Definition of buckling as a sudden deformation

On Wikipedia, buckling is defined as follows:

In engineering, buckling is the sudden change in shape of a structural component under load such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled.

Why do we define buckling as a sudden change in shape? This definition implies that a column is perfectly straight until we apply a load over a critical limit, after which the column suddenly curves sideways. But in real life, columns are not exactly straight and loads are not applied exactly to the center lines of columns so there is a bending moment on the column (and therefore everywhere in the column) for any load, not just a load above a certain limit.

Classical Euler buckling assumes a perfectly ideal column, and for that buckling appears as a bifurcation solution after a critical load is reached (and not before), but a real life column is not ideal and any eccentricity on the column or the load means that a bending moment is technically present for any load, however small.

So are real life instances of buckling not strictly speaking buckling according to this definition?

• Deformation disappears once the load is removed, buckling is permanent. Mar 31, 2020 at 6:59
• @Solar Mike So plastic deformation is needed for buckling? Mar 31, 2020 at 11:29
• Take a drinking plastic straw, place it vertically on the table (with sufficient friction to keep it from slipping). Start adding pressure/weight on the top tip, very slowly... wait for it... there. That’s bucking. Pretty sudden right? It doesn’t have to be a hollow column but it’s more evident in that example. “Slender” columns will fail by buckling before the material would fail under ideal compression stress (usually by shear); that’s in fact the definition of what makes a slender column “slender”. Apr 1, 2020 at 2:33

Yes, buckling is a sudden change in shape. A unevenly made column that bows as a result of increasing stress would just experience deformation. Or a metal bar bent by a "strongman." Buckling is different from regularly occurring plastic deformation.

• But I don't get how this sudden change can happen. As the loads are always eccentric in real life, structures start to bend immediately for any load.. Mar 31, 2020 at 11:40
• @S.Rotos. Structures aren't perfect but they are close enough. This phenomenon was observed in the field and then theory put against it, not the other way around. Mar 31, 2020 at 13:49
• It's not even a matter of whether they are perfect or not. You reach a tipping point where the column deforms to a point that makes it more susceptible to deformation, so it deforms even further. There's an inflection in load vs deflection, and on the far side you get positive instead of negative feedback. A runaway. Apr 1, 2020 at 12:42

Defining buckling as "a sudden change of shape" seems to be mixing up cause and effect IMO.

What happens is that the load-deflection path of the structure bifurcates when the load reaches a critical value, and the structure then follows whichever branch requires the least energy.

Since the stiffness of the two possible paths are usually several orders of magnitude different, it looks as if there is a "sudden deformation". On the other hand if you apply a controlled displacement to deform the structure instead of a load, there may be no sudden deformation at all, but the force you are applying suddenly decreases.

You can also have situations where the post-buckling behaviour is stable and the buckling will reverse when the load is removed - for example "diagonal buckling" in thin plates subjected to shear loads, where the plate buckles (and wrinkles) along the direction of the minimum (compressive) principal stress, but still carries load in the direction of the maximum (tensile) principal stress.

Real structures do suffer from buckling.

Yes, real structures are never perfect. Columns aren't perfectly vertical, cross-sections aren't perfectly consistent throughout the span, the material isn't perfectly homogenous, and the loads aren't perfectly centered. That is all true.

However, for loads lower than the true buckling load (which is much smaller than the result from the Euler equation), all of these imperfections merely lead to deformations. If these deformations are lateral, the deformations will increase the applied bending moment, further increasing deformations. But this feedback cycle has a limit at which the second-order effects stabilize and we have a solid column.

However, a single grain of sand over the true buckling load will cause a sudden and immediate deformation of theoretically infinite amplitude. Obviously, real columns don't deform infinitely, they simply collapse.

For an intuitive visual prop, squeeze a raw strand of spaghetti between two fingers. Depending on how you apply the force from your fingers, you can very well deform it laterally in a controlled manner. As you slowly increase the force you apply, the strand will bow further and further out. And then suddenly it'll snap. In this particular example, the collapse mechanism would probably be via buckling, but I think it's a useful visual distinction between "stable deformations" (including second-order effects) and buckling.

