How Does Beam Support Type Influence Buckling Direction?

[terryken]

Hey guys,

Got a quick question here; A simply supported beam of square cross-section is under compression. Which is the preferred direction for buckling? Why? Does your answer change if the beam is clamped at both ends?

 

[terryken]

any smart souls out there care to explain this?

 

[PhanthomJay]

May we first have your own thoughts on this? Forum rules require some attempt at an answer before we can asist. Thanks.

 

[terryken]

Ok, I'm new here, thanks for telling me the rules. Anyway here is what i think is right for now;

The squared cross-section beam has an identical area moment of inertia in any direction. That means for elastic buckling with clamped ends, a square beam has no preferred direction of buckling. It could buckle by deflecting parallel to one of the sides, or at 45 degrees, or any other direction. I would still obtain the same critical load in all cases.

And according to the formula derived by Euler for columns with no consideration for lateral forces. (Got from wiki)

F = (pi^2 * E * I) / (KL)^2

where
F = maximum or critical force
K = column effective length factor
For both ends clamps, K = 0.50.
For one end clamp and the other end free, K = 2.0

Since E, I & K does not change over length, it does not matter which direction is the preferred side to buckle.

 

[PhanthomJay]

The squared cross-section beam has an identical area moment of inertia in any direction.

It has the same area moment of inertia about its horizontal y and z axes, but not about axes at an angle to the horizontal

That means for elastic buckling with clamped ends, a square beam has no preferred direction of buckling.

That is not completely true, and is there a difference in the direction of buckling if the ends are simply supported versus clamped?

It could buckle by deflecting parallel to one of the sides,

yes

or at 45 degrees, or any other direction

No, that would be true if it was a circle, not a square.

I would still obtain the same critical load in all cases.

And according to the formula derived by Euler for columns with no consideration for lateral forces. (Got from wiki)

F = (pi^2 * E * I) / (KL)^2

where
F = maximum or critical force
K = column effective length factor
For both ends clamps, K = 0.50.
For one end clamp and the other end free, K = 2.0

Since E, I & K does not change over length, it does not matter which direction is the preferred side to buckle.

yes , true, but what is the value of I to use?

 

[terryken]

Thanks Jay for your helpful replay, anyway, the question was just: A simply supported beam of square cross-section is under compression. Which is the preferred direction for buckling? Why? Does your answer change if the beam is clamped at both ends?

There were no values given, just explaining the theory would suffice. So in general, is it safe to say that a square cross-section under compression has no preferred direction for buckling due to the fact that I would still obtain the same critical load in all cases, as shown in the formula above? Am i missing out some critical points in the explanation?

And for the the differences in the direction of buckling if the ends are simply supported versus clamped, there will not be any differences as again, the critical load are the same in all cases.

Please correct me if I'm wrong, thank you

 

[PhanthomJay]

The critical load for a beam with clamped ends is much higher than the critical load for a beam with simply supported ends, but the direction of buckling is the same. The question is, what is that direction? It is not just any direction, because I is not the same in all directions.