Column under a concentric axial load (Buckling)

[okinaw]

Hi everyone,

I have a cylindrical column under a concentric axial load. The load which is applied in the column is less than the critical buckling force (according to Euler´s formula). Due to this load, the column suffers a displacement (buckling effect), and I would like to determine which is this displacement.

I hope you understand the question.

Thanks in advance,

 

[Nidum]

I think I know what you mean but let's be certain . Please draw a diagram .

 

[okinaw]

Thanks for your answer Nidum. I would try to be certain.

The column is fixed in both ends. The load on the column is applied through the center of gravity of its cross section. The applied load (F) is less than the critical buckling force, which is calculated by Eulers formula Fcr = (π^2* E*I) / (αL)^2, where
E = Young´s modulus
I = Area moment of inertia of the cross section,
L = length of column,
α = Column effective length factor.

Due to the applied force, the column suffers a displacement as it is shown in the diagram (d). ¿How can I calculate the displacement d?

 

[Nidum]

Simple answer is that if it doesn't buckle then there is no displacement .

More complete answer though is that it depends on eccentricity of the load . There is a value for this eccentricity above which the column could deflect like a simple beam without actually buckling .

Not always easy to analyse because eccentric loads can cause premature buckling or runaway deflection anyway but it is possible to find cases of stable deflection .

There are other situations where deflection can occur without buckling . These involve beams with initial curvatures and beams with lateral loads .

Generally in this type of problem look for any geometric or loads inputs that could cause an initial lateral beam deflection .

 

[okinaw]

So, to summarize:

If there is no eccentricity and the applied load is less than critical buckling load, there is no displacement. Am I correct?

If a displacement happens, it is because there is an eccentricity and due to this eccentricity the column bends. In this case, the displacement would be calculate by the secant formula?

Thank you very much,

 

[Nidum]

If there is no eccentricity and the applied load is less than critical buckling load, there is no displacement. Am I correct?

Yes .

If a displacement happens, it is because there is an eccentricity and due to this eccentricity the column bends. In this case, the displacement would be calculate by the secant formula?

Yes for simpler problems . More complex problems sometimes require other methods .

 

[mle]

If no imperfections exist (eccentricity or other) and the load is below Euler's: zero displacement
If no imperfections exist and the load is equal or bigger than Euler's: there is displacement but you can't determine it with simple theory, you need to condider large displacement theories.

With imperfections present, you have displacements below Euler's load (actually from the beginning). Their magnitude depends on how close you are to the Euler load, the imperfection type and magnitude. They are not linear relative to load!

 

[Dr.D]

The OP has said nothing (that I saw) about the length of his column. The Euler formula is far too optimistic for short columns. Get a good mechanics of materials book or a machine design book and look up "short columns." I suspect this is where you have a problem.