Hi. This is Module 16 of Mechanics of Materials Part IV. 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 the critical buckling load or the Euler buckling load for a column which was pinned-pinned on both ends. 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 wanted to show you what the results would be in those cases. So here's the pin-pin 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. 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. Then finally, as another example, if we had fixed-free end conditions, this is the critical buckling load that we would come up with. There are other end conditions but these are common ones. You can see in the denominator we have a similar term L squared, here 0.7L squared here, 0.5L squared here, and 2L squared here. We call that the denominator that's squared the effective length of our column. 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. So for the pin-pin condition, we can see that it's the effective length is the length of the column itself. For the fixed-pin condition, here's our inflection point. So our effective length becomes about 0.7L. Then for fixed-fixed, we see that the effective length is about 0.5L. Then for the fixed-free condition, we can see that now it's 2L and you can see when you look at these diagrams, these are the same distances that it looks like in the pin-pin condition. So here's a real world experimental setup of these types of end conditions. So you see here the pin-pin condition with the effective length being L, here the fixed-fixed condition with the effective length being 0.5L, here the fixed-pinned condition with the effective length of 0.7L, and then finally the fixed-free condition which would be 2L for the effective length. So we'll use those for column problems when we have different types of end conditions when we do actual real world examples. That's it for today's module.