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I'm studying Euler's work on structural engineering from a book out of curiosity and it is mentioned that he developed a mathematical theory describing the buckling of columns under a parallel load (the weight-force of the load is directed down along the column). The theory is covered quickly without much motivation.

But this got me thinking; why does a column "buckle" in the first place? If the load presses the column down, why does the column even start deflecting sideways? I know this happens in real life since this fact is easily confirmeable with household objects, but theoretically, why do objects start deflecting sideways instead of just compressing under loads? This might be something obvious and maybe I'm just overthinking but I find this curious nonetheless.

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  • $\begingroup$ +1 for Excellent Question. $\endgroup$ – Mark Jul 13 '18 at 16:49
  • $\begingroup$ Euler buckling is mostly a historical curiosity. It is often used to introduce students to differential equations, as it was one of the motivating problems that led to an orderly study of ODE's. It remains part of the engineering canon for reasons unknown. It is seldom the limiting factor in real designs, but it does crop up is some stayed structures like sailboat masts. But if you have a practical interest in building things, you need a vastly more comprehensive awareness of failure modes and their analysis. $\endgroup$ – Phil Sweet Jul 13 '18 at 20:28
  • $\begingroup$ See if this explanation is more helpful. MIT The Column and Buckling $\endgroup$ – Phil Sweet Jul 13 '18 at 20:28
  • $\begingroup$ "The theory is covered quickly without much motivation." Exactly. It's been entrenched for so long, nobody even thinks about it anymore. That's the best part of this whole question. $\endgroup$ – Phil Sweet Jul 13 '18 at 20:33
  • $\begingroup$ @J... Neither are Euler Buckling examples. But the pop can is a classic real world problem. NASA $\endgroup$ – Phil Sweet Jul 13 '18 at 20:40
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Euler buckling occurs because the world isn't perfect. So that theory assumes that there is an initial infinitesimal deviation along the column (assuming the column is in fact not perfectly vertical*). This deviation causes a bending moment along the beam, which increases the deviation, which increases the bending moment, which increases the deviation...

For loads lower than the Euler load, this vicious cycle eventually stabilizes and the beam doesn't buckle. For the Euler load and above, the cycle never stabilizes and the deflection goes to infinity.

Obviously the real world has initial deviations and other problems which are much higher than "infinitesimal". So in the real world, columns buckle with loads far lower than the theoretical Euler load.

* This is the assumption for Euler buckling, but another possible deviation is that the load is actually not perfectly centered on the column. In the real world, both cases probably happen simultaneously

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Think about a "thin" beam, for example a strip of springy steel. It is very easy to bend the strip into a curve, compared with stretching or compressing it along its length.

When it is bent into a curve, the length of the strip measured around the curve does not change significantly, and that means the straight-line distance between the two ends becomes smaller.

If you try this experimentally with something you can bend easily with your hands, you will find that a graph of the force against distance between the two ends is not a straight line - the effective stiffness reduces as the load increases and the beam curves more.

On the other hand, the stiffness when compressing the beam along its length without bending it is constant (and equal to $EA/L$, as shown in any strength of materials textbook).

Since it is impossible to make a perfectly straight beam in the real world, the beam will buckle when the end load reaches the point where the stiffness in "bending sideways" becomes less than the stiffness in "perfect compression".

Euler's formula gives a fairly good approximation to that load, though it makes a few more assumptions (for example, about the shape of the beam when it bends sideways) which are not completely accurate. But since the tolerances in the beam geometry are also unknown, Euler's formula is good enough to be useful in practice, even though it usually over-estimates the actual buckling load by a factor of a few times (say between 2 and 5 times) compared with real life.

Because the beam becomes more flexible after it buckles, if you apply a constant end load (e.g. the weight of something pressing on the end of the column) the buckling results in catastrophic failure, as the beam curves more and more until it breaks. On the other hand, if you apply a controlled displacement to the end, the process is reversible and when the load is removed the beam will return to its (nominally) straight shape, with no permanent damage.

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  • $\begingroup$ How can you say that Euler's formula is good enough for actual practice if you're also saying the actual load can be 1/5th of the calculated value? Or do you mean that the method can be modified slightly (adding a "scaling factor" or something of the sort) in order for it to be used in practice? That's what the Brazilian code does: it calculates the Euler load and then applies a few reduction factors (not standard safety factors) to bring it down to a better approximation of the real world. $\endgroup$ – Wasabi Jul 14 '18 at 0:45
  • $\begingroup$ In fact,, that's exactly how buckling spring keyboards work - if you assume that any bar is slightly compressible, it effectively becomes a buckling spring! $\endgroup$ – KlaymenDK Jul 14 '18 at 7:11
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Not all the columns fail under compression by buckling. In steel columns shorter than slenderness ration of 50 they fail by direct compression.

It is the principal of stability bifurcation and it appears not only in columns but in failure mode of many other shapes, such as beams, trusses, vessels, and the buckling pattern could be quite complex. Fro example if you cut the cap and bottom of a can of coke and put it under a micro control press, it will buckle along diamond pattern on its wall, twisting around the vertical axis.

In columns it happens because of the elastic behavior of the material leading to bifurcation, be it steel or aluminum, wood, etc.

It is not due to residual imperfection in manufacturing of the column, neither due to load not applied at perfect centre, although those conditions will affect the reaction of column but that belongs to another topic.

As you increase the load applied to the column compression stress develops on the area of the cross section. This stress is applied evenly over the surface of the section, $$ \sigma = P/A$$ But this stress is constantly seeking out ways to force the column to curve so as to release the stress by creating small variation in intensity distribution over the surface area while the total stress is constant, hence creating a lateral momentum, but up to buckling force this virtual stress is just not enough to force the buckling. When the load reaches the buckling level the column will fail by randomly bending on either of two sides which have bigger slenderness ratio.

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If the load is applied through the centre-line of the column then there is no side force, but if the load is offset, but parallel, then there is a sideforce which leads to buckling.

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  • $\begingroup$ No side force is required, if the beam is not perfectly straight and uniform (and of course no real beam is geometrically perfect). $\endgroup$ – alephzero Jul 13 '18 at 16:33
  • $\begingroup$ @alephzero but Euler's formula assumes a perfect beam... $\endgroup$ – Solar Mike Jul 13 '18 at 17:24

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