.

But in our calculations for

our engineering problems, when we use the term maximum we're going to refer to

the stresses with the largest absolute value, or the largest magnitude.

And so, for example, I have sigma sub 1, maybe a 700 megapascals.

Sigma sub 2 is minus 1200 megapascals.

This one is in tension.

This one is in compression.

But if I'm talking about the maximum normal stress, I'm going to refer to

the one that is the maximum absolute value when we do engineering problems.

Okay.

Here's our result again.

We'll see from this that we can end up with what we call a stress invariant.

If I add these two sigma sub 1s and sigma sub 2s together,

I get sigma sub 1 plus sigma sub 2 on the left hand side.

On the right hand side, this plus and minus will cancel out this part.

And so, I'll have sigma sub x plus sigma sub y over two plus sigma sub x plus

sigma sub y over 2, or sigma sub x plus sigma sub y.

That's a very important result.

It's the stress invariance.

So what it's saying is, on any two orthogonal planes

the sum of the normal stresses are going to be constant.

And so, no matter how we turn the block, the sum of the normal stresses on

two orthogonal planes is going to be invariant, or constant.

And so that's where we will leave off this time.

Some important relationships.

And we'll continue on next time.

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