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Â Welcome to Module 5 of Mechanics of Materials I.

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

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

Â a state of stress at a point in 3D.

Â This will be a very general theory.

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

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

Â It could be a bar like this, or

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

Â It could be a structural member in a building, or

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

Â 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,

Â the stress distribution may not be uniform on arbitrary planes.

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

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

Â But for an infinitesimally small point in that member,

Â the stress distribution does approach uniformity.

Â And so, for each point in the member,

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

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

Â perpendicular planes is sufficient to completely describe the state of

Â stress at any point for any orientation.

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

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

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

Â 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,

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

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

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

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

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

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

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

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

Â the second index is the direction of the stress.

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

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

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

Â positive z face in the y direction.

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

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

Â So for sigma xy on the negative face in the negative

Â x direction, it would be down.

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

Â all the other stresses.

Â And so, that's our positive sign convention.

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

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

Â A tensor represents a physical or geometric property or

Â quantity by a mathematical idealization of an array of numbers.

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

Â Advanced Engineering Systems in Motion or 3D Dynamics.

Â 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.

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

Â And now let's look at equilibrium.

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

Â And so, by static equilibrium,

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

Â or about actually the z axis,

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And so I've got about point A,

Â I've got sigma y times the area.

Â Remember, we're summing forces now.

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

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

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

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

Â opposite sigma y down here on the negative face.

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

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

Â Let's do the same thing for dx.

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

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

Â 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.

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

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

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

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

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

Â 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,

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

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

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

Â a counterclockwise rotation, its moment arm is d.

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

Â and then we've got a tau sub yx.

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

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

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

Â this term cancels with this term.

Â The d's cancel, the a's cancel and

Â what I end up in enforcing equilibrium

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

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

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

Â And so here is my state of stress.

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

Â These are equal, and these are equal.

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

Â I put my normal stresses on the diagonal.

Â And I put my sheer stresses on the off diagonal.

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

Â And also to show the positive sign conventions.

Â And we'll see you next time.

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