Critical Load in Buckling problem

[hives97]

I experimented column buckling analysis with different two load conditions in MSC.Nastran.

In the first condition, force was loaded from the downward direction at the top of the column.

In the second case, force was inflicted toward horizontal direction.

But P[critical] of two cases are almost same. (at least same order)

I thought that scale factor of first case must be much larger than second case.

Because horizontal force can bend the column more easily !

Is my intuition wrong ?

 

[MikeLizzi]

Is your intuition wrong? Maybe. Maybe not.

Buckling requires some non-symmetry in geometry or forces for the effect to get started. When you try to buckle a beam horizontally the beam may already have some sag in it do its weight. That sag will not be there in a vertical arrangement. Some other kind of deformation may exist in a vertical arrangement but the magnitude is usually smaller than the sag from a horizontal arrangement. I have not made any numerical analysis but the results of your experiment indicate that, for the material and geometry you used, the sag in the horizontal arrangement is not significant.

 

[Dr.D]

If you impose the load from the side you simply have a bending problem. If you impose the load axially, you have a stability (buckling) problem. Thee should not be any critical load when loading from the side, but there definitely is for axial loading.

Contrary to MikeLizzi says, buckling is not dependent on a lack of symmetry in either the geometry or the loading. It is a loss of stability, not a matter of bending. These are very different situations.

 

[MikeLizzi]

Contrary to what Dr. D says my statement is correct.

A perfectlly symetrical axially loaded material will just compress. The loss of stablility that Dr. D refers to comes from the non-symetry.

 

[Dr.D]

MikeLizzi, by your reasoning, a perfectly symmetrical column can never buckle!

Then I have certainly misled a lot of students over the years. But I am in good company with Timoshenko, Den Hartog, and a host of others.

Would you like to explain why you think as you do?

 

[MikeLizzi]

Take a perfectly symmetrical column, symmetrical down to the crystalline structure, with a perfectly symmetrical axial load.

No force in any direction except the axial means the column gets shorted and fatter. If the crystalline structure has no weakness, no slip-planes to provide that instability you are talking about, where can the buckling come from?

The example of buckling in my dusty old textbook places that the axial loads not quite thru the axis. You probably could recite that example from memory.

We've both used the buckling rules for columns. I don't have to tell you they are empirical rules that combine the always-present errors in geometry and discontinuities in composition to come up with a fudge value for the onset of buckling even when an engineer has done his best to line everything up.

I think we both agree as to what will happen in real life. I was responding to what I understood to be a theoretical question.

 

[hives97]

Dr.D
Would you like to see my attached pictures ?

I made two analysis examples. One is a linear static analysis which depict bending problem, and the other is a buckling analysis at which vertical load was imposed.
On both cases, I used same FE model and imposed same magnitude of force.

According to your opinion, the results of two cases should be same ! But they are not.
Please comment about that.

 

[Dr.D]

hives97:

My first statement above was this:

If you impose the load from the side you simply have a bending problem. If you impose the load axially, you have a stability (buckling) problem. There should not be any critical load when loading from the side, but there definitely is for axial loading.

As I look at your pictures, it looks to me like you have two different problems, one a buckling problem due to axial loading, and one a bending problem due to transverse loading. Where does that contradict what I said?

You say that you made and FEA analysis of the buckling problem. What do you know about the internal workings of your FEA program for buckling? What exactly does it do? Does it solve an eigenproblem? Does it compute the solution for the elastica? In particular, do your lateral deflections in the buckling solution have any absolute value, or are they only relative values? It the drawing is simply a scaled eigensolution then it is relative values only and that is quite different from the lateral displacements computed for the bending problem.

 

[Dr.D]

MikeLizzi:

You said, "I don't have to tell you they are empirical rules that combine the always-present errors in geometry and discontinuities in composition to come up with a fudge value for the onset of buckling even when an engineer has done his best to line everything up."

I would agree that short column theory is largely empirical because we know that for short columns, Euler column theory does not hold up.

On the other hand, the Euler theory of column buckling is a well developed subject of theoretical mechanics with a long history. Some of the earliest work was done by Jacob Bernoulli in 1691. There was later work by Daniel Bernoulli and Leonhard Euler in the 1700s. The problem that they considered is based on the assumption of a completely uniform, homogeneous bar (so that it has no irregularities or internal structure at all) with known elastic modulus and dimensions. The result is a mathematically complex descripton that can only be solved using elliptic integrals. The curve formed by the buckled column is called the elastica. This is considered in considerable detail by A.E.H. Love in The Mathematical Theory of Elasticity. Euler's column theory works very well for long, straight columns.

 

[nvn]

For an ideal, theoretical, perfectly-straight, perfectly-symmetric Euler column, you cannot derive the buckling eigenvalue, i.e., you cannot derive the axial load causing neutral equilibrium (the point of bifurcation of equilibrium), unless you initially deflect (bend) the column laterally by a small bending displacement, y. Go back to Leonhard Euler's derivation in 1744, and you will see that the neutral equilibrium (buckling load) calculation applies to the bent configuration. If the column remains perfectly straight, it remains in stable equilibrium under any axial load P, which is known as the trivial solution, and is not a solution of the buckling load. The column must be initially bent, by a minute amount, to obtain the Euler buckling load, even for an ideal, theoretical, perfect column.

The difference, then, between the theoretical column and a real column is that the theoretical column is initially, laterally displaced by a minute amount, but with no explanation of how it got there, whereas the real column has a reason for being in this position. This is of little consequence, because no real column is perfectly straight, perfectly symmetric, nor has a perfectly symmetric microstructure.

 

[Dr.D]

nvn:

I agree that is how Euler approached the problem, but Euler did not have the last word on it. A.E.H. Love shows in his book (The Mathematical Theory of Elasticity), Art. 265, that the energy stored in the bent column is less than that stored in the column with simple axial compression for axial loads greater than the critical loads. Nature is notorious for preferring the minimum energy state, therefore when the axial load exceeds the critical load, the column will buckle without any load misalignment whatsoever.

I agree that the real columns always have misalignments and imperfections that make these fine points of theoretical interest only.

 

[nvn]

Dr.D: By your own admission, the Love calculations, like Euler, are based on a bent configuration, which means, somewhere in his derivation, he assumes a bending displacement, y, or an initially bent shape, albeit extremely small. Otherwise, there is no way to compute the bending strain energy or bending moment. He likely assumes this y parameter with no explanation of how the column got there, knowing that all real columns would satisfy this fundamental assumption of the derivation.

 

[Dr.D]

The point of Love's analysis is this: At any particular value of axial load, there are two possible deformation modes: (1) simple axial compression, and (2) bending. We can assume that the column is in either mode and compute and compute the associated stored energy. Nature routinely chooses the state with the lower stored energy, or saying this the other way around, the higher stored energy state is unstable.

When the axial load exceeds the Euler critical load, the energy associated with the simple axial compression mode exceeds the energy stored in the bending mode. Therefore nature will choose the lower energy deformation mode and the column will buckle. This will be true for a column of infinite perfection.