• So buckling basically means collapse due to lateral force? Mar 31, 2020 at 11:49
• But then again some sources say that a structure can be stable post buckling. So my confusion lies in why that single grain immediately causes big deformation. Take your spaghetti example, when we press more the spaghetti bends more and then after some force it snaps. But where is the sudden deformation? Doesnt the spaghetti just bow gradually in proportion to the load? Mar 31, 2020 at 12:32

Because it's a failure. Once it starts, it weakens the structure and, assuming the load is consistent, the strength is reduced and it can now buckle easier in a feedback cycle.

Longer spans of wood can 'buckle' and not be a strength issue. Just yesterday I answered the question of what sized piece of wood to use to span between two posts and that would not carry any load, it was purely aesthetic. The failure mode would be for the top of the member, say a 2x4, to roll over. However, this isn't the formal definition of buckling because the member wasn't a compression member. Or is it? The top of the 2x4 is in compression and it has buckled to the side.

• Buckling specifically does not assume this, it assumes strain, not stress, is consistent. Apr 2, 2020 at 1:53

A column or, a bar, even an empty can of soda may have imperfections and or a nonsymmetrical loading and carry a load and deform without buckling.

Buckling happens when the load exceeds the critical loading and the deformation is sudden and large and can continue even without increasing the load anymore.

A column may and usually does have defects, such as drilled holes for bolts connecting to other members, residual stresses from manufacturing, and can still support loads. But it will buckle yet after it reaches its critical load.

A can of soda, if we load it by placing control weights on a metal plate on top of it, will deform imperceptively until the buckling load is reached, then it will be crushed in like an accordion suddenly and will keep wrinkling even if we don't increase the load.

Buckling happens at a stress level called bifurcation point. where the total elastic and plastic energy in the buckled member is less than it would be if the member kept straining.

TL:DR Buckling is a statics phenomena. Collapse is a dynamic phenomena.

So are real life instances of buckling not strictly speaking buckling according to this definition?

They may originate from a buckling event, but then things get complicated at a rate that isn't consistent with how we define buckling.

What's missing from your definition is that buckling is defined as a statics problem, and collapses are inherently dynamic in nature. In classic buckling, the jump to a new geometry happens very fast compared to any change in the geometry at the load points. Buckling in a column happens at a fixed length and some of the stress is instantly released. If forces reappear, the length continues to shorten and the collapse will continue.

The dynamic impact strength of a thin cylinder can be much higher than predicted by buckling theory if the duration of the impact is so short the deformation jump can't get itself organized (think billiard cues on a break and bow arrows hitting a bone). The critical buckling load is very sensitive to the rate of load application and release after buckling has occurred.

• Phil, i have had a lab when in college, I'm 73 now, and we did buckling tests among other things of a can of soup with top and bottom plates cut off carefully. I buckles constantly into spiraling opposing pattern, i rember it took 6 page to write down the cylindrical partial trig differential equations. I think one could search for it. Mar 31, 2020 at 7:53

In the real world, and even in most moderately complex numerical simulations, you are right: columns are not perfectly straight and any load applied will be slightly off center and apply a moment in all situations.

As an exercise take a ruler and push the ends together with your hands. At some point the center of the ruler will bow out in one direction or the other. If you maintain that same force the ruler will remain in the same shape (static equilibrium). Think about this from an energy perspective: while pushing on the ends of the ruler when it is straight there is zero internal energy being stored in the ruler because no displacement occurs. When the stick suddenly flexes your hands are now closer than they were before and you have applied force over that distance, this necessarily means you have done work on the system. That work is conserved as energy stored elastically in the ruler.

The ruler could be bent to the same state purely by applying moments on either end. During the initial phase of loading, by pure compression, before buckling the amount of moment needed to to get to this same state is offset by the increasing compression force. Eventually enough compression occurs that the additional moment needed approaches zero. You can derive theoretical limits to this for perfect geometries to your hearts content. In the real world due to a variety of factors this limit may be significantly less if you actually tested it to its limit.

Some of the potential deformed shapes that are possible can have huge deformations and wild shapes. If your structure was rubber you might be able to see that in the real world. For things like a steel I-beam buckling it will "buckle" according to these theories up until it yields and becomes plastic deformation.

Yes in the real world pure buckling according to the theories exists albeit with some correction factors needed for meaningful engineering decisions .. and overbuilding everything is always the best option.

This is probably something where a video is worth a thousand words. Check out some videos of actual buckling tests. Do you see how the deflection increases slowly at first as they pour in the sand and then suddenly the deflection increases a lot? That's buckling

https://youtu.be/l_HlbF4EoJs