Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bending energy for a rod/column under compression?

  1. Apr 17, 2013 #1
    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.
     
  2. jcsd
  3. Apr 17, 2013 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    2015 Award

    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.
     
  4. Apr 17, 2013 #3
    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).
     
  5. Apr 17, 2013 #4
    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?
     
  6. Apr 17, 2013 #5
    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?
     

    Attached Files:

    • rod.jpg
      rod.jpg
      File size:
      8.7 KB
      Views:
      67
  7. Apr 17, 2013 #6
    Not until critical load is applied (as SteamKing mentioned). Are you familiar with Euler's formula for determining this?
     
  8. Apr 17, 2013 #7
    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)

    ?
     
  9. Apr 17, 2013 #8
    Now you can start plugging your section properties into that to get the critical load.
     
  10. Apr 17, 2013 #9
    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?
     
  11. Apr 17, 2013 #10
    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.
     
  12. Apr 17, 2013 #11
    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?
     
  13. Apr 17, 2013 #12
    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.
     
  14. Apr 17, 2013 #13
    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?
     
  15. Apr 17, 2013 #14
    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.
     
  16. Apr 17, 2013 #15
    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?
     
  17. Apr 17, 2013 #16
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted
Similar Discussions: Bending energy for a rod/column under compression?
  1. Column heading (Replies: 4)

  2. Will it bend? (Replies: 26)

Loading...