Bending energy for a rod/column under compression?

[george95]

Hi,
I would like to calculate a bending energy of a circular rod. It should be elastic bending.
The rod/circular column should be loaded with some point load force at it's top. Any amount could be used, for example F=1,10,100 kN...


I know how to calculate a strain energy of a rod loaded with an axial force. But that formula does not include bending, just normal stress:

U = (σ^2*V)/2*E

Could somebody please help me with this issue?

Thank you.

 

[SteamKing]

I'm confused. Is the rod acting as a column or a beam? Is the load to be applied axially or laterally? If the rod is acting as a column, there will be no appreciable bending action until the critical load is applied axially.

 

[george95]

Hi SteamKing, and thank you for the reply.
My apologies for not being clear.
The rod is vertical, with fixed bottom end, and no restrained (free) upper end. Something like a cantilever beam, but rotated for 90 degrees. The force is suppose to be applied at the top of the rod, vertically (axially).

 

[cake of doom]

If there are no other forces acting your rod/column then trying to get it to generate bending strain will be difficult (as SteamKing stated). Are getting confused with buckling?

Do you have a free body diagram or any attempt at a solution we could look at?

 

[george95]

It may be I am wrong.



I attached the image.

If we take a z coordinate in a way that: z=0 at the ground level and z=L at the top of the column, and if y(z) is the the horizontal deflection of the rod at each point z, could we write an expression for the bending energy coming from the applied force F at the top of the rod?

 

[cake of doom]

Not until critical load is applied (as SteamKing mentioned). Are you familiar with Euler's formula for determining this?

 

[george95]

I think I understand now.
So after the critical load is applied the rod will undergo to buckling, and horizontal deflection y, times F (y*F) will create a momentum?

The F critical (buckling load) would be:

Fcrit = (pi^2*E* I)/(k*L^2)
k(support condition factor) = 2 (for one end fixed, other end free)

Fcrit =(pi^2*E*I)/(2*L^2)

I = pi/(4*r^4) for circular cross-section

Fcrit = (pi^3*E)/(8*L^2*r^4)

?

 

[cake of doom]

Now you can start plugging your section properties into that to get the critical load.

 

[george95]

Ok, and once I get the critical load, how can I connect that with getting the bending energy?
For example, let's say the length of the rod is 2meters. And radius of the cross-section is 0.03m
L=2 m
r=0.03 m
E = 2*10^8 kN/m2

Fcrit = (pi^3*E)/(8*L^2*r^4) = (pi^3*2*10^8)/(8*2^2*0.03^4)= 2.423 kN

Now how can I write an expression for the bending energy?

 

[cake of doom]

That is as good as it gets I'm afraid. Without introducing a lateral load, your column won't be subject to bending. Just buckling under compression.

 

[george95]

What about that horizontal deflection along with Force on the top? It creates some kind of a momentum, and therefor a bending of a rod? So some bending energy must exist?

 

[cake of doom]

That's why your column will need to be stiff enough to resist over the effective length. If a column goes into bending when it should be taking a compression load it means something has gone wrong.

 

[george95]

Thank you for the reply once again Cake of doom.
But this is not an actual project, with real measures. I am just trying to understand how to calculate a bending energy in this case.
Nevertheless even before the rod undergoes buckling, still the force on the top produces some sort of bending, right? So some expression for bending energy could be written?

 

[cake of doom]

Whether real or just theoretical, a load that doesn't exist cannot be accounted for. Using the critical load method; when it bends (buckles), it's gone. Resistance of bending becomes a moot point. You would have to introduce a horizontal load somewhere along the length of the column for it to experience bending. Or you would have to load it eccentrically.

 

[george95]

What about the bending before the buckling? A certain amount of it has to exist above the buckling limit?
I guess even an unloaded rod (that is loaded only with it's dead load - self weight) produces some sort of bending (insignificant bust still there is, right?) due to it's weight?

 

[cake of doom]

Real world example: Have you ever stood a straw on it's end (not one with a bendy middle bit)? Did you ever see it bend under its own wait?

You just haven't created the right conditions for what you are trying to achieve. In a stable column, any bending that does occur will be so small you'd never find it. Equations for columns are the way they are to specifically resist any bending.

To be able to run all the sums you are trying to run you would have to:-
Load the column eccentrically.
Place the load at an angle.
Place the column at an angle.

Thats all you got I'm afraid.