And so, if you remember back to module five, I said that for

more complicated structures, we may have a stress distribution that may

not be uniform, but we can shrink down to an infinitesimally small cube and

the stress distribution will approach uniformity.

So here's a graphic of that cube.

And I've shown the stresses, both the normal and shear stresses acting on

this state of stress at a point in their positive directions.

And then we, in module six, went ahead and we said okay let's

let the out of plane stresses in the z direction equal zero and

we'll come up with the case of plane stress.

And so again back at module four, this was what we did on the normal and

shear stresses on an inclined plane for uniaxial loading for an angle theta.

We found this was the normal stress and the was an expression for

the shear stress.

We found out that the max normal stress occurs at an angle of 0 or 180 degrees.

The max shear stress occurs at an angle of 45 or 135 degrees.

We also noted that as theta became greater than 90 degrees,

the shear stress and the shear force vector changed directions.

And one final note was important that the magnitude of the normal stress,

the maximum normal stress,

was equal to 2 times the magnitude of the maximum shear stress.

And so, now let's extend that for the more general case of plane stress,

where we now have on our cube, in two dimensions,

both shear stresses and normal stresses, and

we're going to want to find the shear stress and

normal stress on an arbitrary plane, at an angle theta.

And in general, if you recall back to module 15

of my course Applications and Engineering Mechanics,

we were able to draw what we called shear force and bending moment diagrams.

And you may wanna go back and review those modules in Applications for

Engineering Mechanics.

But I have a structure, a beam, with different types of vertical loads.

I can also even have an axial load if I'd like, but

I can cut the beam at any point and

find the internal forces acting on an infinitesimally small part.

So for instance, on this beam here,

I could go ahead and have a certain type of loading, I could make a cut.

And I would be able in general to find these stresses,

the normal stresses and the shear stresses, from the external loading.

But I would also like to find what the shear stresses or

the normal stresses are on an arbitrary plane.

And the reason for that is that we may wanna know where the max,

what plane does the max stress, normal stress or

shear stress occur, because we wanna make sure we avoid failure.

And I have a part here.

This is a piston with a connecting rod.

This is actually from a mini Baha vehicle

that one of our teams here at Georgia Tech built.

But you can see that there is a failure.

And whenever we have a situation with loading,

we wanna know what the shear stresses and

normal stresses are on various planes to make sure that we avoid failures.

And so here, again, I take a cut on an inclined plane.

I'm going to go ahead and define coordinates for

the inclined plane in the normal and the tangential direction.