Buckling of columns

Depends on the column length, area, material etc. Yes there is a way to find out, depending on your ability i can't say if it would be easy for you or not. I don't have my books in front of me, but do a quick net search for you...

http://www.engineersedge.com/column_buckling/column_ideal.htm

http://en.wikipedia.org/wiki/Buckling

http://www.efunda.com/formulae/solid_mechanics/columns/theory.cfm

http://www.engrasp.com/doc/etb/mod/stat1/buckle/buckle_help.html

http://www.cs.wright.edu/~jslater/SDTCOutreachWebsite/column_buckle.htm

Many resources on the net, do a bit of research, crunch some numbers, you'll find your answer.

It is not possible to apply a load greater than critical load to a slender column. Critical load, by definition, causes lateral deflection to increase without limit. It is the load required to cause the column to collapse.

Thank you for the answer, I just need to clear up some things.

It says in my book that a structure may be able to withstand the critical load, but this doesnt happen in linear elastic thery?

They thing I am trying to grasp here is: Is there a deflection affiliated with the Euler load? Because when the Euler load is proved u use beam deflection. So let's say that a column has just the Euler load, not more, not less. Will it buckle a special deflection that is allways the same?

I have also read texts trying to explain bucling with a special mechanism with two vertical rods, joined with a horizontal spring in the middle(can you picture it or maybe should I draw a schetch?). If the critical load('Euler'), load is reached here, the rods will only be deflected as much as the disturbing force(not Euler load), made them deflect, and they will stay in this position. Does exactly the same apply for columns, that if the Euler load is reached, the deflection will be decided by how the disturbing force acted, and not specifically on the column itself?

I hope you understood my problem, please tell me if anything was unclear.

There is no deflection associated with the Euler load. When the column load approaches the Euler load, the column is on the point of collapse and will do so suddenly and without warning. Deflection will continue to grow without limit because the column has become unstable after buckling.

I don't fully understand your special mechanism with two vertical rods joined in the middle by a spring. If both rods are subjected to the Euler load, they will fail in unison and the spring will play no role in the failure. If one rod is subjected to the Euler load and the other is unloaded, then the bending rigidity of the unloaded rod will assist the loaded rod and tend to prevent collapse. I am not sure if that answers your question.

Thank you very much Bruce, I think I am starting to understand buckling now. Here is the link to the mechanism:

http://img235.imageshack.us/my.php?image=bukk.jpg

I was maybe a bit unclear, it is the mechanism as a whole that "buckles", not like ordinary columns, but you can picture how in the link.

Just as the "Euler load"(calculated by the propertis of the spring, and lengths of the rods only, not an ordinary Euler load) is reached on this mechanism. It will be in equilibrium in every buckled position.

I read you said that when the Euler load is reached for a column it will collapse. But hypothetically, if it precicely the Euler load, and and we look at the column in a pure theoretical way, will it behave like the above mechanism and be in equlibrium for every buckled position? And not collapse if there arent any disturbing forces?

(Ofcourse we assume positions for which linear theory applies.)

The attached picture is from the reference you provided. You called it a mechanism. It could be a mechanism, but is not necessarily a mechanism. If the spring is very stiff, it is not a mechanism. If the spring is very limber or does not exist at all, it is a mechanism. At what point does this simple structure become a mechanism?

Under a load of P, the upper point moves down a distance, say dy. Simultaneously, point A moves to the right a distance dx and compresses the spring. You can use the so called energy method to determine the critical load for this structure.

Work done by the force P going through a distance dy = P*dy/2.

Work done by the spring being compressed a distance dx = k*dx²/2 where k is the spring constant.

If P*dy/2 < k*dx²/2 then it is not a mechanism, otherwise point A continues to move to the right and the structure becomes a mechanism.

The distance dy may be related to the distance dx by geometry. For a spring with any given spring constant, it is easy to calculate the critical load by simply equating work done by the load P to work done by the spring. When they are equal, P becomes the critical load for that particular structure.

Buckling of a slender column is somewhat similar. You may assume a deflected shape for the column. If its bending stiffness is unable to prevent deflection from growing without limit, the column is unstable and will buckle. For a slender column, it happens at the critical load. For short and intermediate columns, it happens at a lower load because the material begins to yield, so that the theory of elastic buckling is no longer valid.

Is it possible that a column will not collapse at or above critical load in the absence of disturbing forces? It is about the same as asking if the three hinged structure shown above with spring removed and θ = 0 can still carry load. The answer is no because it is unstable. When a column reaches its critical load, it becomes unstable and cannot be relied upon to carry load.