Hi, this is Module 18 of Mechanics and Materials Part 4.

Today's learning outcomes are to define

something called the slenderness ratio,

and then discuss the value of

the slenderness ratio when

the Euler Buckling formula applies.

So as a very quick review,

these were the various conditions

that we've worked with for,

and conditions for Euler Buckling

and the effective lengths in each case.

So this is the one for pin conditions.

When we calculate it now,

when we calculate the critical buckling load,

we have to put in a moment of inertia.

So we got to be careful about

which axis we're bending about

because the axis with a minimum of moment of inertia,

will be the axis that will buckle

about because it has the least resistance to bending.

So let's look at a W365 by 45 beam, wide flange I-beam.

My question to you is if this column

is going to buckle and we're looking like at a top view,

which axis will it buckle about?

Will it buckle about the y-axis or will

it buckle about the x-axis?

You should say the y-axis because the area moment of

inertia about the y-axis is less than

the area moment of inertia about the x-axis.

So now, let's define the radius of gyration about

the axis of bending in terms of I

equals the cross-sectional area times r squared,

where r is this radius of gyration.

We'll substitute that into our equation,

and we can cancel A's,

and we find out that the critical stress

is equal to the critical force over the area,

or in this case Pi squared times Young's modulus

over the quantity L effective divided by r,

where that quantity L effective divided

by r is defined as the slenderness ratio.

So empirically, we can say that

the Euler Buckling load only

agrees well with experimental data if

for steel columns if it's greater than a 140.

So you see this, we

said before that the Euler Buckling load and

theory only applies to long slender columns.

The definition or from experiments we see that when we

say long and slender that means that L effective

over r has to be greater than 140.

So just to check,

let's look at that last example

we solved which was the trust problem.

Let's look at, we

had the dimensions for each of the members,

and so if we were to calculate r,

we'd find out that r is equal

to the square root of I over A, okay.

The radius of gyration.

That would be equal to the square root of,

here's our I for the round bars,

and here's our area,

and so we find out that r would be equal to 0.125 inches.

If we put that into our equation for L effective

over r of the slenderness ratio we see for this problem,

the slenderness ratio came out to be 575.

That's much greater than 140.

So certainly, this was

an example of where we could consider

these members as being easily or

appropriately analyzed using Euler Buckling theory.

So that's it for this module, see you next time.