How to determine displacement due to buckling

[Seppe87]

Hi there,

I was wondering how one can calculate the displacement due to buckling. I already checked the general displacement calculations, but they didn't amount to anything.
So a simply supported beam under compression.

Thanks!

 

[AlephZero]

if you are talking about Euler buckling of a column, it doesn't make sense to "calculate the displacement due to buckling". When the column buckles, it becomes unstable. The math says the displacement increases without limit. In practice, either something breaks, or for a redundant structure the load path changes.

Euler buckling gives you an estimate (usually not very accurate) of the load that will cause buckling. To model what happens after buckling starts, you would have to do a nonlinear analysis including the behavior of the structure for large displacements, nonlinear materials (e.g. plastic behavior of ductile materials or cracking for brittle materials), imperfections in the geometry of the structure (e.g. the effect of manufacturing tolerances), etc, etc.

 

[Seppe87]

wow, okay. I thought as much when I was looking at my books, not a single example of a calculated displacement. But is it possible to counteract the buckling effect? Like applying your compressive force off centre so that there is an induced moment which counteracts the displacement due to buckling. Or is buckling more like: when you reach the critical load, all hell breaks loose and there is nothing you can do :p

 

[SteamKing]

That's why buckling is a phenomenon of structural instability. In order to prevent buckling from occurring, you have a limited set of options:

1. Reduce the compressive loading.
2. Change the geometry of the structure

Option 1 is straightforward. Option 2 can be accomplished in a number of ways. For instance, you can use a member with a higher radius of gyration, you can reduce the unsupported span, you can use a prop, etc.

Eccentrically loading beams in general does not prevent the onset of buckling.

For more information, see: http://people.eng.unimelb.edu.au/montyjp/Mechanics/notes_chapter6.pdf

 

[AlephZero]

But is it possible to counteract the buckling effect?

For a column carrying a load $F$ load in compression, the compressive stress is of the order of $F/A$ where $A$ is the cross section area.

The Euler buckling load is some constant times $EI/L^2$ where I is the second moment of area of the cross section and $L$ is the length. (The constant depends how the ends are restrained).

The best way to eliminate buckling is not to have any columns in compression, but sometimes that is hard to do. You might be able to pre-stress the column with a tension load, like the spokes in a bicycle wheel for example.

If you want to make $EI/L^2$ bigger, sometimes you can make $E$ bigger (change the material) or $L$ smaller (change the geometry of the structure). But the easiest way is often to keep $A$ the same (i.e. use the same amount of material) but make $I$ bigger. A hollow circular or square tube can have a much higher buckling load than a solid rod with the same cross section area.

 

[Seppe87]

Thanks for all your help! No doubt I understand buckling better now. Thanks in particular to SteamKing, your link has provided me with the necessary tools (buckling with eccentric loads can provide a specific amount of displacement). Thanks AlephZero for your help as well, but unfortunately I'm not in the position to change my A, I, E or L, so I'm afraid that didn't help me, but it's good to know!

 

[Er.Paresh]

Hey Seppe87. I wud like to give some more details about Buckling. It is very critical and thus its prevention is necessary rather than cure. Practically, while designing the column when Buckling criteria is considered, the given load is multiplied by a Factor of safety (FOS). suppose FOS is taken as 3, then the Buckling load is taken as 3 times the given load. Then using Euler's relation, as told by AlpheZero, dimensions of the column are found out at this load. Thus the column is designed for 3 times the actual load. However, Euler's relation has limitations. It can be applied only if the slenderness ratio (i.e. ratio of effective length of a column to the least radius of gyration) is less than 80. Else, Rankine's relation is used. Thus Buckling has to be prevented by proper design. If it fails, and column begins to buckle, then you can only go for additional strengthening of column- for eg. if u have a steel column of I-section, then you can strengthen it by welding suitable thickness plates on both its flanges... Hope this helps u understand better.