Column problem 1

[Samplaw]

Hi there,

I have a column with 2 rigid ends (L/4) attached to pinned connections (the upper on being on rollers) then an elastic section in the middle for the other L/2 length of the beam. This is subjected to a load P. I don't even know where to start as other column buckling problems have just been dealing with fully elastic beams.

Any help towards this would be appreciated.

The solution is numerical as stated in the question

 

[BAretired]

I think I understand the problem, although your description leaves something to be desired. I believe you are dealing with a column of length L supported by a hinge at the lower end and a vertical roller at the upper end (if horizontal roller, the column would be unstable). The value of EI is finite for the central half of the column but the two ends are rigid, so EI is infinite for a length of L/4 at each end. The load P is applied axially at each end and your task is to find the value of P which causes buckling. Is that correct?

By inspection, the answer is going to be between π²EI/L² and 4π²EI/L². You can assume a deflected shape, say a sin curve for the central portion and a straight line extension at each end for the two rigid portions, then use the techniques you have been taught to find the critical load.

If the end rotation is θ at the onset of buckling, the moment at L/4 is PθL/4, so you might consider treating the problem as a straight column of length L/2 with an axial load P and an applied moment of PθL/4 at each end.

BA

 

[Samplaw]

yea sorry i was writing this is a rush,

a bit more information, it says to also prove that (alpha).L/2 = 1.72 numerically, as well as the critical buckling load.

a question, why is the moment at L/4 pθL/4. I got PsinθL/4?

would you then treat this as an eccentrically loaded column?

thanks for the help this is really racking my brain.

 

[CANPRO]

You should go to your Library and find a copy of Stephen Timoshenko's "Theory of Elastic Stability". Its a great reference and will probably provide more info on the topic than your typical Mechanics of Materials text.

By the way, I don't think you're supposed to be asking homework questions on here.

 

[BAretired]

Samplaw, sinθ is correct but for very small angles, it is usual in engineering problems to take sinθ or tanθ = θ.

I agree with CANEIT. "Theory of Elastic Stability" is an excellent reference which covers buckling of columns with variable cross section and specifically treats buckling of bars with changes in cross section using Newmark's Numerical Procedures.

BA

 

[johnbridge231]

You should be able to locate the corresponding buckling load (for Euler or bending buckling about the weak axis of the section), using appropriate tables or by simple hand calculations. If you can use a software, it is very easy to model this element using 4 nodes and performing a modal analysis in order to identify the critical loads for buckling.

Analysis and Design of arbitrary cross sections
Reinforcement design to all major codes
Moment Curvature analysis

http://www.engissol.com/cross-section-analysis-design.html