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Welcome to Module 5 of Mechanics of Materials I.

Today's learning outcomes are to define the state of stress for

a point in three dimensions and to define the sign convention for

a state of stress at a point in 3D.

This will be a very general theory.

It's a general 3D state of stress for an arbitrarily loaded member.

Here, I show an arbitrarily loaded member with a number of forces acting on it.

It could be a bar like this, or

it might even be some sort of a tubular section like this.

It could be a structural member in a building, or

a strut in an aircraft, or perhaps a part in a mechanical device.

So this theory's going to apply for any kind of arbitrarily loaded member.

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When we have more complicated members like this,

the stress distribution may not be uniform on arbitrary planes.

If you remember, up until this point, we said that the stress on a plane,

the normal stress, was uniform across the cross-section.

But for an infinitesimally small point in that member,

the stress distribution does approach uniformity.

And so, for each point in the member,

an infinite number of planes can be passed through the point.

But it can be shown, and we'll show later in the class that three mutually

perpendicular planes is sufficient to completely describe the state of

stress at any point for any orientation.

And so hence, we'll use a cube as our infinitesimally small point for

the state of stress because indeed, a cube has three mutually perpendicular planes.

And so here's our member, and we can cut into this member at any point and

shrink down and look at a cube, and I've shown a little cube in there.

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Okay, here is my state of stress,

my 3D state of stress at a point shown by a cube.

I've shown the stresses on the cube in what's the positive sign convention.

So I have a positive normal stress in the x direction on the positive x direction.

Likewise, and I haven't shown it, I've only shown stresses on the positive

phases, but likewise, I have a positive stress in this direction, sigma x.

Because it's in the negative x direction on the negative x face,

and so a negative times a negative is the positive sign convention.

For sheer stresses, I have, the first index is the face,

the second index is the direction of the stress.

So I have the sheer stress on the positive x face In the positive y direction.

Here's the stress on the positive y face in the positive x direction.

For the y face, we have positive y face in the z direction,

positive z face in the y direction.

And on here, we have positive z face in the positive x direction, positive

x face in the z direction, and again, the likewise would be true on the backside.

So for sigma xy on the negative face in the negative

x direction, it would be down.

And again, a negative times a negative is a positive, same thing for

all the other stresses.

And so, that's our positive sign convention.

Here, it's shown again, stress now is a tensor.

These normal and sheer stresses can be expressed as a tensor, they are a tensor.

A tensor represents a physical or geometric property or

quantity by a mathematical idealization of an array of numbers.

And I talked quite a little bit about tensors in Module 20 of my earlier course,

Advanced Engineering Systems in Motion or 3D Dynamics.

And so I'd like you to go back to that and do a review of tensors.

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by d, just arbitrarily calling that distance d for the cube.

Let's call this point down here A on the z axis.

And now let's look at equilibrium.

We want this state of stress to be in static equilibrium.

And so, by static equilibrium,

let's go ahead and sum moments about point A,

or about actually the z axis,

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And so I've got about point A,

I've got sigma y times the area.

Remember, we're summing forces now.

Stress is not a force, stress is a force per unit area.

So we've got sigma y times a, which will be the force on the top face

due to the normal stress, times its moment arm, which is going to be d/2.

And then remember, I'm going to have a equal and

opposite sigma y down here on the negative face.

And so that's going to be, that's going to cause a clockwise rotation, so

that's going to be -(sigma y A) d/2.

Let's do the same thing for dx.

Okay, for dx or sigma x, we have clockwise rotation here for this sigma sub x.

So that's going to be -(sigma sub x A), so

it's converted into a force times, again, d/2 is the moment arm.

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And then we have a corresponding sigma x on the back side.

And so, it's going to tend to cause a counterclockwise rotation,

so that's going to be + (sigma sub x A) d/2.

And as far as the sheer stresses y sub z, z sub y,

z sub x, and x sub z, they're not going to cause a moment about the z axis.

The only sheer stresses that are going to cause a moment about the z axis

are this tau sub yx and this tau sub xy.

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The corresponding sheer stresses on the back faces, or the negative sides,

on the y and the x plane are going to be, their force is going to go through point

A, so they're not going to have a tendency to cause a moment.

So we're going to have tau sub xy, it's going to tend to cause

a counterclockwise rotation, its moment arm is d.

And so we've got plus tau sub xy times the area times d,

and then we've got a tau sub yx.

It's going to be negative, because it's going to cause a clockwise rotation,

(tau sub yx A) d, and all of that has to equal 0 for equilibrium.

Well, we can see here that this term cancels with this term,

this term cancels with this term.

The d's cancel, the a's cancel and

what I end up in enforcing equilibrium

is that tau xy has to be equal to tau yx.

And you can do a similar type of equilibrium for the tau yz and tau zy.

They have to be equal and tau xz and tau zx have to be equal.

And so here is my state of stress.

I see that this sheer stress, this sheer stress are equal.

These are equal, and these are equal.

I can express as a tensor the state of stress in matrix notation.

I put my normal stresses on the diagonal.

And I put my sheer stresses on the off diagonal.

And so that's, as a wrap up, the way to express a 3D point of stress.

And also to show the positive sign conventions.

And we'll see you next time.

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