L4 Column Buckling

追加: 2026-06-01

[00:00:00]hello everyone and welcome back in this video we will be looking at a quick visual demonstration of column buckling in a two-story building to start let's put Euler's buckling formula up on the screen remember that this is the formula that determines the capacity of our column as a structural engineer our goal is to determine both the demand on a member and its capacity if the capacity of the member exceeds the demand then the design is adequate if not we need to change some parameters now it's important to note at the start here that when analyzing compressive members such as columns we need to broaden our perspective from a one-dimensional analysis to two-dimensional so what do I mean by that well let's first take a look at a member in tension and let's recall what that failure mode looks like and we're going to do that by pulling on this piece of electrical tape so you can see as I pull on this it starts to yield and then eventually I continue pulling it ruptures into two separate pieces and just in case you missed that I'm gonna do it again this time with a bit longer piece of tape and again as I pull on it note that there isn't really anything much going on here aside from it elongating in the direction that I'm pulling on it it starts to yield and then it ruptures so that wasn't really too exciting there wasn't a whole lot going on with that piece of tape as I said it didn't pop out of plane it didn't suddenly change direction it just continued everything was in one line and this would be true of every piece of electrical tape that I pull on or any material that I pull on you would see the same failure happen again and again and again so calculating the load capacity of a piece of tape in tension is very simple it's just the cross-sectional area times the yield strength and done now let's compare that to when I pushed down on some of these columns here with this model so I'm going to apply compression to this First Column here and you can see the buckling shape that forms is sort of like an s now let's jump to this column here now note that it's the same overall length it's the same material this sort of springy metal so theoretically If This Were the same type of analysis as intention we would see the same sort of shape occur however you can see we don't get that same s shape here instead we get this C shape so clearly there are some other factors at play here Beyond just the material and its cross-sectional area so hopefully you can see visually that compressive analysis is going to be a bit trickier than tensile failure analysis there are more than just those two variables tensile failure as I showed always happens in line with the direction you're pulling compressive failure however depends on some other factors and it can change so what are those other factors well again let's look back at our formula pi squared of course is a constant so that isn't going to be changing anything modulus of elasticity is a material property so that only changes if we're swapping to a different material so that leaves us with moment of inertia the unsupported length and this K Factor now recall that the K factor is a term used to modify that unsupported length of the column depending on its boundary conditions or in other words whether the top and bottom of the column are pinned or fixed and note again that these three variables did not show up at all in our tensile analysis when we measure the capacity of a member in tension we didn't care at all about its moment of inertia its length or its boundary conditions but let's get back to talking about columns um let's first talk about the first change in variable moment of inertia recall that in simple terms moment of inertia is a geometric value that expresses a given shapes resistance to bending so when I look at the cross section of this piece of foam I should notice that there are some axes that I can draw through the center of this shape that would yield larger moments of inertia or larger resistance to bending than others in fact the largest possible moment of inertia I could calculate would be about this axis so bending about this axis here would generate the largest moment of inertia and on the other hand the smallest possible moment of inertia would be calculated about this axis in other words I've just identified the strong and the weak axis of this particular shape therefore if I were given a scenario where the other two variables that unsupported length and the K Factor were the same in all directions around the column it would be pretty easy to calculate the overall capacity of that column I would just calculate the smaller moment of inertia plug it into Euler's buckling equation and done that's my overall capacity but if it's possible to have two different values in different directions for one variable is it possible to do the same with the other two variables as well well yes it is let's take a look at this model again now I'm going to start with these two exterior columns and then we'll take a look at what's going on with that interior column so again as I push down on these two exterior columns you can see that they Buckle in this sort of s shape so what is causing that to occur well as you can see these two columns have a beam framing into them along the front face of the building and they both also have a beam framing into them along this short face of the building so in other words they are braced in this direction and in this direction so their unsupported length is really only equal to one story now let's compare that to the interior column here as you can see it does have beams framing into it in this direction along the long side of the building however if I turn it this way hopefully you can see that there is no beam in this direction so there is nothing bracing this column in and out in this direction so the question is is this brace here enough to cut the unsupported length of this column and increase its capacity like it did with these two and of course as we saw earlier it does not now note that the buckling here for this column when I push down on it does not and cannot occur in that shape this way because it is braced by these beams here and in fact it would never Buckle along this front plane it's always going to Buckle in this direction right because in this direction along the front face of the building its unsupported length is just one story height same as this column however in this direction now it's unsupported height is two stories or in other words it's been doubled and if you pay particularly close attention to Euler's buckling formula you'll know that when you double the unsupported length of a column what you're really doing is cutting that capacity by four times because that length is squared in the formula so in other words when we provide these two beams for this column what we've done is we've improved the buckling capacity for this column in this direction however as I said column analysis is a two-dimensional game so if you improve its capacity in One Direction you really haven't affected the overall capacity of that column at all so this all leads us to the final question if we can have different values for the same variable about two different axes is it possible to create a scenario where we don't know for certain which axis will fail first well let's look at this column again it Buckles in this direction right buckles into the long side of the building now it's doing that of course because it's unsupported life is double what its unsupported line is in this direction but what if we were to provide a rectangular column here instead of this round circular column and what if we were to orient it such that it's strong axis is into and out of your screen in other words the strong axis bending is oriented in this longer unsupported length Direction well now we have created a scenario where we have a weak axis with a small moment of inertia but a shorter unsupported length and we have a strong axis with a large moment of inertia but also a larger unsupported line so which way would that column buckle well in that scenario we would need to run the numbers about both axes the small number of course would be the controlling value that determines the overall strength of the column so remember column analysis is a two-dimensional game pay attention to both primary axes and always choose the smaller number when determining capacity also be careful when you remove sections of floor near a column you may think that by removing floor area near a column you are removing load from that column and therefore removing demand and making that column's life easier but you may also be removing bracing for that column and significantly increasing its unsupported length and therefore decreasing its capacity which would obviously make the life of that column much more difficult so hopefully this visual demonstration of column failure buckling gives you a clear picture in your mind of how columns fail so that you can use the Euler's buckling formula more effectively