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.