0:00

[MUSIC]

Welcome to Module 7 of Mechanics of Materials Part III.

Now that we've finished up a review of how to find the sheer or moment at any

particular point in a beam, let's continue on with a theory for beam bending.

And today's learning outcome is to derive the strain-curvature relationship for

pure beam bending.

0:27

And so we'll start with a simply supported beam and

we'll apply moments to either end so that this beam is in pure bending.

It's under flexure with a constant bending moment and

there's no shear forces being applied and so we make some assumptions.

The first assumption is that our beam is symmetric

about the xy plane or the plane of bending.

Let me give you some examples.

The first example is a square or rectangular beam where if you

draw a line down the XY plane, you can see, if you put a mirror there,

that on the left and right hand side, it would be symmetric.

The same thing with this cross section.

We're symmetric about the XY plane.

Again, if I draw a line down the XY plane,

I can see that this is a symmetric cross section,

and this is a typical I beam that's symmetric about the plane of bending.

1:25

And we also make the assumption that plane sections remain plane.

And so, here's a demonstration of that.

I've drawn on this beam that I can deform quite a little bit,

so I can exaggerate the bending.

I've shown with a magic marker here, the plane sections, and you can see as I bend

this beam, that those plains remain plain, they don't warp in and out of the plain.

And so there's no twisting, so there's no twisting like this.

Actually if you twist like this, then the sections do not remain plain.

2:25

And so, here's my beam with the constant bending moment applied,

let's now look at a little differential element.

So, we'll take a slice out of that beam, and we'll go ahead and bend it.

And so, what I've got here is I've got my constant moment applied on either side and

here's my original position of my beam and an exaggerated position once it's bent.

And it's bent through an angle d theta and

when I bend this, I want to go back and look at my demo again.

When I bend this beam with constant moment on either side,

you can see that the top above the dash line, that section

3:08

of the beam is in compression, those get compressed together.

And at the bottom, the beam is in tension.

But there is a line or a surface or

an axis right along the center where the beam neither contracts or extends,

so it's not in compression or tension and we call that the neutral surface.

And I've identified the neutral surface here on my diagram as well.

And we say that the radius of curvature is from the point where we intersect for

the curvature down to the neutral surface.

3:48

And so the curvature is equal to Kappa,

it's given the symbol Kappa and it's equal to one over the radius of curvature.

And so, the next thing I want you to do is to tell me what this length,

this arch length dx is, in terms of d theta and the radius of curvature.

4:12

And you should say that okay, dx, this arch length, is equal to row,

the radius, d theta, the angle.

4:40

And now, I want to look at an axis or

a plane that's a little above the neutral surface or the neutral access.

It's up a distance y.

And so what I want you to do now is what is this distance for

this purple line from one end to the other of the beam?

5:00

And again by arc length you said, okay the radius is row,

the radius curvature minus y, so that's row-y times d theta.

That gives us the length of that line, that purple line.

And so I can substitute in for d theta from a relationship above,

which is dx over row and simplifying that, I get dx- y over rho dx.

And so let's recall back now to Part 1 of my mechanics materials course,

where I defined normal strain as the elongation per unit length,

okay, the change in length over the original length.

And so here for my example of this differential element,

we see that the original length of my purple

line before it deforms would be DX, but

then I subtract Y over row DX, so

it shortens by a distance, negative y over row dx.

And so that's my delta and therefore my strain-curvature relationship says that

epsilon in the x direction is that change in length, which is minus

y over row dx over its original length of purpose line which is dx.

And so that's the strain curvature relationship.

I can cancel dx in the numerator and the denominator, and

I end up with this relationship.

And so, we know that one over row is Kappa, the curvature.

And so I can express it either way.

6:40

We see now that strain is proportional to curvature, and

it varies linearly with the distance y from the neutral axis.

So the further we get from the neutral axis either above or

below, we're going to get more and more strain, and

that strain is proportional to the curvature of the beam.

6:58

And the Strain Sign Convention is the same way as used before.

If we're in tension, then it elongates, that's positive.

Here, where we took y being above the neutral axis, and

we were in compression, we saw that the strain was negative.

7:15

And this is independent of material.

I haven't talked anything about material so far.

And so, there are strains actually in the y and the z direction, due to Poisson's

effect which we talked about in the first course for mechanics and materials.

But there's no stresses because the beam is aloud to be free to deform laterally.

And so, therefore what we find is

pure bending in beams produces this uniaxial stress in the X direction.