Welcome to Module 19 of Mechanics of Materials, Part 4.

Today's learning outcome is to

graph modes of failure between yielding and buckling.

So for modes of failure,

we're looking at when we have buckling,

the failure stress is going to be when we

reach the critical buckling load divided by the area,

and we put that in terms of

our slenderness ratio or L effective

divided by r. If we find failure due to yielding,

that's just the force in the member divided by the area,

and those are the two conditions

or the two modes of failure.

So here they are again.

We know that the failure yield stress

for steel is 36 ksi.

The Young's modulus is 29,000 ksi.

So I can plot these two failure conditions.

For yield, the failure will

occur when P over A is greater than

36 ksi regardless of the L effective over r. However,

when I plot this equation on for buckling,

I get this purple curve here.

We see that if we're to the right of that,

then our column is going to fail because of buckling.

So as long as we're inside this area

down here, we're okay.

The ideal column will not fail.

So we see there's a splitting point here for

the L effective divided by r, the slenderness ratio.

Where on the left-hand side,

if our slenderness ratio is less than this value,

we will fail due to yielding,

and we call the short compressive members.

To the right, when buckling governs,

we call those long compressive members.

Now, for actual columns,

if you actually plotted this and

looked at the actual conditions,

you would find out that for

real columns, not ideal columns,

that we don't have such a defined cutoff point here,

that we move in this direction.

Anything above and to the right is failure.

Anything below and to the left is safe.

We don't have failure.

So we define two points here,

one at 140 for the slenderness ratio,

and a one at a little bit less than 50.

Remember now, we said for

a slenderness ratio of 140 or greater,

all those buckling load occur, can be applied.

So these can be considered long compressive members,

and we're perfectly fine with using

the buckling theory that we've defined to this point.

For slenderness ratio is

a little bit less than 50 or smaller,

those columns are going to

always fail in terms of yielding.

We call those short compressive yieldings.

Now, for real columns,

we get something in between,

which is a medium length compressive member.

In those cases, if you go

on with more advanced looks at a column buckling,

you would have to use empirical formulas or

industrial codes to make sure that you stay

within the allowable stresses

and don't experience failure.

So that's a good overview of how to

treat steel columns and whether

they will be subjected

to yielding or buckling type failure.

So that's it. We'll see you next time.