[SOUND] Hi, this is module 16 of

Mechanics of Materials Part IV.

And today's learning outcome is to develop the critical buckling load or

what's called the Euler buckling load for columns with various end conditions.

So last time we developed a critical buckling load, or

the Euler buckling load for a column which was

pinned-pinned on both ends, and this was the equation we came up with.

We can develop a differential equation and appropriate boundary conditions for

other end conditions to find the Euler buckling load in those situation, and

I want to show you what the results would be in those cases.

And so, here is the pinned-pinned condition and

there is the Euler buckling load that we came up with in that case.

Another end condition would be pinned and then fixed, and if we did the mathematics,

this is the critical buckling load we would come up with.

If we had fixed-fixed end conditions on both ends of our column, this

is the Euler buckling load or the critical buckling load we would come up with.

And then finally, as another example, if we had fixed-free end conditions,

this is the critical buckling load that we would come up with.

And there are other end conditions, but these are common ones.

You can see in the denominator, we have a similar term, L squared here,

point seven L squared here, point five L squared here, and two L squared here.

And we call that that denominator,

that squared, the effective length of our column.

And so the effective length is the distance between two successive inflection

points or points of zero moment, and they must be modified for

actual end conditions so we're doing an ideal situation here.

And so, for the pinned-pinned condition,

we can see that it's the effective length is the length of the column itself.

For the fixed-pinned condition, here's our inflection point,

and so our effective length becomes about 0.7 L.

And then, for fixed-fixed, we see that the effective length is about 0.5 L.

And then, for the fixed-free condition, we can see that now it's 2 L.

And you can see, when you look at these diagrams, these are the same

distances that it looks like in the pinned-pinned condition.

And so here's a real world experimental set up of these types of end conditions.

And so you see here the pinned-pinned condition

with the effective length being L.

Here the fixed-fixed condition with the effective length being 0.5 L.

Here, the fixed-pinned condition with the effective length of 0.7 L and then

finally the fixed-free condition which would be 2 L for the effective length.

And so, we'll use those for column problems when we have different types of

end conditions when we do actual real world examples.

And that's it for today's module.

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