So we're looking at column buckling of a simple column,

long straight prismatic bar,

subjected to compressive axial loads.

Here it isn't it's deflected shape.

Buckling is when a stable equilibrium becomes unstable.

So during the initial compression,

if we do a slight perturbation laterally,

the load could be removed

and the column would return

to its straight configuration.

But there is a point at which it

becomes unstable so that when buckling occurs,

a critical value is reached which when

the beam or the column is perturbed laterally,

it will not return to its straight configuration.

So for long slender columns,

the critical buckling occurs at

stress levels below

the proportional limit of the material.

So therefore, this type of

buckling is in elastic phenomenon.

We call this simple column buckling

Euler buckling for long slender columns

and it's named after Leonhard Euler,

who was again a very famous Swiss mathematician.

Here, I show being configuration pinned on both ends.

If we put it under compressive loads,

this shows the column

and the shape is exaggerated when loaded.

We want to find,

what is the minimum axial compressive load

that will cause buckling?

So here again is the situation.

I've added a set of coordinates at

the center of the beam with X up and Y to the left,

and I said that the beam

above the origin here is l over two.

We're going to say that the deflection

at the end of the beam,

at the top is going to be delta.

Now let's look at a slice.

Let's go ahead and cut this column.

We'll cut it at a distance x.

At x, we'll have some deflection y.

So here is a free body diagram

of the top portion above that cut.

We have our P force down.

We're going to have to have an equal P force

up and we're going to have

to have a moment reaction to

keep this in static equilibrium.

You should be able to go ahead and

solve for the moment reaction using

your static equilibrium techniques on

your own and then come on back and see how you did.

So first of all, I'm going to say, "Okay.

Well, the distance from here to here,

for the cut is l over two minus x and the distance

from where this cut is on

our beam out to where the P force is applied,

is going to be delta minus y here.

So that's a good sketch.

I can now sum moments about this point down here.

I have the differential equation for

the elastic curve that we talked about before.

If we sum moments,

we have the moment that's balancing MR due

to the force P is P

times its moment arm or delta minus y.

So we can rearrange that equation and now

you can see that we

have a differential equation for column buckling that is

in terms of the coordinate y.